Exercises - Surface Integrals
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- Surface Integrals with Parametric Surfaces
- Computation Examples of Parametric Surfaces
- General formula for Calculating Surface Integrals
- Connection between General Formula and Parametric Surfaces Form
- Surface Integrals for Piecewise Smooth Surfaces
- Orientation of Surfaces
- Surface Integrals over Vector Fields
- Surface Integrals over Vector Fields - Example
- Useful Formula for Computing Surface Integrals of Vector Fields
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10 part a
- Exercise 10 part b
- Exercise 11
- Exercise 12

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[{"Name":"Exercises - Surface Integrals","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Surface Integrals with Parametric Surfaces","Duration":"17m 45s","ChapterTopicVideoID":8781,"CourseChapterTopicPlaylistID":4974,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8781.jpeg","UploadDate":"2020-02-26T12:26:47.1900000","DurationForVideoObject":"PT17M45S","Description":null,"MetaTitle":"Surface Integrals with Parametric Surfaces: Video + Workbook | Proprep","MetaDescription":"Surface Integrals - General Calculations with Surface Integrals. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/calculus-i%2c-ii-and-iii/surface-integrals/general-calculations-with-surface-integrals/vid9644","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.110","Text":"In this section we\u0027ll be talking about surface integrals."},{"Start":"00:04.110 ","End":"00:06.420","Text":"But before we can get into the integrals part,"},{"Start":"00:06.420 ","End":"00:08.685","Text":"we have to talk some more about surfaces."},{"Start":"00:08.685 ","End":"00:10.710","Text":"We have seen them before."},{"Start":"00:10.710 ","End":"00:17.340","Text":"For example, the graph of a function of 2 variables,"},{"Start":"00:17.340 ","End":"00:20.130","Text":"z as a function of x and y, for example."},{"Start":"00:20.130 ","End":"00:27.960","Text":"but I want to talk now about parametric surfaces and the analogy to curves."},{"Start":"00:27.960 ","End":"00:35.025","Text":"Curves in parametric form were given by r of t,"},{"Start":"00:35.025 ","End":"00:39.570","Text":"which is just 3 functions,"},{"Start":"00:39.570 ","End":"00:43.815","Text":"x of t, y of t,"},{"Start":"00:43.815 ","End":"00:49.190","Text":"z of t. I could have given them letters f, g,"},{"Start":"00:49.190 ","End":"00:51.200","Text":"h and said x equals f of t and so on,"},{"Start":"00:51.200 ","End":"00:57.140","Text":"but we just say x equals x of t. This is in 3D."},{"Start":"00:57.140 ","End":"01:02.775","Text":"In 2-dimensions you wouldn\u0027t have the z of t. This is a 3D curve."},{"Start":"01:02.775 ","End":"01:11.865","Text":"By analogy, a 3D surface we\u0027re going to need 2 parameters."},{"Start":"01:11.865 ","End":"01:17.920","Text":"Instead of t, lets say u and v. For a surface,"},{"Start":"01:24.520 ","End":"01:33.980","Text":"x is some function of u and v and y is another function of u and v and z is"},{"Start":"01:33.980 ","End":"01:37.865","Text":"a function of u and v. The difference between"},{"Start":"01:37.865 ","End":"01:42.830","Text":"a curve and a surface is that a surface needs 2 parameters."},{"Start":"01:42.830 ","End":"01:45.560","Text":"Even though it bends, it\u0027s like a 2-dimensional thing,"},{"Start":"01:45.560 ","End":"01:49.110","Text":"it\u0027s like a sheet of paper as this is just a line."},{"Start":"01:49.330 ","End":"01:53.270","Text":"The easiest example or class of"},{"Start":"01:53.270 ","End":"01:56.885","Text":"examples would be when you have a function of 2 variables."},{"Start":"01:56.885 ","End":"02:04.505","Text":"Suppose I had the z equals f of x and y,"},{"Start":"02:04.505 ","End":"02:12.275","Text":"then we could automatically describe the graph of it as the surface,"},{"Start":"02:12.275 ","End":"02:17.105","Text":"which is parameterized if we use x and y as parameters."},{"Start":"02:17.105 ","End":"02:21.065","Text":"I\u0027ll show you a diagram."},{"Start":"02:21.065 ","End":"02:23.270","Text":"I found this on the Internet."},{"Start":"02:23.270 ","End":"02:24.970","Text":"It\u0027s not quite what I wanted."},{"Start":"02:24.970 ","End":"02:27.760","Text":"They\u0027re using x_1, x_2, x_3 for coordinates."},{"Start":"02:27.760 ","End":"02:35.885","Text":"If we just replace x_1 and x_2 by x and y,"},{"Start":"02:35.885 ","End":"02:38.210","Text":"just change everything and you\u0027ll see"},{"Start":"02:38.210 ","End":"02:45.785","Text":"that the diagrams pretty much describes the surface."},{"Start":"02:45.785 ","End":"02:51.440","Text":"Notice also that we could restrict x and y not to be everywhere in the whole plane,"},{"Start":"02:51.440 ","End":"02:55.110","Text":"but just say in a rectangle as is here."},{"Start":"02:55.110 ","End":"02:58.240","Text":"Or natural restrictions because of the function,"},{"Start":"02:58.240 ","End":"03:03.640","Text":"the function might have a domain and you might be forced to restrict x and y."},{"Start":"03:03.640 ","End":"03:07.535","Text":"This can also be written in parametric form."},{"Start":"03:07.535 ","End":"03:13.180","Text":"What we can say is that in parametric form,"},{"Start":"03:13.180 ","End":"03:18.105","Text":"we need u and v,"},{"Start":"03:18.105 ","End":"03:22.049","Text":"we could let x equal just u,"},{"Start":"03:22.049 ","End":"03:25.650","Text":"we could let y equals v,"},{"Start":"03:25.650 ","End":"03:33.970","Text":"and z would be f of u and v. Just basically taking this and replacing x and y by u and v,"},{"Start":"03:33.970 ","End":"03:36.965","Text":"then here we\u0027d have u, v,"},{"Start":"03:36.965 ","End":"03:42.890","Text":"and the function of u and v that gives us a point in 3D on this surface."},{"Start":"03:42.890 ","End":"03:44.975","Text":"Let me give an actual example."},{"Start":"03:44.975 ","End":"03:55.055","Text":"Let\u0027s take f of x and y to be a hemisphere of radius 2 and upper hemisphere."},{"Start":"03:55.055 ","End":"03:58.730","Text":"The reason we can\u0027t take a whole sphere is then it wouldn\u0027t be a function."},{"Start":"03:58.730 ","End":"04:01.805","Text":"I want z as a function of x and y."},{"Start":"04:01.805 ","End":"04:05.120","Text":"In this case, if the radius is 2,"},{"Start":"04:05.120 ","End":"04:12.680","Text":"this would be the square root of 4 minus x squared minus y squared."},{"Start":"04:12.680 ","End":"04:14.375","Text":"If you\u0027re not sure about this,"},{"Start":"04:14.375 ","End":"04:18.175","Text":"just think that the equation of a sphere,"},{"Start":"04:18.175 ","End":"04:20.685","Text":"if you remember quadric surfaces,"},{"Start":"04:20.685 ","End":"04:28.100","Text":"would be x squared plus y squared plus z squared is equal to r squared,"},{"Start":"04:28.100 ","End":"04:31.225","Text":"or in this case 2 squared which is 4."},{"Start":"04:31.225 ","End":"04:35.040","Text":"Just writing that, it\u0027s 2 squared because a general."},{"Start":"04:35.040 ","End":"04:37.770","Text":"r squared and then if you bring x squared plus"},{"Start":"04:37.770 ","End":"04:40.895","Text":"y squared over to the other side and take the square root,"},{"Start":"04:40.895 ","End":"04:45.200","Text":"then the square root is positive,"},{"Start":"04:45.200 ","End":"04:47.300","Text":"so it\u0027s just the upper hemisphere."},{"Start":"04:47.300 ","End":"04:50.715","Text":"If you wanted the lower hemisphere, you\u0027d make a minus."},{"Start":"04:50.715 ","End":"04:55.735","Text":"In parametric form, this would be"},{"Start":"04:55.735 ","End":"05:03.435","Text":"r of u and v is equal to u,"},{"Start":"05:03.435 ","End":"05:07.200","Text":"u is like x, v is like y,"},{"Start":"05:07.200 ","End":"05:15.635","Text":"and z would be the square root of 4 minus u squared minus v squared."},{"Start":"05:15.635 ","End":"05:20.150","Text":"But here we have to restrict the domain because we can\u0027t just"},{"Start":"05:20.150 ","End":"05:24.620","Text":"take any u and v. For the original function,"},{"Start":"05:24.620 ","End":"05:27.590","Text":"we have to have x squared plus y squared less"},{"Start":"05:27.590 ","End":"05:30.935","Text":"than or equal to 4 or else we\u0027ll get something negative here."},{"Start":"05:30.935 ","End":"05:38.270","Text":"It\u0027s actually defined on x squared plus y squared less than or equal to 4,"},{"Start":"05:38.270 ","End":"05:44.255","Text":"which is the circle together with its interior and the boundary,"},{"Start":"05:44.255 ","End":"05:45.830","Text":"a disk if you like,"},{"Start":"05:45.830 ","End":"05:50.500","Text":"of radius 2, that\u0027s the domain."},{"Start":"05:50.500 ","End":"05:53.550","Text":"It\u0027s not defined for all x, y."},{"Start":"05:53.550 ","End":"05:58.310","Text":"Likewise here, we would have to say that u"},{"Start":"05:58.310 ","End":"06:05.315","Text":"squared plus v squared less than or equal to 4 is the domain for this."},{"Start":"06:05.315 ","End":"06:08.910","Text":"In fact, since we\u0027ve gone over to u and v, let me erase this."},{"Start":"06:08.930 ","End":"06:12.290","Text":"Here\u0027s a sketch of the domain,"},{"Start":"06:12.290 ","End":"06:16.370","Text":"the region where this is defined."},{"Start":"06:16.370 ","End":"06:24.005","Text":"This is the circle of radius 2 and I moved this restriction on u and v to here."},{"Start":"06:24.005 ","End":"06:27.310","Text":"This is u and this is v,"},{"Start":"06:27.310 ","End":"06:30.260","Text":"so we get a disc including the boundary,"},{"Start":"06:30.260 ","End":"06:32.250","Text":"just like here in this picture,"},{"Start":"06:32.250 ","End":"06:35.840","Text":"it looks like it was defined over a rectangle."},{"Start":"06:35.840 ","End":"06:39.980","Text":"Now suppose we wanted the whole sphere,"},{"Start":"06:39.980 ","End":"06:43.430","Text":"not just the upper hemisphere. How would we do that?"},{"Start":"06:43.430 ","End":"06:45.950","Text":"We can\u0027t do it as a function,"},{"Start":"06:45.950 ","End":"06:49.265","Text":"but we can do it in parametric form."},{"Start":"06:49.265 ","End":"06:54.290","Text":"Let\u0027s see if we can get a sphere of radius 2 in parametric form."},{"Start":"06:54.290 ","End":"06:58.590","Text":"First, maybe I\u0027ll give a sketch of the sphere."},{"Start":"06:58.750 ","End":"07:01.190","Text":"Here\u0027s our sphere."},{"Start":"07:01.190 ","End":"07:03.545","Text":"How would we parameterize it?"},{"Start":"07:03.545 ","End":"07:06.095","Text":"We know it\u0027s Cartesian coordinates."},{"Start":"07:06.095 ","End":"07:08.195","Text":"It\u0027s just up here."},{"Start":"07:08.195 ","End":"07:12.175","Text":"Let me just copy that over here."},{"Start":"07:12.175 ","End":"07:14.750","Text":"It\u0027s not immediately clear."},{"Start":"07:14.750 ","End":"07:16.490","Text":"I mean, it\u0027s quite difficult to,"},{"Start":"07:16.490 ","End":"07:19.070","Text":"unless you know how to parameterize it."},{"Start":"07:19.070 ","End":"07:21.290","Text":"But here\u0027s the trick we\u0027re going to use."},{"Start":"07:21.290 ","End":"07:27.035","Text":"If you remember, we had something called spherical coordinates."},{"Start":"07:27.035 ","End":"07:30.560","Text":"You might want to review spherical coordinates,"},{"Start":"07:30.560 ","End":"07:32.540","Text":"I wont to do it here."},{"Start":"07:32.540 ","End":"07:36.800","Text":"You should also remember that we did the equation"},{"Start":"07:36.800 ","End":"07:41.765","Text":"of a sphere centered at the origin with radius a."},{"Start":"07:41.765 ","End":"07:45.005","Text":"The equation was Rho equals a,"},{"Start":"07:45.005 ","End":"07:48.380","Text":"but in our case the radius is 2."},{"Start":"07:48.380 ","End":"07:56.850","Text":"Rho equals 2 is the equation of the sphere in spherical coordinates."},{"Start":"07:56.850 ","End":"08:01.290","Text":"If I just substitute Rho equals 2 here, I\u0027ll get x,"},{"Start":"08:01.290 ","End":"08:03.920","Text":"y, and z, and instead of writing it in parametric form,"},{"Start":"08:03.920 ","End":"08:06.450","Text":"we\u0027ll write it in vector form."},{"Start":"08:08.870 ","End":"08:11.360","Text":"I\u0027m not going to use u and v,"},{"Start":"08:11.360 ","End":"08:13.505","Text":"I\u0027m going to use Theta and Phi."},{"Start":"08:13.505 ","End":"08:16.350","Text":"There\u0027s no reason we have to use u and v,"},{"Start":"08:16.350 ","End":"08:18.170","Text":"although those are common."},{"Start":"08:18.170 ","End":"08:22.300","Text":"We\u0027ll get 2 parameters, Theta and Phi,"},{"Start":"08:22.300 ","End":"08:32.955","Text":"and this will equal the x part is 2 sine Phi cosine Theta,"},{"Start":"08:32.955 ","End":"08:39.905","Text":"then 2 sine Phi sine Theta,"},{"Start":"08:39.905 ","End":"08:45.255","Text":"and then 2 cosine of Phi."},{"Start":"08:45.255 ","End":"08:49.430","Text":"This is a parametric equation in 2 parameters."},{"Start":"08:49.430 ","End":"08:53.315","Text":"I don\u0027t have to restrict Theta and Phi."},{"Start":"08:53.315 ","End":"08:56.260","Text":"They could be anything,"},{"Start":"08:56.260 ","End":"09:01.160","Text":"but then we would get the sphere several times."},{"Start":"09:01.160 ","End":"09:04.255","Text":"Suppose we don\u0027t want to get duplication,"},{"Start":"09:04.255 ","End":"09:10.670","Text":"for example, Theta would be the longitude angle."},{"Start":"09:11.490 ","End":"09:15.640","Text":"As we go around to every 2Pi, it repeats itself,"},{"Start":"09:15.640 ","End":"09:22.945","Text":"so we might want to restrict Theta to be between 0 and 2Pi."},{"Start":"09:22.945 ","End":"09:29.260","Text":"Strictly speaking, I should make it not up to and not including the 2Pi."},{"Start":"09:29.260 ","End":"09:35.215","Text":"But often, you let that slide even though the 2Pi and the 0 give the same thing,"},{"Start":"09:35.215 ","End":"09:38.905","Text":"and Phi is the latitude,"},{"Start":"09:38.905 ","End":"09:41.395","Text":"except that we start from the North pole and"},{"Start":"09:41.395 ","End":"09:45.535","Text":"0 degrees and a South pole is a 180 degrees or Pi."},{"Start":"09:45.535 ","End":"09:52.060","Text":"We would say that 0 less than or equal to Phi,"},{"Start":"09:52.060 ","End":"09:54.565","Text":"less than or equal to Pi."},{"Start":"09:54.565 ","End":"09:59.905","Text":"If I sketch the domain or region of definition,"},{"Start":"09:59.905 ","End":"10:01.990","Text":"then here\u0027s what it looks like."},{"Start":"10:01.990 ","End":"10:04.450","Text":"I\u0027ll just label the axes."},{"Start":"10:04.450 ","End":"10:09.640","Text":"This is the Theta axis and this is the Phi axis."},{"Start":"10:09.640 ","End":"10:20.050","Text":"This goes from 0 to Pi and we need to shade the inside because it includes the inside."},{"Start":"10:20.050 ","End":"10:26.155","Text":"Before I continue, there\u0027s something I forgot to mention earlier,"},{"Start":"10:26.155 ","End":"10:29.230","Text":"just a bit of a generalization."},{"Start":"10:29.230 ","End":"10:32.740","Text":"We said that when we have z as a function of x and y,"},{"Start":"10:32.740 ","End":"10:36.670","Text":"then we can easily parameterize it like this."},{"Start":"10:36.670 ","End":"10:45.340","Text":"I just wanted to say that a very similar thing happens when y might be"},{"Start":"10:45.340 ","End":"10:55.960","Text":"a function of x and z or you might have x is a function of y and z,"},{"Start":"10:55.960 ","End":"11:00.940","Text":"and a very similar thing would happen analogously."},{"Start":"11:00.940 ","End":"11:03.700","Text":"I just wanted to mention that\u0027s already 3 kinds of surfaces."},{"Start":"11:03.700 ","End":"11:06.700","Text":"When we can isolate 1 variable in terms of the other,"},{"Start":"11:06.700 ","End":"11:09.640","Text":"then it\u0027s easy to get a parametrization."},{"Start":"11:09.640 ","End":"11:12.655","Text":"Now, let\u0027s continue."},{"Start":"11:12.655 ","End":"11:19.120","Text":"We did the example with the sphere and we used spherical coordinates."},{"Start":"11:19.120 ","End":"11:23.260","Text":"I give 1 more example like this."},{"Start":"11:23.260 ","End":"11:26.950","Text":"We\u0027ll do a cylinder in cylindrical coordinates."},{"Start":"11:26.950 ","End":"11:30.340","Text":"Let me first bring in a sketch."},{"Start":"11:30.340 ","End":"11:41.110","Text":"Let\u0027s say the cylinder is of radius 3 and that it is centered on the z-axis."},{"Start":"11:41.110 ","End":"11:47.230","Text":"We know that the equation of such a cylinder in Cartesian coordinates,"},{"Start":"11:47.230 ","End":"11:55.210","Text":"just ignore z and make a requirement that x squared plus y squared equals 3 squared,"},{"Start":"11:55.210 ","End":"12:01.220","Text":"which is 9, and z is unbounded as it can be anything."},{"Start":"12:01.380 ","End":"12:05.049","Text":"The trick here is to work in cylindrical coordinates."},{"Start":"12:05.049 ","End":"12:07.000","Text":"Let me remind you."},{"Start":"12:07.000 ","End":"12:11.650","Text":"These are the conversion equations from cylindrical to Cartesian."},{"Start":"12:11.650 ","End":"12:22.550","Text":"In cylindrical coordinates, this cylinder comes out to be just r equals the constant 3."},{"Start":"12:23.040 ","End":"12:29.800","Text":"Notice that if you did x squared plus y squared from here and here,"},{"Start":"12:29.800 ","End":"12:35.110","Text":"we would get r squared because cosine squared plus sine squared is 1,"},{"Start":"12:35.110 ","End":"12:41.380","Text":"and this corresponds with this because from here also r squared equals 9 or 3 squared."},{"Start":"12:41.380 ","End":"12:48.200","Text":"The parametric equation of the surface comes from just replacing r with 3."},{"Start":"12:48.270 ","End":"12:51.640","Text":"Well, I\u0027m going to reuse the letter r vector r,"},{"Start":"12:51.640 ","End":"12:53.710","Text":"not the same as this."},{"Start":"12:53.710 ","End":"12:56.200","Text":"We have 2 variables now,"},{"Start":"12:56.200 ","End":"13:04.639","Text":"Theta and z and x is 3 cosine Theta,"},{"Start":"13:04.710 ","End":"13:10.660","Text":"y is 3 sine Theta,"},{"Start":"13:10.660 ","End":"13:14.920","Text":"and z is just z."},{"Start":"13:14.920 ","End":"13:17.710","Text":"It\u0027s 2 parameters, Theta and z."},{"Start":"13:17.710 ","End":"13:23.450","Text":"Again, we don\u0027t use u and v. Any 2 letters will do."},{"Start":"13:23.460 ","End":"13:26.305","Text":"We could write this unrestricted."},{"Start":"13:26.305 ","End":"13:30.700","Text":"But since we don\u0027t want to repeat ourselves with the Theta going round and round,"},{"Start":"13:30.700 ","End":"13:36.620","Text":"we will restrict Theta to be between 0 and 2Pi."},{"Start":"13:36.660 ","End":"13:40.600","Text":"Let\u0027s also suppose that we want the finite cylinder,"},{"Start":"13:40.600 ","End":"13:41.980","Text":"not the infinite cylinder."},{"Start":"13:41.980 ","End":"13:45.565","Text":"As it is with z unbounded it would be,"},{"Start":"13:45.565 ","End":"13:51.250","Text":"and I marked in some values,"},{"Start":"13:51.250 ","End":"13:55.060","Text":"I\u0027m just guessing this is a 3 and this is minus 2,"},{"Start":"13:55.060 ","End":"14:05.260","Text":"and so we would restrict z to be between minus 2 and 3."},{"Start":"14:05.260 ","End":"14:10.340","Text":"This gives us again a rectangle."},{"Start":"14:10.590 ","End":"14:16.120","Text":"Here we have a Theta axis,"},{"Start":"14:16.120 ","End":"14:23.815","Text":"here we would have a z-axis."},{"Start":"14:23.815 ","End":"14:33.310","Text":"Let\u0027s say that here is minus 2 and here is 3, and this is 0,"},{"Start":"14:33.310 ","End":"14:35.720","Text":"and this is 2Pi,"},{"Start":"14:35.790 ","End":"14:43.315","Text":"and we get this domain or region R as a shaded and there we are."},{"Start":"14:43.315 ","End":"14:45.999","Text":"Now, I want to do reverse example."},{"Start":"14:45.999 ","End":"14:47.380","Text":"What do I mean by that?"},{"Start":"14:47.380 ","End":"14:49.675","Text":"Well, in the previous examples,"},{"Start":"14:49.675 ","End":"14:52.645","Text":"we were given a description or an equation"},{"Start":"14:52.645 ","End":"14:56.755","Text":"of a surface and we had to find a parametric surface."},{"Start":"14:56.755 ","End":"14:57.970","Text":"Let\u0027s do the opposite."},{"Start":"14:57.970 ","End":"15:03.070","Text":"Let\u0027s start with a parametric description and then try and see what the surface is."},{"Start":"15:03.070 ","End":"15:06.565","Text":"For this example, let\u0027s take the following."},{"Start":"15:06.565 ","End":"15:14.530","Text":"We\u0027ll take r position vector of u and"},{"Start":"15:14.530 ","End":"15:24.220","Text":"v is equal to u for the x and then u cosine v for y,"},{"Start":"15:24.220 ","End":"15:29.510","Text":"and u sine v for the z position."},{"Start":"15:29.820 ","End":"15:40.480","Text":"I could write it in parametric form also just like this, x equals u,"},{"Start":"15:40.480 ","End":"15:44.575","Text":"y equals u cosine v,"},{"Start":"15:44.575 ","End":"15:52.525","Text":"and z equals u sine v. Parametric and vector forms are very closely related."},{"Start":"15:52.525 ","End":"15:55.795","Text":"Let\u0027s see if we can figure out what this surface is."},{"Start":"15:55.795 ","End":"15:58.945","Text":"We need to do a bit of algebra and manipulation."},{"Start":"15:58.945 ","End":"16:01.930","Text":"But I see there\u0027s a cosine and a sine and I"},{"Start":"16:01.930 ","End":"16:05.245","Text":"know that cosine squared plus sine squared is 1."},{"Start":"16:05.245 ","End":"16:11.410","Text":"If I compute y squared plus z squared,"},{"Start":"16:11.410 ","End":"16:18.445","Text":"what I\u0027ll get is u squared cosine squared"},{"Start":"16:18.445 ","End":"16:25.540","Text":"v plus u squared sine squared v. Now,"},{"Start":"16:25.540 ","End":"16:28.780","Text":"I take u squared outside the brackets and cosine"},{"Start":"16:28.780 ","End":"16:33.415","Text":"squared plus sine squared of anything is 1,"},{"Start":"16:33.415 ","End":"16:36.835","Text":"so this is just equal to u squared."},{"Start":"16:36.835 ","End":"16:38.710","Text":"But u is equal to x,"},{"Start":"16:38.710 ","End":"16:40.975","Text":"so that is x squared."},{"Start":"16:40.975 ","End":"16:44.950","Text":"What we\u0027ve essentially got is the surface,"},{"Start":"16:44.950 ","End":"16:47.410","Text":"and it\u0027s a quadric surface,"},{"Start":"16:47.410 ","End":"16:55.750","Text":"which is that x squared equals y squared plus z squared."},{"Start":"16:55.750 ","End":"17:01.750","Text":"We\u0027ve had something similar before in the section on quadric surfaces."},{"Start":"17:01.750 ","End":"17:06.250","Text":"We had z squared equals x squared plus"},{"Start":"17:06.250 ","End":"17:11.155","Text":"y squared and that was a cone that opened up along the z-axis."},{"Start":"17:11.155 ","End":"17:15.400","Text":"Here, we just get a cone that opens up along the x-axis."},{"Start":"17:15.400 ","End":"17:19.510","Text":"Let me illustrate it. Here it is."},{"Start":"17:19.510 ","End":"17:25.090","Text":"This here is the x-axis that it opens along because the x squared is the 1 on its own,"},{"Start":"17:25.090 ","End":"17:32.800","Text":"and this would be the z-axis and this is the y-axis."},{"Start":"17:32.800 ","End":"17:36.430","Text":"Of course, you wouldn\u0027t be expected to provide a sketch like this,"},{"Start":"17:36.430 ","End":"17:38.770","Text":"but you should be able to identify from this"},{"Start":"17:38.770 ","End":"17:41.650","Text":"that it\u0027s a quadric surface and that it\u0027s a cone."},{"Start":"17:41.650 ","End":"17:45.770","Text":"Now, it\u0027s time for a break."}],"ID":9644},{"Watched":false,"Name":"Computation Examples of Parametric Surfaces","Duration":"21m 32s","ChapterTopicVideoID":8782,"CourseChapterTopicPlaylistID":4974,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.170 ","End":"00:02.880","Text":"I cleared the board because we\u0027re going"},{"Start":"00:02.880 ","End":"00:04.950","Text":"to continue now with a bit of theory."},{"Start":"00:04.950 ","End":"00:07.140","Text":"We just did a few examples."},{"Start":"00:07.140 ","End":"00:12.360","Text":"I kept up on the board the parametric equation"},{"Start":"00:12.360 ","End":"00:14.685","Text":"of a curve and of the surface."},{"Start":"00:14.685 ","End":"00:16.860","Text":"When we had a curve,"},{"Start":"00:16.860 ","End":"00:20.310","Text":"we only need 1 parameter and usually it\u0027s t"},{"Start":"00:20.310 ","End":"00:22.020","Text":"and when we have a surface,"},{"Start":"00:22.020 ","End":"00:24.675","Text":"we have 2 parameters,"},{"Start":"00:24.675 ","End":"00:29.490","Text":"typically u and v. I want to remind you that in a curve,"},{"Start":"00:29.490 ","End":"00:33.180","Text":"we had something called a tangent vector."},{"Start":"00:33.180 ","End":"00:38.910","Text":"The tangent vector was simply the derivative."},{"Start":"00:38.910 ","End":"00:40.590","Text":"To get the derivative,"},{"Start":"00:40.590 ","End":"00:44.540","Text":"we just differentiated each of the components separately."},{"Start":"00:44.540 ","End":"00:46.385","Text":"This is a tangent vector,"},{"Start":"00:46.385 ","End":"00:48.200","Text":"provided it\u0027s not 0,"},{"Start":"00:48.200 ","End":"00:49.865","Text":"I\u0027m going to assume it\u0027s not 0."},{"Start":"00:49.865 ","End":"00:52.860","Text":"Now in the case of a surface,"},{"Start":"00:53.440 ","End":"00:57.320","Text":"we don\u0027t have 1 tangent vector"},{"Start":"00:57.320 ","End":"01:02.285","Text":"because a surface can contain many directions."},{"Start":"01:02.285 ","End":"01:06.710","Text":"Actually, what we\u0027re more interested in is the normal vector,"},{"Start":"01:06.710 ","End":"01:09.305","Text":"but getting ahead of ourselves,"},{"Start":"01:09.305 ","End":"01:14.270","Text":"let me just go back a bit and take the analog of what is"},{"Start":"01:14.270 ","End":"01:19.665","Text":"the derivative in the case of 2 parameters,"},{"Start":"01:19.665 ","End":"01:22.755","Text":"where we have 2 partial derivatives."},{"Start":"01:22.755 ","End":"01:32.705","Text":"What we have is we could take the derivative of this,"},{"Start":"01:32.705 ","End":"01:35.465","Text":"but with respect to u,"},{"Start":"01:35.465 ","End":"01:38.260","Text":"in which case I wouldn\u0027t call it prime,"},{"Start":"01:38.260 ","End":"01:41.645","Text":"I would just call it r_u,"},{"Start":"01:41.645 ","End":"01:48.725","Text":"which would also be in the other notation dr by du."},{"Start":"01:48.725 ","End":"01:52.700","Text":"Some people even keep in the prime here, anyway I don\u0027t."},{"Start":"01:52.700 ","End":"01:59.660","Text":"Then we also have another which is with respect to v. We have,"},{"Start":"01:59.660 ","End":"02:02.810","Text":"instead of a derivative which is a tangent,"},{"Start":"02:02.810 ","End":"02:11.075","Text":"we have 2 partial derivatives of a vector function with 2 parameters."},{"Start":"02:11.075 ","End":"02:16.865","Text":"It turns out that both of these are tangent to the surface."},{"Start":"02:16.865 ","End":"02:23.310","Text":"We take the cross product of these 2"},{"Start":"02:23.310 ","End":"02:27.705","Text":"and what we will get will be a normal,"},{"Start":"02:27.705 ","End":"02:33.410","Text":"and this normal vector\u0027s also a function of u and v."},{"Start":"02:33.410 ","End":"02:34.760","Text":"If it happens to be 0,"},{"Start":"02:34.760 ","End":"02:39.710","Text":"then we\u0027re stuck as far as finding the equation of"},{"Start":"02:39.710 ","End":"02:45.340","Text":"the tangent at a point which is the main use for the normal here."},{"Start":"02:45.340 ","End":"02:48.140","Text":"I think we should just get to an example"},{"Start":"02:48.140 ","End":"02:49.790","Text":"and then things will be clearer."},{"Start":"02:49.790 ","End":"02:53.240","Text":"In the example, we\u0027re given a parametric surface"},{"Start":"02:53.240 ","End":"02:54.830","Text":"and the point on the surface,"},{"Start":"02:54.830 ","End":"02:57.350","Text":"and we have to find the equation"},{"Start":"02:57.350 ","End":"03:00.355","Text":"of the tangent plane at that point."},{"Start":"03:00.355 ","End":"03:04.520","Text":"Let\u0027s first of all see if this even makes sense,"},{"Start":"03:04.520 ","End":"03:06.860","Text":"if this point is on this surface."},{"Start":"03:06.860 ","End":"03:09.020","Text":"What we do is compare components"},{"Start":"03:09.020 ","End":"03:11.945","Text":"and we get a system of equations."},{"Start":"03:11.945 ","End":"03:16.340","Text":"Actually 3 equations and 2 unknowns"},{"Start":"03:16.340 ","End":"03:18.485","Text":"which usually doesn\u0027t have a solution."},{"Start":"03:18.485 ","End":"03:20.870","Text":"We get that u equals 2."},{"Start":"03:20.870 ","End":"03:25.880","Text":"We get that 2v squared equals 2."},{"Start":"03:25.880 ","End":"03:33.340","Text":"We get that u squared plus v equals 3."},{"Start":"03:34.640 ","End":"03:39.204","Text":"U equals 2, we have already."},{"Start":"03:39.204 ","End":"03:43.790","Text":"Now if I tried to solve this middle 1,"},{"Start":"03:43.790 ","End":"03:46.910","Text":"I would get that v squared is 1,"},{"Start":"03:46.910 ","End":"03:50.590","Text":"so v is plus or minus 1."},{"Start":"03:50.590 ","End":"03:53.735","Text":"It\u0027s probably better to go with this 1."},{"Start":"03:53.735 ","End":"03:57.560","Text":"If I put u equals 2 into this 1,"},{"Start":"03:57.560 ","End":"04:04.725","Text":"then I get that 2 squared plus v equals 3,"},{"Start":"04:04.725 ","End":"04:09.420","Text":"2 squared is 4, so v equals minus 1."},{"Start":"04:09.420 ","End":"04:12.230","Text":"Basically, if I just summarize that and say"},{"Start":"04:12.230 ","End":"04:20.869","Text":"that my UV for this point is at 2, minus 1."},{"Start":"04:20.869 ","End":"04:24.340","Text":"Let\u0027s compute the partial derivatives."},{"Start":"04:24.340 ","End":"04:28.895","Text":"The partial derivative with respect to u,"},{"Start":"04:28.895 ","End":"04:32.480","Text":"this will equal just component-wise."},{"Start":"04:32.480 ","End":"04:35.705","Text":"First 1 with respect to u is 1."},{"Start":"04:35.705 ","End":"04:42.875","Text":"The second 1 with respect to u is 0 because v is a constant,"},{"Start":"04:42.875 ","End":"04:44.885","Text":"as far as u goes."},{"Start":"04:44.885 ","End":"04:48.950","Text":"The derivative of this 1 with respect to u would be 2u."},{"Start":"04:49.600 ","End":"04:54.440","Text":"The derivative with respect to v, what do I get?"},{"Start":"04:54.440 ","End":"05:02.525","Text":"Here 0, here 4v, and here 1."},{"Start":"05:02.525 ","End":"05:10.650","Text":"Now, if I substitute u equals 2,"},{"Start":"05:10.650 ","End":"05:12.725","Text":"v equals minus 1,"},{"Start":"05:12.725 ","End":"05:18.270","Text":"then this becomes 1,"},{"Start":"05:18.270 ","End":"05:23.070","Text":"0 and 2u is 4."},{"Start":"05:23.070 ","End":"05:25.335","Text":"That\u0027s what this 1 becomes."},{"Start":"05:25.335 ","End":"05:29.445","Text":"This 1 becomes 0,"},{"Start":"05:29.445 ","End":"05:34.230","Text":"4v is minus 4 and 1."},{"Start":"05:34.230 ","End":"05:44.055","Text":"These are my 2 tangent vectors at this point."},{"Start":"05:44.055 ","End":"05:48.160","Text":"Where is it? 2, 2, 3."},{"Start":"05:48.620 ","End":"05:52.850","Text":"Hopefully, the normal is not going to be 0"},{"Start":"05:52.850 ","End":"05:55.025","Text":"because then we\u0027ll be stuck."},{"Start":"05:55.025 ","End":"06:04.095","Text":"We get that the normal vector at that point is equal to 1, 0,"},{"Start":"06:04.095 ","End":"06:10.544","Text":"4 cross with 0 minus 4, 1,"},{"Start":"06:10.544 ","End":"06:19.280","Text":"and this turns out to be 16 minus 1 minus 4."},{"Start":"06:19.280 ","End":"06:21.740","Text":"I\u0027m not going to do the computation."},{"Start":"06:21.740 ","End":"06:23.750","Text":"There\u0027s several ways of doing it."},{"Start":"06:23.750 ","End":"06:25.700","Text":"One with determinant, 1 without,"},{"Start":"06:25.700 ","End":"06:30.380","Text":"and just, I don\u0027t want to waste time with this computation."},{"Start":"06:30.380 ","End":"06:34.220","Text":"One thing I do usually do is check"},{"Start":"06:34.220 ","End":"06:36.170","Text":"to make sure that this thing is at least"},{"Start":"06:36.170 ","End":"06:38.690","Text":"perpendicular to both of these because that\u0027s easy to"},{"Start":"06:38.690 ","End":"06:42.570","Text":"do using the dot product of this with each of these separately."},{"Start":"06:42.570 ","End":"06:43.730","Text":"If I do with the dot product,"},{"Start":"06:43.730 ","End":"06:47.120","Text":"for the first 1, get 16 times 1,"},{"Start":"06:47.120 ","End":"06:51.900","Text":"then minus 0, and then minus 16."},{"Start":"06:51.900 ","End":"06:54.480","Text":"Basically, I get 16 minus 16 is 0."},{"Start":"06:54.480 ","End":"06:57.110","Text":"That\u0027s okay, perpendicular to this."},{"Start":"06:57.110 ","End":"06:59.794","Text":"As for this 16 times 0, nothing."},{"Start":"06:59.794 ","End":"07:02.450","Text":"I get plus 4 from this times this,"},{"Start":"07:02.450 ","End":"07:04.355","Text":"and minus 4 from this times this."},{"Start":"07:04.355 ","End":"07:06.170","Text":"At least it\u0027s perpendicular to both,"},{"Start":"07:06.170 ","End":"07:08.395","Text":"good chance that this is right."},{"Start":"07:08.395 ","End":"07:11.890","Text":"Now remember, we need to find the tangent plane."},{"Start":"07:11.890 ","End":"07:14.105","Text":"To find the tangent plane,"},{"Start":"07:14.105 ","End":"07:19.940","Text":"all we\u0027ll need is a normal and a point on the plane."},{"Start":"07:19.940 ","End":"07:20.870","Text":"Now this is the point."},{"Start":"07:20.870 ","End":"07:23.860","Text":"Let me just write its position vector."},{"Start":"07:23.860 ","End":"07:29.400","Text":"I\u0027ll call the position vector r-naught."},{"Start":"07:29.400 ","End":"07:36.930","Text":"Just this but with the same numbers but in an angular bracket."},{"Start":"07:37.760 ","End":"07:42.215","Text":"In case you forgotten the formula for the plane,"},{"Start":"07:42.215 ","End":"07:44.960","Text":"let me refresh your memory."},{"Start":"07:44.960 ","End":"07:48.095","Text":"There\u0027s actually 2 forms of the equation."},{"Start":"07:48.095 ","End":"07:50.360","Text":"If you\u0027d like to have 0 on the right,"},{"Start":"07:50.360 ","End":"07:51.860","Text":"you\u0027d use this 1,"},{"Start":"07:51.860 ","End":"07:53.660","Text":"and if you\u0027d like to have your x, y,"},{"Start":"07:53.660 ","End":"07:56.510","Text":"z on the left and a number on the right you use this 1."},{"Start":"07:56.510 ","End":"07:59.010","Text":"I\u0027m going to use this 1."},{"Start":"08:00.050 ","End":"08:05.120","Text":"I\u0027ll just highlight the important information"},{"Start":"08:05.120 ","End":"08:07.985","Text":"where we have r-naught."},{"Start":"08:07.985 ","End":"08:13.965","Text":"We have n, and of course r,"},{"Start":"08:13.965 ","End":"08:20.395","Text":"the other [inaudible] here is just x, y, z."},{"Start":"08:20.395 ","End":"08:22.430","Text":"Although we have to be careful"},{"Start":"08:22.430 ","End":"08:26.659","Text":"because I\u0027ve used the letter r twice,"},{"Start":"08:26.659 ","End":"08:30.695","Text":"once for the surface,"},{"Start":"08:30.695 ","End":"08:32.930","Text":"and once r for the plane."},{"Start":"08:32.930 ","End":"08:38.345","Text":"Perhaps I should mark it,"},{"Start":"08:38.345 ","End":"08:41.080","Text":"maybe I\u0027ll just make it a different color."},{"Start":"08:41.080 ","End":"08:43.639","Text":"What we get from here,"},{"Start":"08:43.639 ","End":"08:50.630","Text":"and let\u0027s see what is n dot with r. This is n and dot with x,"},{"Start":"08:50.630 ","End":"09:00.510","Text":"y, z just gives 16x minus y minus 4z equals."},{"Start":"09:00.510 ","End":"09:06.900","Text":"Then I want n dot with r-naught, this point."},{"Start":"09:06.900 ","End":"09:09.260","Text":"We can do this mentally."},{"Start":"09:09.260 ","End":"09:13.685","Text":"16 times 2 is 32,"},{"Start":"09:13.685 ","End":"09:18.000","Text":"and then we\u0027ve got minus 2 minus 12,"},{"Start":"09:18.000 ","End":"09:21.435","Text":"18, I make it."},{"Start":"09:21.435 ","End":"09:24.860","Text":"This is the answer. This is the equation of the plane"},{"Start":"09:24.860 ","End":"09:27.120","Text":"that we were looking for."},{"Start":"09:29.630 ","End":"09:35.570","Text":"We\u0027re done with this application,"},{"Start":"09:35.570 ","End":"09:43.745","Text":"which is the tangent plane to a parametric surface at a given point."},{"Start":"09:43.745 ","End":"09:47.360","Text":"The second thing I\u0027d like to talk about is"},{"Start":"09:47.360 ","End":"09:51.180","Text":"the surface area"},{"Start":"09:52.720 ","End":"10:00.000","Text":"of a parametric surface."},{"Start":"10:00.780 ","End":"10:03.859","Text":"I\u0027ll give an example."},{"Start":"10:04.290 ","End":"10:07.075","Text":"What I\u0027d like to do is find"},{"Start":"10:07.075 ","End":"10:16.670","Text":"the surface area of a sphere."},{"Start":"10:16.830 ","End":"10:25.285","Text":"Let\u0027s do it in general of radius, let\u0027s say a."},{"Start":"10:25.285 ","End":"10:27.280","Text":"There\u0027s many ways to do this."},{"Start":"10:27.280 ","End":"10:30.970","Text":"I want to do it using parametric surfaces."},{"Start":"10:30.970 ","End":"10:33.790","Text":"In fact, we\u0027ve actually done a sphere before."},{"Start":"10:33.790 ","End":"10:37.459","Text":"Let me flash back to that lesson."},{"Start":"10:37.710 ","End":"10:40.990","Text":"Here is that lesson."},{"Start":"10:40.990 ","End":"10:47.480","Text":"I\u0027m just going to copy this back to the present,"},{"Start":"10:47.640 ","End":"10:54.490","Text":"and here we are but then the radius was 2"},{"Start":"10:54.490 ","End":"10:55.929","Text":"and we want to make it general."},{"Start":"10:55.929 ","End":"11:05.155","Text":"I\u0027ll replace these 2s with a, a, a, a."},{"Start":"11:05.155 ","End":"11:11.380","Text":"We restricted Theta to go between 0 and 2Pi"},{"Start":"11:11.380 ","End":"11:14.140","Text":"so we don\u0027t keep going round and round."},{"Start":"11:14.140 ","End":"11:17.500","Text":"We just basically covered the sphere once"},{"Start":"11:17.500 ","End":"11:21.160","Text":"the Phi naturally goes from 0 to Pi,"},{"Start":"11:21.160 ","End":"11:25.900","Text":"0 would be the North Pole and Pi is the South Pole."},{"Start":"11:25.900 ","End":"11:29.680","Text":"It\u0027s like a line of latitude,"},{"Start":"11:29.680 ","End":"11:31.659","Text":"but we don\u0027t start from the equator."},{"Start":"11:31.659 ","End":"11:38.110","Text":"Anyway, this means that Theta and Phi are defined over a finite,"},{"Start":"11:38.110 ","End":"11:42.220","Text":"or the bounded region, I think I called it are there,"},{"Start":"11:42.220 ","End":"11:44.200","Text":"but I\u0027ll use D now for domain,"},{"Start":"11:44.200 ","End":"11:46.990","Text":"we sometimes use D sometimes R."},{"Start":"11:46.990 ","End":"11:48.940","Text":"Now, all the missing is a theory."},{"Start":"11:48.940 ","End":"11:52.900","Text":"I\u0027m just going to give you a formula for how to solve in general,"},{"Start":"11:52.900 ","End":"11:57.115","Text":"when we have a parametric surface over a region D,"},{"Start":"11:57.115 ","End":"12:01.045","Text":"then the formula is as follows."},{"Start":"12:01.045 ","End":"12:09.169","Text":"The area of D"},{"Start":"12:09.450 ","End":"12:13.360","Text":"is equal to a double integral."},{"Start":"12:13.360 ","End":"12:17.510","Text":"Here it is, the formula."},{"Start":"12:18.480 ","End":"12:22.750","Text":"Actually, if you look back above,"},{"Start":"12:22.750 ","End":"12:24.970","Text":"you\u0027ll see that this cross-product"},{"Start":"12:24.970 ","End":"12:27.220","Text":"is what we defined as the normal vector."},{"Start":"12:27.220 ","End":"12:29.740","Text":"I guess you could get an alternative form"},{"Start":"12:29.740 ","End":"12:32.365","Text":"where this is the double integral over"},{"Start":"12:32.365 ","End":"12:40.820","Text":"the region of the magnitude of the normal vector, also dA."},{"Start":"12:41.760 ","End":"12:47.950","Text":"That\u0027s in general, and now,"},{"Start":"12:47.950 ","End":"12:52.189","Text":"back to our particular case."},{"Start":"12:52.260 ","End":"12:57.430","Text":"What we need now is to compute the 2 partial derivatives."},{"Start":"12:57.430 ","End":"13:01.810","Text":"This is r, or write Theta."},{"Start":"13:01.810 ","End":"13:04.120","Text":"I mean, in essence, in our case,"},{"Start":"13:04.120 ","End":"13:08.350","Text":"we\u0027re just replacing u with Theta"},{"Start":"13:08.350 ","End":"13:12.640","Text":"and v with Phi because the parameters don\u0027t"},{"Start":"13:12.640 ","End":"13:14.920","Text":"have to be called u and v."},{"Start":"13:14.920 ","End":"13:21.310","Text":"Our Theta is first component Thetas here."},{"Start":"13:21.310 ","End":"13:24.444","Text":"Derivative of cosine is minus sine,"},{"Start":"13:24.444 ","End":"13:29.860","Text":"so we have minus a sine Phi."},{"Start":"13:29.860 ","End":"13:32.170","Text":"The rest of it is as is because it\u0027s constant"},{"Start":"13:32.170 ","End":"13:33.670","Text":"as far as Theta goes,"},{"Start":"13:33.670 ","End":"13:41.200","Text":"sine Theta and derivative of sine is cosine, but it\u0027s plus,"},{"Start":"13:41.200 ","End":"13:49.640","Text":"so a sine Phi cosine Theta."},{"Start":"13:49.890 ","End":"13:55.270","Text":"This, as far as Theta goes, is a constant,"},{"Start":"13:55.270 ","End":"13:57.535","Text":"so this is 0,"},{"Start":"13:57.535 ","End":"13:59.935","Text":"so that\u0027s with respect to Theta."},{"Start":"13:59.935 ","End":"14:06.085","Text":"Then the partial derivative with respect to Phi is equal to,"},{"Start":"14:06.085 ","End":"14:12.805","Text":"this time we get a cosine Phi."},{"Start":"14:12.805 ","End":"14:15.535","Text":"Cosine Theta stays."},{"Start":"14:15.535 ","End":"14:25.915","Text":"Here, sine Phi again, still cosine Phi."},{"Start":"14:25.915 ","End":"14:33.550","Text":"This time this sine Theta is what sticks and cosine,"},{"Start":"14:33.550 ","End":"14:35.350","Text":"we get minus sine,"},{"Start":"14:35.350 ","End":"14:40.730","Text":"so minus a sine of Phi."},{"Start":"14:41.370 ","End":"14:47.180","Text":"Now, we have to figure out what is the cross-product."},{"Start":"14:47.550 ","End":"14:51.940","Text":"I need to compute the cross-product."},{"Start":"14:51.940 ","End":"14:54.400","Text":"Let\u0027s say that this is n,"},{"Start":"14:54.400 ","End":"14:57.110","Text":"which is the cross-product."},{"Start":"14:57.780 ","End":"15:02.650","Text":"I\u0027ll just say that n equals a,"},{"Start":"15:02.650 ","End":"15:05.020","Text":"it\u0027s a messy computation, not too bad."},{"Start":"15:05.020 ","End":"15:08.360","Text":"I\u0027m just going to quote the answer."},{"Start":"15:08.490 ","End":"15:20.330","Text":"Minus a squared sine squared Phi cosine Theta"},{"Start":"15:20.640 ","End":"15:29.320","Text":"and then minus a squared sine"},{"Start":"15:29.320 ","End":"15:35.035","Text":"squared Phi sine Theta."},{"Start":"15:35.035 ","End":"15:37.780","Text":"Then they\u0027re all minuses,"},{"Start":"15:37.780 ","End":"15:40.735","Text":"they\u0027re all minus a squared something."},{"Start":"15:40.735 ","End":"15:49.670","Text":"Sine Phi cosine Theta."},{"Start":"15:50.370 ","End":"15:57.240","Text":"That\u0027s the n or the ru cross rv."},{"Start":"15:57.240 ","End":"16:02.900","Text":"Now, the double integral dA,"},{"Start":"16:04.080 ","End":"16:07.240","Text":"if it was with x and y, it would be dx,"},{"Start":"16:07.240 ","End":"16:09.939","Text":"dy but in our case,"},{"Start":"16:09.939 ","End":"16:18.160","Text":"dA is just, it\u0027s either d Theta d Phi or d Phi d Theta."},{"Start":"16:18.160 ","End":"16:22.690","Text":"It doesn\u0027t really matter which order we do it in."},{"Start":"16:22.690 ","End":"16:26.470","Text":"I\u0027ll write it as d Phi d Theta."},{"Start":"16:26.470 ","End":"16:33.100","Text":"We\u0027ll do the integral in the vertical direction first,"},{"Start":"16:33.100 ","End":"16:38.450","Text":"so need more space here."},{"Start":"16:40.740 ","End":"16:50.935","Text":"What we get is that the surface area is equal to"},{"Start":"16:50.935 ","End":"16:57.850","Text":"double integral of the magnitude"},{"Start":"16:57.850 ","End":"17:02.215","Text":"of n. The a squared comes out."},{"Start":"17:02.215 ","End":"17:04.870","Text":"I can take minus a squared outside"},{"Start":"17:04.870 ","End":"17:07.570","Text":"the brackets and for magnitude,"},{"Start":"17:07.570 ","End":"17:09.144","Text":"I need absolute value."},{"Start":"17:09.144 ","End":"17:12.265","Text":"It will be just a squared."},{"Start":"17:12.265 ","End":"17:15.505","Text":"Then I need the magnitude of this."},{"Start":"17:15.505 ","End":"17:19.600","Text":"The magnitude would just be the square root"},{"Start":"17:19.600 ","End":"17:23.410","Text":"of this squared plus this squared plus this squared."},{"Start":"17:23.410 ","End":"17:25.840","Text":"I\u0027m not going to do all the computation,"},{"Start":"17:25.840 ","End":"17:27.700","Text":"but we get a lot of sine squared plus"},{"Start":"17:27.700 ","End":"17:29.785","Text":"cosine squared canceling out."},{"Start":"17:29.785 ","End":"17:39.565","Text":"In the end, we get the square root of sine squared Phi."},{"Start":"17:39.565 ","End":"17:47.080","Text":"Then dA is d Phi d Theta"},{"Start":"17:47.080 ","End":"17:53.515","Text":"but the right side here,"},{"Start":"17:53.515 ","End":"17:58.345","Text":"the square root of sine squared Phi,"},{"Start":"17:58.345 ","End":"18:01.510","Text":"the square root of a squared in"},{"Start":"18:01.510 ","End":"18:03.250","Text":"general or something squared"},{"Start":"18:03.250 ","End":"18:05.710","Text":"is the thing itself but in absolute value."},{"Start":"18:05.710 ","End":"18:11.410","Text":"It\u0027s the absolute value of sine of Phi but Phi"},{"Start":"18:11.410 ","End":"18:18.460","Text":"is in the region from 0 to Pi, or here it is."},{"Start":"18:18.460 ","End":"18:24.295","Text":"The sign is always non-negative from 0-180 degrees."},{"Start":"18:24.295 ","End":"18:30.080","Text":"I can actually drop the bars and it\u0027s equal to sine Phi,"},{"Start":"18:30.840 ","End":"18:40.090","Text":"and I just erase this and instead put sine of Phi,"},{"Start":"18:40.090 ","End":"18:44.020","Text":"have to put in the limits of integration."},{"Start":"18:44.020 ","End":"18:47.875","Text":"This d Phi goes with this integral,"},{"Start":"18:47.875 ","End":"18:52.000","Text":"and that goes from 0 to Pi."},{"Start":"18:52.000 ","End":"19:00.440","Text":"The d Theta goes with this integral and the Theta is from 0-2 Pi."},{"Start":"19:02.160 ","End":"19:08.020","Text":"What we get is a"},{"Start":"19:08.020 ","End":"19:17.170","Text":"squared integral from 0-2 Pi."},{"Start":"19:17.170 ","End":"19:24.220","Text":"Now, the integral of sine is minus cosine."},{"Start":"19:24.220 ","End":"19:29.410","Text":"I\u0027ve got minus cosine Phi."},{"Start":"19:29.410 ","End":"19:35.930","Text":"This has to be taken between the limits 0-Pi."},{"Start":"19:37.980 ","End":"19:42.160","Text":"This will be d Theta."},{"Start":"19:42.160 ","End":"19:46.720","Text":"Now, whenever I have a minus,"},{"Start":"19:46.720 ","End":"19:52.390","Text":"and the trick to do is just to change the order,"},{"Start":"19:52.390 ","End":"19:58.420","Text":"to take cosine of 0 minus cosine of Phi."},{"Start":"19:58.420 ","End":"20:00.910","Text":"I\u0027ll just do that bit at the side,"},{"Start":"20:00.910 ","End":"20:03.630","Text":"then I change the order and I can get rid of the minus,"},{"Start":"20:03.630 ","End":"20:12.660","Text":"so what I get is cosine of 0 minus cosine of Phi, 180 degrees."},{"Start":"20:12.660 ","End":"20:19.645","Text":"Now, cosine of 0 is equal to 1,"},{"Start":"20:19.645 ","End":"20:23.950","Text":"and cosine of Phi or 180 degrees is minus 1."},{"Start":"20:23.950 ","End":"20:27.445","Text":"Actually comes out to be 2."},{"Start":"20:27.445 ","End":"20:30.800","Text":"If I plug that back in here,"},{"Start":"20:30.800 ","End":"20:35.260","Text":"then the 2 comes out,"},{"Start":"20:35.260 ","End":"20:45.140","Text":"so I get 2a squared times the integral from 0-2 Pi of d Theta."},{"Start":"20:45.750 ","End":"20:51.650","Text":"You could think of it as 1d Theta because"},{"Start":"20:51.650 ","End":"20:57.540","Text":"then the integral of 1d Theta is just Theta."},{"Start":"20:57.540 ","End":"21:01.745","Text":"I need to evaluate it between 0 and 2Pi."},{"Start":"21:01.745 ","End":"21:06.380","Text":"So it just comes out to be 2Pi minus 0, which is 2Pi."},{"Start":"21:06.380 ","End":"21:11.340","Text":"I\u0027ve got 2a squared times 2Pi."},{"Start":"21:12.130 ","End":"21:19.510","Text":"Finally, that gives me 2 times 2 is 4,"},{"Start":"21:19.510 ","End":"21:23.515","Text":"I\u0027ll put the Pi next and then the a squared,"},{"Start":"21:23.515 ","End":"21:28.280","Text":"and this is a well-known formula."},{"Start":"21:29.070 ","End":"21:32.990","Text":"I\u0027m concluding this clip."}],"ID":9645},{"Watched":false,"Name":"General formula for Calculating Surface Integrals","Duration":"13m 50s","ChapterTopicVideoID":8783,"CourseChapterTopicPlaylistID":4974,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.580","Text":"Continuing with surface integrals."},{"Start":"00:02.580 ","End":"00:06.060","Text":"In the previous clip, we talked about parametric surfaces."},{"Start":"00:06.060 ","End":"00:12.450","Text":"Now I\u0027m going to give definition of several kinds of surface integrals."},{"Start":"00:12.450 ","End":"00:14.130","Text":"We\u0027ll start with 1 kind,"},{"Start":"00:14.130 ","End":"00:15.945","Text":"which is the simplest."},{"Start":"00:15.945 ","End":"00:20.520","Text":"For this I need basically 2 ingredients."},{"Start":"00:20.520 ","End":"00:24.390","Text":"What I need is a surface,"},{"Start":"00:24.390 ","End":"00:26.415","Text":"I\u0027ll explain what I mean,"},{"Start":"00:26.415 ","End":"00:29.260","Text":"and I need a function."},{"Start":"00:32.030 ","End":"00:37.350","Text":"For the function, I\u0027ll take a scalar function and 3 variables,"},{"Start":"00:37.350 ","End":"00:43.070","Text":"x, y, z, in future they will be also vector fields and so on."},{"Start":"00:43.070 ","End":"00:47.224","Text":"Surface has also several kind we could describe,"},{"Start":"00:47.224 ","End":"00:49.310","Text":"as we will in this example,"},{"Start":"00:49.310 ","End":"00:54.450","Text":"z as some function of x and y."},{"Start":"00:54.450 ","End":"00:58.205","Text":"That would be one kind where we have an explicit function."},{"Start":"00:58.205 ","End":"01:00.470","Text":"There\u0027s also parametric surfaces,"},{"Start":"01:00.470 ","End":"01:02.885","Text":"which we did in the previous clip."},{"Start":"01:02.885 ","End":"01:07.620","Text":"There\u0027s also level surfaces of 3D functions,"},{"Start":"01:07.620 ","End":"01:08.700","Text":"there is various kinds."},{"Start":"01:08.700 ","End":"01:13.730","Text":"Here I\u0027m going to talk about z as a function of x and y and later I\u0027ll talk"},{"Start":"01:13.730 ","End":"01:16.160","Text":"about the symmetry we get of x as a function of y and"},{"Start":"01:16.160 ","End":"01:19.800","Text":"z or y as a function of x and z and so on."},{"Start":"01:19.960 ","End":"01:23.585","Text":"Let me draw a sketch."},{"Start":"01:23.585 ","End":"01:27.530","Text":"Well, I didn\u0027t draw it, I borrowed it from the Internet."},{"Start":"01:27.530 ","End":"01:32.135","Text":"The surface is S and that\u0027s what\u0027s pictured here."},{"Start":"01:32.135 ","End":"01:39.835","Text":"The function g is defined on some domain or region D,"},{"Start":"01:39.835 ","End":"01:44.160","Text":"so we can say that x and y are from here."},{"Start":"01:44.510 ","End":"01:49.530","Text":"F is some function of 3 variables,"},{"Start":"01:49.530 ","End":"01:53.390","Text":"is defined in the region around S. It could be defined"},{"Start":"01:53.390 ","End":"01:57.680","Text":"everywhere but it has to be defined in some 3-dimensional region"},{"Start":"01:57.680 ","End":"02:02.630","Text":"containing S. What we\u0027re going to do now is"},{"Start":"02:02.630 ","End":"02:07.950","Text":"define the integral of this function,"},{"Start":"02:07.950 ","End":"02:12.090","Text":"of x, y, and z over"},{"Start":"02:12.090 ","End":"02:19.085","Text":"the surface S. For this kind of integral, we write ds."},{"Start":"02:19.085 ","End":"02:20.960","Text":"It\u0027s not the same, S is this,"},{"Start":"02:20.960 ","End":"02:23.255","Text":"this is always ds,"},{"Start":"02:23.255 ","End":"02:26.585","Text":"but this could be, maybe I\u0027ll change the color on this."},{"Start":"02:26.585 ","End":"02:30.800","Text":"So this indicates what the surface is and this is just a symbol,"},{"Start":"02:30.800 ","End":"02:32.270","Text":"just like we had da,"},{"Start":"02:32.270 ","End":"02:33.605","Text":"and you\u0027ll see that in the moment."},{"Start":"02:33.605 ","End":"02:34.910","Text":"So we define it."},{"Start":"02:34.910 ","End":"02:42.240","Text":"This is actually equal by definition to the double integral and this is the"},{"Start":"02:42.240 ","End":"02:50.070","Text":"regular double integral we learned over the 2D region D. D is in the xy plane."},{"Start":"02:50.070 ","End":"02:52.060","Text":"Happens to be a rectangle here."},{"Start":"02:52.060 ","End":"02:55.840","Text":"It\u0027s nice to work with, but doesn\u0027t necessarily have to be."},{"Start":"02:55.840 ","End":"02:59.730","Text":"Of f, of x, y,"},{"Start":"02:59.730 ","End":"03:02.910","Text":"and z but I don\u0027t want z,"},{"Start":"03:02.910 ","End":"03:04.905","Text":"I want only 2 variables."},{"Start":"03:04.905 ","End":"03:11.650","Text":"In the z position I am putting g of x and y times,"},{"Start":"03:11.650 ","End":"03:14.040","Text":"and it\u0027s getting a bit nasty,"},{"Start":"03:14.040 ","End":"03:18.370","Text":"something and then at the end, da."},{"Start":"03:18.560 ","End":"03:28.980","Text":"This something is g with respect to x squared of x and y. I didn\u0027t write that in,"},{"Start":"03:28.980 ","End":"03:32.620","Text":"otherwise it\u0027ll be too complicated, too messy."},{"Start":"03:32.620 ","End":"03:37.974","Text":"G with respect to y squared plus 1."},{"Start":"03:37.974 ","End":"03:40.910","Text":"I highlighted it."},{"Start":"03:41.040 ","End":"03:45.395","Text":"We\u0027ll do an example."},{"Start":"03:45.395 ","End":"03:49.600","Text":"In the example, I\u0027ll give you first what f is."},{"Start":"03:49.600 ","End":"03:51.400","Text":"We\u0027ll take f of x,"},{"Start":"03:51.400 ","End":"03:59.975","Text":"y and z to equal xy plus z, is defined everywhere."},{"Start":"03:59.975 ","End":"04:06.740","Text":"The surface S, I\u0027ll describe as follows: it\u0027s going to be"},{"Start":"04:06.740 ","End":"04:12.810","Text":"part of a plane x"},{"Start":"04:12.810 ","End":"04:17.310","Text":"plus y plus z equals 2,"},{"Start":"04:17.310 ","End":"04:19.130","Text":"we know this is an equation of a plane,"},{"Start":"04:19.130 ","End":"04:21.760","Text":"but in the first octant."},{"Start":"04:21.760 ","End":"04:25.025","Text":"In case you\u0027re wondering what is an octant,"},{"Start":"04:25.025 ","End":"04:26.990","Text":"just like in 2D,"},{"Start":"04:26.990 ","End":"04:28.745","Text":"we have 4 quadrants."},{"Start":"04:28.745 ","End":"04:32.360","Text":"In 3D we have 8 octants."},{"Start":"04:32.360 ","End":"04:41.360","Text":"The first octant just means basically where the x is non-negative,"},{"Start":"04:41.360 ","End":"04:46.325","Text":"y is non-negative and z is non-negative."},{"Start":"04:46.325 ","End":"04:48.695","Text":"Then you\u0027ll see in the picture."},{"Start":"04:48.695 ","End":"04:52.020","Text":"Here\u0027s the picture."},{"Start":"04:52.220 ","End":"04:58.670","Text":"This is the surface S and the first octant"},{"Start":"04:58.670 ","End":"05:05.855","Text":"is this part of space where all the coordinates are positive."},{"Start":"05:05.855 ","End":"05:09.290","Text":"So the plane intersected with this actually gives us a triangle,"},{"Start":"05:09.290 ","End":"05:13.295","Text":"S and D in the picture."},{"Start":"05:13.295 ","End":"05:15.920","Text":"I won\u0027t shade it, it\u0027ll be a mess."},{"Start":"05:15.920 ","End":"05:19.475","Text":"I\u0027ll just draw the outline of D,"},{"Start":"05:19.475 ","End":"05:22.590","Text":"it\u0027s this bit here,"},{"Start":"05:22.870 ","End":"05:29.330","Text":"and I\u0027ll label it D. This point of course,"},{"Start":"05:29.330 ","End":"05:30.410","Text":"is where y is 2."},{"Start":"05:30.410 ","End":"05:32.750","Text":"This is y, this is x,"},{"Start":"05:32.750 ","End":"05:34.850","Text":"and this is z."},{"Start":"05:34.850 ","End":"05:40.200","Text":"Each of the axes is cut at the point where it\u0027s value is 2."},{"Start":"05:41.180 ","End":"05:46.475","Text":"What I want to do is describe the surface in this form."},{"Start":"05:46.475 ","End":"05:54.710","Text":"Well, that\u0027s easy because I\u0027m on the plane and in particular on Earth,"},{"Start":"05:54.710 ","End":"05:57.350","Text":"we have this equation that holds,"},{"Start":"05:57.350 ","End":"06:06.620","Text":"and I can write it as z equals 2 minus x minus y."},{"Start":"06:06.620 ","End":"06:11.970","Text":"So this is going to be my g of x, y."},{"Start":"06:14.680 ","End":"06:18.815","Text":"Our task is to compute the surface integral,"},{"Start":"06:18.815 ","End":"06:21.335","Text":"which is equal to this,"},{"Start":"06:21.335 ","End":"06:23.660","Text":"and I just copied it down here."},{"Start":"06:23.660 ","End":"06:26.825","Text":"So what do we get in our case?"},{"Start":"06:26.825 ","End":"06:29.460","Text":"Let\u0027s get some space here."},{"Start":"06:32.870 ","End":"06:38.115","Text":"F is equal to xy plus z,"},{"Start":"06:38.115 ","End":"06:41.720","Text":"so what we get is the double"},{"Start":"06:41.720 ","End":"06:49.770","Text":"integral on the region D. Now f of x,"},{"Start":"06:49.770 ","End":"06:51.975","Text":"y and z is xy plus z,"},{"Start":"06:51.975 ","End":"06:56.235","Text":"so we get xy plus,"},{"Start":"06:56.235 ","End":"06:58.805","Text":"now z we take from here,"},{"Start":"06:58.805 ","End":"07:00.910","Text":"the g of x, y."},{"Start":"07:00.910 ","End":"07:04.999","Text":"So that\u0027s 2 minus x minus y."},{"Start":"07:06.830 ","End":"07:10.120","Text":"Next, let me take care of this bit."},{"Start":"07:10.120 ","End":"07:17.845","Text":"In fact, this bit is actually the bit that is ds in the formula in general."},{"Start":"07:17.845 ","End":"07:19.555","Text":"Now in our case,"},{"Start":"07:19.555 ","End":"07:21.959","Text":"g with respect to x,"},{"Start":"07:21.959 ","End":"07:23.845","Text":"and I\u0027m reading it from here."},{"Start":"07:23.845 ","End":"07:28.480","Text":"With respect to x, it\u0027s minus 1 squared,"},{"Start":"07:28.480 ","End":"07:30.550","Text":"and with respect to y,"},{"Start":"07:30.550 ","End":"07:35.630","Text":"it\u0027s also minus 1 squared."},{"Start":"07:35.730 ","End":"07:41.925","Text":"Then plus 1 as is, and then there\u0027s a dA."},{"Start":"07:41.925 ","End":"07:47.860","Text":"So basically, let me just say that this square root comes out to be the square root of,"},{"Start":"07:47.860 ","End":"07:50.450","Text":"1 plus 1 plus 1 is 3."},{"Start":"07:50.450 ","End":"07:52.870","Text":"This square root of 3 is a constant,"},{"Start":"07:52.870 ","End":"07:55.510","Text":"so I can write it in front,"},{"Start":"07:55.510 ","End":"07:58.910","Text":"and this is dA."},{"Start":"08:01.190 ","End":"08:09.270","Text":"What we have to do now is to compute in 2 dimensions a double integral."},{"Start":"08:09.270 ","End":"08:14.760","Text":"So we have to describe D in a more convenient form."},{"Start":"08:15.500 ","End":"08:21.420","Text":"Here\u0027s the picture of the part in the x,y plane."},{"Start":"08:21.420 ","End":"08:24.675","Text":"This is the D from here,"},{"Start":"08:24.675 ","End":"08:30.440","Text":"this is the x-axis,"},{"Start":"08:30.440 ","End":"08:32.900","Text":"and this is the y-axis,"},{"Start":"08:32.900 ","End":"08:36.510","Text":"and this is the point where they\u0027re both 2."},{"Start":"08:36.510 ","End":"08:40.850","Text":"We can get the equation of this line quite"},{"Start":"08:40.850 ","End":"08:46.400","Text":"simply because if I take z equals 2 minus x minus y,"},{"Start":"08:46.400 ","End":"08:51.300","Text":"but I let z equals 0 in here,"},{"Start":"08:51.300 ","End":"08:53.105","Text":"then if z is 0,"},{"Start":"08:53.105 ","End":"09:00.350","Text":"I get just that this line is x plus y equals 2."},{"Start":"09:00.350 ","End":"09:04.910","Text":"I\u0027d like to put 1 variable in terms of the other,"},{"Start":"09:04.910 ","End":"09:12.300","Text":"either x in terms of y or y in terms of x. I\u0027ll go with y equals 2 minus x,"},{"Start":"09:12.300 ","End":"09:13.950","Text":"I could have gone the other way."},{"Start":"09:13.950 ","End":"09:19.520","Text":"That means that we\u0027re going to do this integral first dy and then dx."},{"Start":"09:19.520 ","End":"09:23.275","Text":"So what we get is the integral."},{"Start":"09:23.275 ","End":"09:34.595","Text":"Now, first dy means y is going to go from 0-2 minus x."},{"Start":"09:34.595 ","End":"09:40.805","Text":"It\u0027s like each vertical strip is going to go from 0-2 minus x."},{"Start":"09:40.805 ","End":"09:45.455","Text":"Then afterwards we\u0027re going to take x from 0-2."},{"Start":"09:45.455 ","End":"09:52.090","Text":"That would describe this region D. Now we need this function,"},{"Start":"09:52.090 ","End":"09:56.010","Text":"which is, I\u0027ll just copy it as is,"},{"Start":"09:56.010 ","End":"10:01.595","Text":"xy plus 2 minus x minus y."},{"Start":"10:01.595 ","End":"10:05.799","Text":"Now this integral is dy,"},{"Start":"10:05.799 ","End":"10:07.635","Text":"that\u0027s what y goes from."},{"Start":"10:07.635 ","End":"10:11.740","Text":"Then x goes from 0-2."},{"Start":"10:12.490 ","End":"10:16.340","Text":"Now let me do the inner integral,"},{"Start":"10:16.340 ","End":"10:19.850","Text":"as a side exercise over here."},{"Start":"10:19.850 ","End":"10:28.920","Text":"What we get if we integrate with respect to y is a 1/2 xy squared,"},{"Start":"10:28.920 ","End":"10:31.440","Text":"because x is a constant,"},{"Start":"10:31.440 ","End":"10:40.070","Text":"and then plus 2y minus xy"},{"Start":"10:40.070 ","End":"10:45.675","Text":"minus 1/2y squared and"},{"Start":"10:45.675 ","End":"10:54.190","Text":"this taken between 0 and 2 minus x."},{"Start":"10:54.560 ","End":"10:57.980","Text":"Now, if I plug in 0,"},{"Start":"10:57.980 ","End":"11:02.585","Text":"this whole thing is 0 because everything contains y."},{"Start":"11:02.585 ","End":"11:05.555","Text":"This is of course limits on y."},{"Start":"11:05.555 ","End":"11:16.920","Text":"Notice that this bit is like 2 minus x times y."},{"Start":"11:16.920 ","End":"11:20.345","Text":"If I substitute y equals 2 minus x,"},{"Start":"11:20.345 ","End":"11:24.980","Text":"I\u0027m going to get 2 minus x squared and then similarly everywhere I see y squared,"},{"Start":"11:24.980 ","End":"11:28.805","Text":"I\u0027m going to get 2 minus x squared."},{"Start":"11:28.805 ","End":"11:34.625","Text":"So basically I can get 2 minus x squared outside the brackets."},{"Start":"11:34.625 ","End":"11:37.280","Text":"Let\u0027s see what I\u0027m left with."},{"Start":"11:37.280 ","End":"11:40.140","Text":"I\u0027m left with 1/2x,"},{"Start":"11:40.540 ","End":"11:42.910","Text":"I\u0027ll write it 1/2x,"},{"Start":"11:42.910 ","End":"11:45.329","Text":"and then from here plus 1,"},{"Start":"11:45.329 ","End":"11:49.185","Text":"and from here minus 1/2,"},{"Start":"11:49.185 ","End":"11:53.760","Text":"and the 1 minus 1/2 is just 1/2."},{"Start":"11:53.760 ","End":"11:57.845","Text":"So getting back to here,"},{"Start":"11:57.845 ","End":"12:05.475","Text":"what we have is the square root of 3 and I guess I could take the 1/2 out,"},{"Start":"12:05.475 ","End":"12:08.535","Text":"over 2 times the integral from"},{"Start":"12:08.535 ","End":"12:16.575","Text":"0-2 of 2 minus x"},{"Start":"12:16.575 ","End":"12:23.175","Text":"squared times x plus 1,"},{"Start":"12:23.175 ","End":"12:29.590","Text":"because we took the 1/2 out, this dx."},{"Start":"12:29.780 ","End":"12:36.605","Text":"To save time, I computed this at the side and it comes out to be x"},{"Start":"12:36.605 ","End":"12:45.100","Text":"cubed minus 3x squared plus 4."},{"Start":"12:45.620 ","End":"12:49.935","Text":"So we get, let\u0027s see,"},{"Start":"12:49.935 ","End":"12:54.280","Text":"the integral of this is"},{"Start":"12:54.710 ","End":"13:05.280","Text":"x^4 over 4 minus x cubed plus 4x from 0-2,"},{"Start":"13:05.280 ","End":"13:10.385","Text":"0 gives nothing so I just have to substitute 2."},{"Start":"13:10.385 ","End":"13:20.700","Text":"I get that this equals square root of 3 over 2 times, now,"},{"Start":"13:20.700 ","End":"13:25.560","Text":"if I put in 2, 2^4 is 16 over"},{"Start":"13:25.560 ","End":"13:34.469","Text":"4 is 4 minus 8 plus 8."},{"Start":"13:34.469 ","End":"13:37.710","Text":"So this is just 4,"},{"Start":"13:37.710 ","End":"13:39.705","Text":"4 over 2 is 2,"},{"Start":"13:39.705 ","End":"13:46.140","Text":"and so the answer is 2 times the square root of 3."},{"Start":"13:46.140 ","End":"13:51.250","Text":"I\u0027ll highlight it. It\u0027s time for a break."}],"ID":9646},{"Watched":false,"Name":"Connection between General Formula and Parametric Surfaces Form","Duration":"16m 21s","ChapterTopicVideoID":8784,"CourseChapterTopicPlaylistID":4974,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.344","Text":"Back from the break."},{"Start":"00:03.344 ","End":"00:06.195","Text":"Let\u0027s continue."},{"Start":"00:06.195 ","End":"00:08.970","Text":"I\u0027ll clear the board."},{"Start":"00:08.970 ","End":"00:13.425","Text":"I\u0027m going to talk about parametric surfaces,"},{"Start":"00:13.425 ","End":"00:18.720","Text":"but I want to remind you that this kind of surface where z is a function of x,"},{"Start":"00:18.720 ","End":"00:25.755","Text":"and y is really 3 in 1 because suppose we had y is some function,"},{"Start":"00:25.755 ","End":"00:28.695","Text":"also call it g, of x and z,"},{"Start":"00:28.695 ","End":"00:33.160","Text":"then we just make the corresponding changes in this formula."},{"Start":"00:33.590 ","End":"00:36.110","Text":"This is what we would get,"},{"Start":"00:36.110 ","End":"00:39.200","Text":"except that in this 1,"},{"Start":"00:39.200 ","End":"00:41.060","Text":"I use the other notation,"},{"Start":"00:41.060 ","End":"00:45.425","Text":"dg by dx is the same as gx,"},{"Start":"00:45.425 ","End":"00:47.540","Text":"dg by dz is gz."},{"Start":"00:47.540 ","End":"00:53.145","Text":"We keep alternating between the various notations,"},{"Start":"00:53.145 ","End":"00:58.355","Text":"and similarly, if you had z is g of x and y and so on."},{"Start":"00:58.355 ","End":"01:01.045","Text":"Now, for parametric."},{"Start":"01:01.045 ","End":"01:08.440","Text":"Now, I\u0027m going to replace this kind in the surface by a parametric surface,"},{"Start":"01:08.440 ","End":"01:11.875","Text":"also call it S, and in general,"},{"Start":"01:11.875 ","End":"01:18.725","Text":"be given by a vector function r of 2 variables, u and v,"},{"Start":"01:18.725 ","End":"01:23.375","Text":"which would be some function x of u and v,"},{"Start":"01:23.375 ","End":"01:26.450","Text":"some function y of u and v,"},{"Start":"01:26.450 ","End":"01:30.200","Text":"and some function z of u and v,"},{"Start":"01:30.200 ","End":"01:34.160","Text":"and u and v might be restricted to"},{"Start":"01:34.160 ","End":"01:41.065","Text":"some domain D. In this case,"},{"Start":"01:41.065 ","End":"01:43.790","Text":"we would get a different formula,"},{"Start":"01:43.790 ","End":"01:48.150","Text":"more general 1 as follows."},{"Start":"01:49.060 ","End":"01:52.865","Text":"In this case, I just copied the left-hand side."},{"Start":"01:52.865 ","End":"01:59.375","Text":"The surface integral, is the regular,"},{"Start":"01:59.375 ","End":"02:01.640","Text":"double it to grow over D,"},{"Start":"02:01.640 ","End":"02:03.290","Text":"which we talked about."},{"Start":"02:03.290 ","End":"02:05.270","Text":"That\u0027s the domain for u and"},{"Start":"02:05.270 ","End":"02:14.945","Text":"v of f. Instead of writing xuv,"},{"Start":"02:14.945 ","End":"02:19.980","Text":"yuv, and zuv, I\u0027ll just write f of r of u and"},{"Start":"02:19.980 ","End":"02:24.860","Text":"v. Angular brackets and round brackets is almost the same thing."},{"Start":"02:24.860 ","End":"02:27.690","Text":"The position vector is like a point."},{"Start":"02:28.130 ","End":"02:31.470","Text":"That corresponds to this part,"},{"Start":"02:31.470 ","End":"02:35.705","Text":"and this part corresponds to the magnitude"},{"Start":"02:35.705 ","End":"02:42.380","Text":"of the partial derivative of r with respect to u,"},{"Start":"02:42.380 ","End":"02:50.270","Text":"cross product with the partial derivative of r with respect to v. All this is da,"},{"Start":"02:50.270 ","End":"02:52.984","Text":"where a is for the area."},{"Start":"02:52.984 ","End":"02:58.285","Text":"It turns out that actually this is a special case of this,"},{"Start":"02:58.285 ","End":"03:00.900","Text":"and I highlighted this too."},{"Start":"03:00.900 ","End":"03:05.015","Text":"I\u0027d like to just briefly show you why this is a special case of this."},{"Start":"03:05.015 ","End":"03:07.680","Text":"It\u0027s good practice."},{"Start":"03:07.840 ","End":"03:18.185","Text":"Supposing that we have this kind of a situation where z is the function g of x and y,"},{"Start":"03:18.185 ","End":"03:21.955","Text":"I could always take a parameterization r,"},{"Start":"03:21.955 ","End":"03:23.630","Text":"and instead of u and v,"},{"Start":"03:23.630 ","End":"03:32.735","Text":"I could just stick to x and y and say that r of xy is just equal to x,y,"},{"Start":"03:32.735 ","End":"03:36.460","Text":"and g of xy in the place of z."},{"Start":"03:36.460 ","End":"03:39.690","Text":"Now, if I compute this,"},{"Start":"03:39.690 ","End":"03:42.755","Text":"I have to replace u and v by x and y."},{"Start":"03:42.755 ","End":"03:49.870","Text":"What I get is that r with respect to x is equal to 1,"},{"Start":"03:49.870 ","End":"03:53.795","Text":"0, because y is a constant as far as x goes."},{"Start":"03:53.795 ","End":"03:58.205","Text":"Then gx, don\u0027t bother writing dx, y,"},{"Start":"03:58.205 ","End":"04:03.410","Text":"and r with respect to y would be equal to similarly,"},{"Start":"04:03.410 ","End":"04:09.990","Text":"this would be 0, 1, gy."},{"Start":"04:10.490 ","End":"04:16.140","Text":"If you compute rx cross ry,"},{"Start":"04:16.140 ","End":"04:18.640","Text":"which is what I want here,"},{"Start":"04:18.700 ","End":"04:23.090","Text":"I\u0027m not going to do the computation of the cross-product."},{"Start":"04:23.090 ","End":"04:27.605","Text":"I\u0027ll just give you the answer that it comes out as"},{"Start":"04:27.605 ","End":"04:35.410","Text":"minus gx minus gy, 1."},{"Start":"04:35.410 ","End":"04:45.590","Text":"Then if I take the magnitude of this,"},{"Start":"04:45.590 ","End":"04:47.525","Text":"I\u0027ll just write it up here,"},{"Start":"04:47.525 ","End":"04:53.680","Text":"same thing, that is rx cross ry,"},{"Start":"04:53.680 ","End":"04:55.340","Text":"then this will just equal"},{"Start":"04:55.340 ","End":"04:59.690","Text":"the square root of this squared plus this squared plus this squared,"},{"Start":"04:59.690 ","End":"05:04.705","Text":"which is gx squared plus gy squared plus 1 squared,"},{"Start":"05:04.705 ","End":"05:09.020","Text":"and that is exactly what is written here."},{"Start":"05:09.020 ","End":"05:13.885","Text":"This is a special case of this."},{"Start":"05:13.885 ","End":"05:18.255","Text":"Let\u0027s do an example of this formula."},{"Start":"05:18.255 ","End":"05:21.425","Text":"I guess I don\u0027t need this stuff."},{"Start":"05:21.425 ","End":"05:27.080","Text":"The example I\u0027m going to take is to compute the double"},{"Start":"05:27.080 ","End":"05:34.955","Text":"integral of zds over the surface S,"},{"Start":"05:34.955 ","End":"05:43.430","Text":"where S is described in words as the upper half a sphere or"},{"Start":"05:43.430 ","End":"05:50.310","Text":"hemisphere of radius 2 and the"},{"Start":"05:50.310 ","End":"05:58.465","Text":"upper means where the z is positive or non-negative."},{"Start":"05:58.465 ","End":"06:00.885","Text":"I\u0027ll give you a sketch."},{"Start":"06:00.885 ","End":"06:03.650","Text":"Here\u0027s the picture, it\u0027s hard to see."},{"Start":"06:03.650 ","End":"06:05.690","Text":"This is the x-axis,"},{"Start":"06:05.690 ","End":"06:08.090","Text":"this here is the y-axis,"},{"Start":"06:08.090 ","End":"06:12.620","Text":"and this here is the z-axis and radius is 2,"},{"Start":"06:12.620 ","End":"06:16.935","Text":"and it\u0027s the part above the x-y plane."},{"Start":"06:16.935 ","End":"06:21.090","Text":"Of course, this is r function f. I mean, f of x,"},{"Start":"06:21.090 ","End":"06:25.760","Text":"y, z just ignores x and y and is equal to z."},{"Start":"06:25.760 ","End":"06:31.850","Text":"That\u0027s f. That\u0027s S. What we need is a parametric form of S. Of course,"},{"Start":"06:31.850 ","End":"06:37.055","Text":"we could write z as a function of x and y"},{"Start":"06:37.055 ","End":"06:44.480","Text":"because a sphere is x squared plus y squared plus z squared equals radius squared,"},{"Start":"06:44.480 ","End":"06:46.445","Text":"in this case, 2 squared is 4."},{"Start":"06:46.445 ","End":"06:50.030","Text":"We could bring the x squared and y squared over to"},{"Start":"06:50.030 ","End":"06:54.430","Text":"the other side and then take the positive square root,"},{"Start":"06:54.430 ","End":"06:57.650","Text":"but then we wouldn\u0027t be practicing the parametric."},{"Start":"06:57.650 ","End":"06:59.150","Text":"I want a parametric,"},{"Start":"06:59.150 ","End":"07:03.430","Text":"and the obvious thing to do is the spherical coordinates."},{"Start":"07:03.430 ","End":"07:08.540","Text":"Now, we\u0027ve seen this sphere of radius 2 before,"},{"Start":"07:08.540 ","End":"07:10.460","Text":"but it was the whole sphere."},{"Start":"07:10.460 ","End":"07:13.630","Text":"I brought the equations in for the whole sphere."},{"Start":"07:13.630 ","End":"07:15.750","Text":"How do I change this?"},{"Start":"07:15.750 ","End":"07:17.670","Text":"It\u0027s just a hemisphere."},{"Start":"07:17.670 ","End":"07:21.090","Text":"All we need is for the latitude angle."},{"Start":"07:21.090 ","End":"07:27.810","Text":"This Phi, instead of going from 0-180 degrees to stop at 90 degrees,"},{"Start":"07:27.810 ","End":"07:30.580","Text":"if I just put a Pi/2 here,"},{"Start":"07:30.580 ","End":"07:33.960","Text":"then that will make it an upper hemisphere."},{"Start":"07:34.150 ","End":"07:37.610","Text":"That\u0027s the parametrization, the function."},{"Start":"07:37.610 ","End":"07:40.264","Text":"Now, we want to use this formula."},{"Start":"07:40.264 ","End":"07:45.905","Text":"What I\u0027m still missing is the 2 partial derivatives."},{"Start":"07:45.905 ","End":"07:48.920","Text":"Only instead of u and v,"},{"Start":"07:48.920 ","End":"07:53.480","Text":"I\u0027m going to have Theta and Phi."},{"Start":"07:53.480 ","End":"08:02.730","Text":"In our case, the partial derivative with respect to Theta will equal,"},{"Start":"08:02.730 ","End":"08:05.845","Text":"in this case, Phi is like a constant."},{"Start":"08:05.845 ","End":"08:09.440","Text":"Derivative of cosine is minus sine,"},{"Start":"08:09.440 ","End":"08:16.800","Text":"so I have minus 2 sine of Phi sine Theta,"},{"Start":"08:16.800 ","End":"08:19.245","Text":"that\u0027s minus sine Theta."},{"Start":"08:19.245 ","End":"08:21.930","Text":"Derivative of sine is cosine,"},{"Start":"08:21.930 ","End":"08:28.575","Text":"so this is just 2 sine of Phi cosine of Theta."},{"Start":"08:28.575 ","End":"08:33.365","Text":"Then this is a constant as far as Theta goes, so that\u0027s zero."},{"Start":"08:33.365 ","End":"08:37.315","Text":"The other partial derivative with respect to Phi,"},{"Start":"08:37.315 ","End":"08:41.780","Text":"this time, I treat Theta as the constant."},{"Start":"08:41.780 ","End":"08:46.440","Text":"I get 2 cosine Phi,"},{"Start":"08:46.480 ","End":"08:50.090","Text":"this Cosine Theta just sticks."},{"Start":"08:50.090 ","End":"08:55.740","Text":"Then again, 2 cosine Phi,"},{"Start":"08:55.740 ","End":"08:59.055","Text":"this time with a sine Theta."},{"Start":"08:59.055 ","End":"09:02.480","Text":"The derivative of cosine is minus sine,"},{"Start":"09:02.480 ","End":"09:06.730","Text":"so it\u0027s minus 2 sine Phi."},{"Start":"09:06.730 ","End":"09:11.800","Text":"What I need is the cross-product."},{"Start":"09:11.820 ","End":"09:18.700","Text":"I\u0027m not going to spend a lot of time computing cross-products."},{"Start":"09:18.700 ","End":"09:20.440","Text":"You should know how to do this."},{"Start":"09:20.440 ","End":"09:23.065","Text":"I\u0027m just going to quote the result."},{"Start":"09:23.065 ","End":"09:25.540","Text":"This is the result,"},{"Start":"09:25.540 ","End":"09:27.745","Text":"I just quoted it."},{"Start":"09:27.745 ","End":"09:32.785","Text":"What I need though is the magnitude of this thing."},{"Start":"09:32.785 ","End":"09:41.965","Text":"The magnitude of r Theta cross r Phi is equal to it\u0027s going to be the square root."},{"Start":"09:41.965 ","End":"09:43.630","Text":"Again, I\u0027m going to take a shortcut."},{"Start":"09:43.630 ","End":"09:46.750","Text":"It would be this squared plus this squared plus this squared."},{"Start":"09:46.750 ","End":"09:49.600","Text":"After messing around with trigonometry and"},{"Start":"09:49.600 ","End":"09:53.875","Text":"using identities like sine squared plus cosine squared is 1."},{"Start":"09:53.875 ","End":"10:02.035","Text":"All that\u0027s left here is 16 sine squared of Phi."},{"Start":"10:02.035 ","End":"10:06.160","Text":"You have to be careful when taking the square root of something squared."},{"Start":"10:06.160 ","End":"10:12.445","Text":"The temptation would be to say that this would be just for sine Phi."},{"Start":"10:12.445 ","End":"10:17.215","Text":"But normally, the square root of something squared need an absolute value."},{"Start":"10:17.215 ","End":"10:23.920","Text":"However, because Phi is between 0 and Pi over 2,"},{"Start":"10:23.920 ","End":"10:26.425","Text":"in fact, even between 0 and Pi,"},{"Start":"10:26.425 ","End":"10:27.925","Text":"the sine is positive."},{"Start":"10:27.925 ","End":"10:31.555","Text":"I can go back and drop the absolute value."},{"Start":"10:31.555 ","End":"10:35.094","Text":"Now, I\u0027m going to have to use this formula here."},{"Start":"10:35.094 ","End":"10:38.635","Text":"If I scroll, I\u0027m going to lose it."},{"Start":"10:38.635 ","End":"10:41.545","Text":"Let me just record some of the things up here."},{"Start":"10:41.545 ","End":"10:46.100","Text":"This part was 4 sine Phi."},{"Start":"10:47.040 ","End":"10:53.680","Text":"This part f of it is just the z part."},{"Start":"10:53.680 ","End":"10:56.290","Text":"The z is 2 cosine Phi."},{"Start":"10:56.290 ","End":"11:00.595","Text":"This part is 2 cosine Phi."},{"Start":"11:00.595 ","End":"11:06.280","Text":"Note that this d is actually what\u0027s written here."},{"Start":"11:06.280 ","End":"11:11.860","Text":"That\u0027s my D, which is actually a rectangle in terms of Theta Phi."},{"Start":"11:11.860 ","End":"11:16.585","Text":"It\u0027s a rectangle from 0 to 2Pi and this one from 0 to Pi over 2."},{"Start":"11:16.585 ","End":"11:19.640","Text":"Now, we can proceed."},{"Start":"11:19.890 ","End":"11:25.300","Text":"Let\u0027s see what we get for the integral."},{"Start":"11:25.300 ","End":"11:33.895","Text":"Just write it over here again is the integral over d. Now d,"},{"Start":"11:33.895 ","End":"11:39.220","Text":"I can take it as first of all with respect to Theta,"},{"Start":"11:39.220 ","End":"11:41.140","Text":"then with respect to Phi."},{"Start":"11:41.140 ","End":"11:45.370","Text":"I think we\u0027ll do that. We\u0027ll take Theta from 0 to"},{"Start":"11:45.370 ","End":"11:53.365","Text":"2 Pi and Phi from 0 to Pi over 2."},{"Start":"11:53.365 ","End":"11:57.460","Text":"Then the dA will be d Theta,"},{"Start":"11:57.460 ","End":"12:01.450","Text":"That\u0027s the inner one, and d Phi."},{"Start":"12:01.450 ","End":"12:02.995","Text":"That\u0027s the outer one."},{"Start":"12:02.995 ","End":"12:05.380","Text":"It doesn\u0027t really matter, actually could\u0027ve done it the other way"},{"Start":"12:05.380 ","End":"12:07.945","Text":"round of this times this."},{"Start":"12:07.945 ","End":"12:14.290","Text":"What I\u0027d like to do is take the 4 outside the integral and inside I"},{"Start":"12:14.290 ","End":"12:22.180","Text":"want to leave 2 sine Phi cosine Phi."},{"Start":"12:22.180 ","End":"12:25.120","Text":"The 4 I took out 2 times this times this,"},{"Start":"12:25.120 ","End":"12:26.620","Text":"I just changed the order."},{"Start":"12:26.620 ","End":"12:34.000","Text":"The reason I did that is that there\u0027s a famous formula in trigonometry that"},{"Start":"12:34.000 ","End":"12:42.745","Text":"the sine of 2 Alpha is 2 sine Alpha cosine Alpha."},{"Start":"12:42.745 ","End":"12:48.500","Text":"Here, I\u0027ll change Alpha with Phi and read it from right to left."},{"Start":"12:48.600 ","End":"12:54.650","Text":"I get the integral."},{"Start":"12:55.110 ","End":"12:59.140","Text":"Same limits, 0 to 2 Pi,"},{"Start":"12:59.140 ","End":"13:03.770","Text":"0 to Pi over 2"},{"Start":"13:03.900 ","End":"13:11.155","Text":"of sine of 2 Phi."},{"Start":"13:11.155 ","End":"13:16.015","Text":"Then d Theta d Phi."},{"Start":"13:16.015 ","End":"13:19.420","Text":"Now, if I first do the inner integral,"},{"Start":"13:19.420 ","End":"13:24.710","Text":"this is just a constant as far as Theta is concerned."},{"Start":"13:27.210 ","End":"13:33.835","Text":"What I get is 4"},{"Start":"13:33.835 ","End":"13:41.200","Text":"times the integral from 0 to Pi over 2 of this constant."},{"Start":"13:41.200 ","End":"13:43.600","Text":"This is like times 1,"},{"Start":"13:43.600 ","End":"13:46.345","Text":"it\u0027s just this times Theta."},{"Start":"13:46.345 ","End":"13:48.550","Text":"I\u0027ll put the Theta in front."},{"Start":"13:48.550 ","End":"13:54.580","Text":"Theta sine of 2 Phi."},{"Start":"13:54.580 ","End":"14:01.165","Text":"This has to be evaluated between 0 and 2 Pi."},{"Start":"14:01.165 ","End":"14:04.150","Text":"I still need the d Phi."},{"Start":"14:04.150 ","End":"14:09.805","Text":"Now, this is the limit for Theta."},{"Start":"14:09.805 ","End":"14:12.355","Text":"When Theta is 0,"},{"Start":"14:12.355 ","End":"14:15.160","Text":"it\u0027s just 0, and when Theta is 2 Pi,"},{"Start":"14:15.160 ","End":"14:17.440","Text":"it\u0027s just 2 Pi."},{"Start":"14:17.440 ","End":"14:25.010","Text":"What I get is 2 Pi sine 2 Phi."},{"Start":"14:26.670 ","End":"14:30.145","Text":"I\u0027ll take the Pi upfront."},{"Start":"14:30.145 ","End":"14:34.490","Text":"But it suits me to leave the 2 inside."},{"Start":"14:34.670 ","End":"14:45.425","Text":"It\u0027s 2 and then sine of 2 Phi d Phi."},{"Start":"14:45.425 ","End":"14:49.480","Text":"Now, the reason I left the 2 in is it makes it easier to integrate"},{"Start":"14:49.480 ","End":"14:53.785","Text":"this because the integral of sine is minus cosine."},{"Start":"14:53.785 ","End":"14:56.575","Text":"The 2 is the inner derivative already."},{"Start":"14:56.575 ","End":"15:02.260","Text":"I get 4 Pi times minus cosine"},{"Start":"15:02.260 ","End":"15:09.775","Text":"sine of 2 Phi between 0 and Pi over 2."},{"Start":"15:09.775 ","End":"15:15.010","Text":"Let me do this bit at the side and only use the famous trick that when you have a minus,"},{"Start":"15:15.010 ","End":"15:18.200","Text":"you can throw out the minus."},{"Start":"15:23.820 ","End":"15:27.865","Text":"Instead of Pi 2 over to 0,"},{"Start":"15:27.865 ","End":"15:31.900","Text":"we\u0027ll switch it to 0 here and Pi over 2 here."},{"Start":"15:31.900 ","End":"15:34.420","Text":"Because if we do that then the subtraction in reverse,"},{"Start":"15:34.420 ","End":"15:36.250","Text":"it\u0027s like getting rid of a minus."},{"Start":"15:36.250 ","End":"15:41.620","Text":"What that gives us is cosine of twice 0"},{"Start":"15:41.620 ","End":"15:47.200","Text":"is 0 minus cosine of twice Pi over 2 is cosine Pi."},{"Start":"15:47.200 ","End":"15:50.215","Text":"Now cosine of 0 is 1,"},{"Start":"15:50.215 ","End":"15:52.750","Text":"and cosine of Pi is minus 1."},{"Start":"15:52.750 ","End":"15:57.170","Text":"I get 1 minus minus 1 is 2."},{"Start":"15:58.560 ","End":"16:04.930","Text":"What I get is 4 Pi times 2."},{"Start":"16:04.930 ","End":"16:12.070","Text":"In other words, the answer is 8 Pi."},{"Start":"16:12.070 ","End":"16:15.350","Text":"I\u0027ll highlight that."},{"Start":"16:16.230 ","End":"16:20.870","Text":"That\u0027s the end of this example."}],"ID":9647},{"Watched":false,"Name":"Surface Integrals for Piecewise Smooth Surfaces","Duration":"24m 6s","ChapterTopicVideoID":8789,"CourseChapterTopicPlaylistID":4974,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.940","Text":"In this clip, we\u0027re continuing with surface integrals."},{"Start":"00:03.940 ","End":"00:11.340","Text":"I want to just extend it to something called piecewise smooth surfaces,"},{"Start":"00:11.340 ","End":"00:14.669","Text":"which is analogous to line integrals."},{"Start":"00:14.669 ","End":"00:18.450","Text":"We also have lines which were piecewise smooth."},{"Start":"00:18.450 ","End":"00:21.165","Text":"There were made up of several bits of line,"},{"Start":"00:21.165 ","End":"00:23.085","Text":"each one of them being smooth."},{"Start":"00:23.085 ","End":"00:26.400","Text":"I\u0027ll just mention the concept of smooth."},{"Start":"00:26.400 ","End":"00:28.560","Text":"In the case of lines,"},{"Start":"00:28.560 ","End":"00:35.100","Text":"we typically had a vector function r of t and it was"},{"Start":"00:35.100 ","End":"00:42.050","Text":"called smooth if it had continuous second order derivatives."},{"Start":"00:42.050 ","End":"00:43.670","Text":"But that\u0027s too technical."},{"Start":"00:43.670 ","End":"00:45.440","Text":"But the more importantly,"},{"Start":"00:45.440 ","End":"00:54.695","Text":"there was an extra condition that r prime of t was never 0."},{"Start":"00:54.695 ","End":"01:02.705","Text":"Similarly for surfaces, here we have r as a function of 2 parameters."},{"Start":"01:02.705 ","End":"01:07.944","Text":"We call them u and v and it\u0027s smooth."},{"Start":"01:07.944 ","End":"01:14.175","Text":"Also if it has continuous first and second-order derivatives,"},{"Start":"01:14.175 ","End":"01:15.900","Text":"but that\u0027s too technical,"},{"Start":"01:15.900 ","End":"01:17.935","Text":"so we\u0027d ignore that."},{"Start":"01:17.935 ","End":"01:20.645","Text":"There\u0027s also a condition."},{"Start":"01:20.645 ","End":"01:23.510","Text":"But this time we don\u0027t have a single derivative,"},{"Start":"01:23.510 ","End":"01:25.790","Text":"we have 2 partial derivatives."},{"Start":"01:25.790 ","End":"01:32.510","Text":"We have r with respect to u and we have r with respect to"},{"Start":"01:32.510 ","End":"01:37.590","Text":"v. The condition for smooth is that"},{"Start":"01:37.590 ","End":"01:45.365","Text":"this cross product is not equal to 0."},{"Start":"01:45.365 ","End":"01:48.710","Text":"I can actually rewrite this in terms of the magnitude because"},{"Start":"01:48.710 ","End":"01:53.210","Text":"a vector is 0 only if its magnitude is 0."},{"Start":"01:53.210 ","End":"01:56.150","Text":"In fact, let me write it with magnitude."},{"Start":"01:56.150 ","End":"02:03.080","Text":"In fact, this bit here is actually the normal vector."},{"Start":"02:03.080 ","End":"02:09.020","Text":"Besides being differentiable, it has to have a non-zero normal then it\u0027s smooth."},{"Start":"02:09.020 ","End":"02:11.610","Text":"Now as for piecewise,"},{"Start":"02:11.740 ","End":"02:17.555","Text":"I\u0027ll explain with an example that sometimes a surface is made up of more than 1 bit."},{"Start":"02:17.555 ","End":"02:21.409","Text":"I\u0027ll take the example of a cone with a base."},{"Start":"02:21.409 ","End":"02:27.800","Text":"I\u0027ll begin with a picture and then I\u0027ll explain what it all means."},{"Start":"02:27.800 ","End":"02:32.870","Text":"In words. What I want to take is the surface of a cone."},{"Start":"02:32.870 ","End":"02:36.335","Text":"But I\u0027m going to include the base as part of the cone,"},{"Start":"02:36.335 ","End":"02:38.255","Text":"like a closed cone."},{"Start":"02:38.255 ","End":"02:44.910","Text":"The conditions on the cone is centered on the z-axis."},{"Start":"02:46.180 ","End":"02:52.220","Text":"The base is in the x,"},{"Start":"02:52.220 ","End":"02:58.270","Text":"y plane, and it\u0027s circle of radius."},{"Start":"02:58.270 ","End":"03:09.330","Text":"Of course, the base of the cone is a circle with a radius of 2,"},{"Start":"03:09.330 ","End":"03:17.135","Text":"and the height of the cone is 3."},{"Start":"03:17.135 ","End":"03:19.240","Text":"That pretty much describes the cone."},{"Start":"03:19.240 ","End":"03:24.310","Text":"Meaning we have here a circle of radius 2 in the x,"},{"Start":"03:24.310 ","End":"03:26.425","Text":"y plane, and this height,"},{"Start":"03:26.425 ","End":"03:32.685","Text":"the z-axis runs through the apex and its height is 3,"},{"Start":"03:32.685 ","End":"03:36.210","Text":"and we want both bits,"},{"Start":"03:36.210 ","End":"03:39.405","Text":"the cone and the base."},{"Start":"03:39.405 ","End":"03:41.420","Text":"That\u0027s just the surface."},{"Start":"03:41.420 ","End":"03:43.610","Text":"The surface is made up of 2 bits."},{"Start":"03:43.610 ","End":"03:46.310","Text":"We maybe we\u0027ll call the surface of"},{"Start":"03:46.310 ","End":"03:52.850","Text":"the cone part S_1 and the surface at the bottom that we don\u0027t see,"},{"Start":"03:52.850 ","End":"03:57.245","Text":"you can imagine that it\u0027s a circle here."},{"Start":"03:57.245 ","End":"03:59.760","Text":"That would be S_2,"},{"Start":"03:59.760 ","End":"04:02.505","Text":"the bass part of the surface."},{"Start":"04:02.505 ","End":"04:06.320","Text":"I need give you a function to take the integral over,"},{"Start":"04:06.320 ","End":"04:12.420","Text":"and we\u0027ll take the function f of x,"},{"Start":"04:12.420 ","End":"04:16.959","Text":"y, z is equal to"},{"Start":"04:16.959 ","End":"04:24.425","Text":"the square root of x squared plus y squared and z doesn\u0027t come into it."},{"Start":"04:24.425 ","End":"04:33.200","Text":"I want to find the integral over the whole S. Say that the S is the complete cone,"},{"Start":"04:33.200 ","End":"04:38.615","Text":"which is basically the union of S_1 with S_2."},{"Start":"04:38.615 ","End":"04:41.720","Text":"I want to point out something that we\u0027re not going to use,"},{"Start":"04:41.720 ","End":"04:45.335","Text":"but just to be technically correct."},{"Start":"04:45.335 ","End":"04:48.080","Text":"The base is certainly a disk,"},{"Start":"04:48.080 ","End":"04:51.350","Text":"it\u0027s a smooth surface,"},{"Start":"04:51.350 ","End":"04:55.760","Text":"and the sides of the cone is smooth except for this point here,"},{"Start":"04:55.760 ","End":"04:58.730","Text":"where if we actually do the computation and maybe later,"},{"Start":"04:58.730 ","End":"05:02.765","Text":"we will see that this thing comes out to be 0."},{"Start":"05:02.765 ","End":"05:06.590","Text":"But a single point doesn\u0027t matter and we\u0027re not going to get,"},{"Start":"05:06.590 ","End":"05:07.910","Text":"as I said, too technical,"},{"Start":"05:07.910 ","End":"05:14.030","Text":"but I just wanted to point out that the cone is not smooth at the tip, but that\u0027s okay."},{"Start":"05:14.030 ","End":"05:19.710","Text":"We have a piecewise almost smooth 2 surfaces together,"},{"Start":"05:19.710 ","End":"05:21.525","Text":"making 1 large surface,"},{"Start":"05:21.525 ","End":"05:27.360","Text":"and we\u0027re going to compute the integral over the whole of S"},{"Start":"05:27.750 ","End":"05:38.190","Text":"of f of x, y, z dS."},{"Start":"05:38.190 ","End":"05:40.210","Text":"Just like we did with line integrals,"},{"Start":"05:40.210 ","End":"05:42.190","Text":"we\u0027re going to break this up into 2 bits."},{"Start":"05:42.190 ","End":"05:45.985","Text":"It\u0027s going to be the integral over S_1,"},{"Start":"05:45.985 ","End":"05:48.775","Text":"the cone part of the same thing,"},{"Start":"05:48.775 ","End":"05:55.500","Text":"plus the integral over the base which is S_2 of the same thing."},{"Start":"05:55.500 ","End":"05:59.805","Text":"We really have 2 in 1 and then we have to at the results."},{"Start":"05:59.805 ","End":"06:07.920","Text":"Let\u0027s start with S_1 and we need a parametrization of S_1,"},{"Start":"06:07.920 ","End":"06:10.305","Text":"which is this part of the cone."},{"Start":"06:10.305 ","End":"06:18.074","Text":"The parameters I\u0027m going to choose will be the height."},{"Start":"06:18.074 ","End":"06:20.955","Text":"I\u0027ll call it z naturally,"},{"Start":"06:20.955 ","End":"06:28.655","Text":"and the angle like in polar or cylindrical coordinates Theta."},{"Start":"06:28.655 ","End":"06:31.510","Text":"Because once I know the height and once I know the angle,"},{"Start":"06:31.510 ","End":"06:34.320","Text":"I know which point on the cone I\u0027m at,"},{"Start":"06:34.320 ","End":"06:36.850","Text":"and this is going to equal."},{"Start":"06:36.850 ","End":"06:44.850","Text":"Well I need a side calculation to tell me what the radius is at every given height."},{"Start":"06:44.850 ","End":"06:46.780","Text":"If you think about it,"},{"Start":"06:46.780 ","End":"06:55.164","Text":"I\u0027m looking at for a straight line function that gives me the radius in terms of z,"},{"Start":"06:55.164 ","End":"06:57.855","Text":"such that when z is 0,"},{"Start":"06:57.855 ","End":"07:01.865","Text":"the radius is 2,"},{"Start":"07:01.865 ","End":"07:05.790","Text":"and when z is 3 which is the height,"},{"Start":"07:05.790 ","End":"07:08.270","Text":"the radius is 0 an,"},{"Start":"07:08.270 ","End":"07:12.340","Text":"d I want to straight line function like this."},{"Start":"07:12.340 ","End":"07:23.110","Text":"An easy computation shows us that r is equal to 2 minus 2/3 z."},{"Start":"07:23.110 ","End":"07:26.104","Text":"Now, once I know this,"},{"Start":"07:26.104 ","End":"07:28.010","Text":"then I can tell what my x,"},{"Start":"07:28.010 ","End":"07:30.450","Text":"y, and z are."},{"Start":"07:31.160 ","End":"07:36.690","Text":"The x is just r cosine Theta,"},{"Start":"07:36.690 ","End":"07:41.250","Text":"so r cosine Theta is this thing"},{"Start":"07:41.250 ","End":"07:46.950","Text":"2 minus 2/3 z cosine Theta."},{"Start":"07:46.950 ","End":"07:49.220","Text":"I just make a note this bit is r,"},{"Start":"07:49.220 ","End":"07:55.330","Text":"and then I need r sine Theta."},{"Start":"07:55.330 ","End":"08:00.795","Text":"So r which is 2 minus 2/3 z,"},{"Start":"08:00.795 ","End":"08:04.170","Text":"again, that\u0027s r sine Theta."},{"Start":"08:04.170 ","End":"08:08.260","Text":"The last component is just z itself."},{"Start":"08:08.510 ","End":"08:14.420","Text":"This is my parametrization for S_1."},{"Start":"08:14.420 ","End":"08:17.810","Text":"I\u0027ll just write down that is S_1."},{"Start":"08:17.810 ","End":"08:22.645","Text":"Now we can use the formula for the integral,"},{"Start":"08:22.645 ","End":"08:31.260","Text":"and what we get is the double integral."},{"Start":"08:33.990 ","End":"08:37.030","Text":"I wrote in the formula to remind you,"},{"Start":"08:37.030 ","End":"08:38.680","Text":"so double integral over D,"},{"Start":"08:38.680 ","End":"08:40.450","Text":"and we haven\u0027t really said what D is yet,"},{"Start":"08:40.450 ","End":"08:45.955","Text":"we\u0027ll get to that, of the function,"},{"Start":"08:45.955 ","End":"08:49.990","Text":"which is the square root of x squared plus y squared."},{"Start":"08:49.990 ","End":"08:52.525","Text":"We take x and y from here,"},{"Start":"08:52.525 ","End":"08:54.790","Text":"this whole thing is x,"},{"Start":"08:54.790 ","End":"08:57.460","Text":"this whole thing is y."},{"Start":"08:57.460 ","End":"09:05.515","Text":"Basically, what we get is r squared cosine squared plus r squared sine squared,"},{"Start":"09:05.515 ","End":"09:07.525","Text":"and that\u0027s just r squared,"},{"Start":"09:07.525 ","End":"09:09.115","Text":"and then when we take the square root,"},{"Start":"09:09.115 ","End":"09:10.351","Text":"we just get r."},{"Start":"09:10.351 ","End":"09:13.075","Text":"Well, the absolute value of r,"},{"Start":"09:13.075 ","End":"09:15.520","Text":"but this is positive."},{"Start":"09:15.520 ","End":"09:26.425","Text":"This whole thing comes out to just 2 minus 2/3 z."},{"Start":"09:26.425 ","End":"09:29.785","Text":"That\u0027s the f bit, that\u0027s this."},{"Start":"09:29.785 ","End":"09:35.200","Text":"Now, we need the cross-product, but before that,"},{"Start":"09:35.200 ","End":"09:37.600","Text":"let me just explain about the region"},{"Start":"09:37.600 ","End":"09:42.810","Text":"D. I borrowed this picture from somewhere and they used the letter R,"},{"Start":"09:42.810 ","End":"09:45.600","Text":"so I\u0027ll just use the letter D instead,"},{"Start":"09:45.600 ","End":"09:50.760","Text":"but we do often use R. Also instead of u and v,"},{"Start":"09:50.760 ","End":"09:59.260","Text":"what I want is z instead of u that goes from 0-3,"},{"Start":"09:59.260 ","End":"10:07.045","Text":"and instead of v, I want Theta."},{"Start":"10:07.045 ","End":"10:12.355","Text":"Our region is actually a rectangle in Theta and z,"},{"Start":"10:12.355 ","End":"10:13.870","Text":"which will make things simpler."},{"Start":"10:13.870 ","End":"10:19.495","Text":"Anyway, back to the formula with the ru cross rv,"},{"Start":"10:19.495 ","End":"10:26.470","Text":"so let\u0027s see, ru,"},{"Start":"10:26.470 ","End":"10:30.910","Text":"partial derivative of r with respect to u is not really u,"},{"Start":"10:30.910 ","End":"10:33.385","Text":"in our case it\u0027s z,"},{"Start":"10:33.385 ","End":"10:38.035","Text":"and the partial derivative with respect to z is just equal to,"},{"Start":"10:38.035 ","End":"10:40.300","Text":"let\u0027s see, Theta is a constant,"},{"Start":"10:40.300 ","End":"10:44.230","Text":"so the derivative of this with respect to z is minus 2/3,"},{"Start":"10:44.230 ","End":"10:49.150","Text":"so we get minus 2/3 cosine Theta."},{"Start":"10:49.150 ","End":"10:51.565","Text":"The next bit, similarly,"},{"Start":"10:51.565 ","End":"10:56.875","Text":"minus 2/3 sine Theta."},{"Start":"10:56.875 ","End":"10:59.290","Text":"This is getting in the way."},{"Start":"10:59.290 ","End":"11:03.580","Text":"The last component derivative of z is just 1."},{"Start":"11:03.580 ","End":"11:08.230","Text":"Now, what about with respect to Theta?"},{"Start":"11:08.230 ","End":"11:10.990","Text":"In this case, z is the constant,"},{"Start":"11:10.990 ","End":"11:14.980","Text":"and the derivative of cosine is minus sine,"},{"Start":"11:14.980 ","End":"11:18.805","Text":"so we get minus"},{"Start":"11:18.805 ","End":"11:26.540","Text":"2 minus 2/3 z sine Theta,"},{"Start":"11:27.030 ","End":"11:34.735","Text":"and then the derivative of sine is cosine,"},{"Start":"11:34.735 ","End":"11:44.095","Text":"but this bit stays 2 minus 2/3 z cosine Theta,"},{"Start":"11:44.095 ","End":"11:49.330","Text":"and this doesn\u0027t have any Theta at all, so that\u0027s 0."},{"Start":"11:49.330 ","End":"11:53.600","Text":"Now, we have to compute the cross product,"},{"Start":"11:53.940 ","End":"11:57.940","Text":"rz cross r Theta,"},{"Start":"11:57.940 ","End":"12:01.120","Text":"and this is equal to,"},{"Start":"12:01.120 ","End":"12:05.185","Text":"as usual, I\u0027m not going to waste valuable time computing,"},{"Start":"12:05.185 ","End":"12:09.175","Text":"hope you know how to compute cross products so I\u0027ll just put the result,"},{"Start":"12:09.175 ","End":"12:11.470","Text":"it\u0027s minus"},{"Start":"12:11.470 ","End":"12:20.335","Text":"2 minus 2/3 z cosine Theta."},{"Start":"12:20.335 ","End":"12:22.810","Text":"Next component,"},{"Start":"12:22.810 ","End":"12:32.185","Text":"minus 2 minus 2/3 z sine Theta,"},{"Start":"12:32.185 ","End":"12:36.760","Text":"and the last component"},{"Start":"12:36.760 ","End":"12:45.190","Text":"is minus 2/3,"},{"Start":"12:45.190 ","End":"12:55.465","Text":"2 minus 2/3 z close brackets."},{"Start":"12:55.465 ","End":"13:01.520","Text":"That\u0027s not all, which is we still have to find the magnitude,"},{"Start":"13:03.870 ","End":"13:08.170","Text":"and so I\u0027m going to do the computation for you again,"},{"Start":"13:08.170 ","End":"13:13.600","Text":"we don\u0027t want to waste valuable time on these computation."},{"Start":"13:13.600 ","End":"13:16.810","Text":"You take this squared, plus this squared, plus this squared,"},{"Start":"13:16.810 ","End":"13:22.195","Text":"and use the fact that cosine squared plus sine squared is equal to 1,"},{"Start":"13:22.195 ","End":"13:30.490","Text":"and what we end up with is the square root of 13/3"},{"Start":"13:30.490 ","End":"13:39.970","Text":"times 2 minus 2/3 of z."},{"Start":"13:39.970 ","End":"13:42.490","Text":"Now, this is this bit here,"},{"Start":"13:42.490 ","End":"13:43.900","Text":"this bit here is this,"},{"Start":"13:43.900 ","End":"13:46.555","Text":"I\u0027m going to copy this to here,"},{"Start":"13:46.555 ","End":"13:52.825","Text":"and then just add a dA on the end."},{"Start":"13:52.825 ","End":"13:58.090","Text":"Very well. I\u0027ll get rid of this formula now."},{"Start":"13:58.090 ","End":"14:00.580","Text":"Now, we can do this double integral."},{"Start":"14:00.580 ","End":"14:06.925","Text":"I\u0027ll write the dA as dz d Theta."},{"Start":"14:06.925 ","End":"14:10.375","Text":"I might as well integrate it with respect to z first,"},{"Start":"14:10.375 ","End":"14:13.825","Text":"so we get the integral."},{"Start":"14:13.825 ","End":"14:16.570","Text":"Now, if I\u0027m doing z first,"},{"Start":"14:16.570 ","End":"14:21.040","Text":"z is going from 0-3,"},{"Start":"14:21.040 ","End":"14:28.760","Text":"and then that would mean that Theta is here 0 to 2 Pi."},{"Start":"14:30.600 ","End":"14:36.505","Text":"I\u0027ve got the square root of 13/3,"},{"Start":"14:36.505 ","End":"14:38.725","Text":"and this and this makes it squared,"},{"Start":"14:38.725 ","End":"14:48.800","Text":"2 minus 2/3 z squared dz d Theta."},{"Start":"14:49.620 ","End":"14:59.900","Text":"Now, the integral of this with respect to z will come out."},{"Start":"15:00.330 ","End":"15:03.280","Text":"I can see that,"},{"Start":"15:03.280 ","End":"15:04.810","Text":"but this is a constant, it stays,"},{"Start":"15:04.810 ","End":"15:06.985","Text":"this thing is something squared,"},{"Start":"15:06.985 ","End":"15:13.720","Text":"so what I can do is raise the power by 1,"},{"Start":"15:13.720 ","End":"15:20.845","Text":"so I\u0027ve got 2 minus 2/3 z to the power of 3."},{"Start":"15:20.845 ","End":"15:24.160","Text":"Then I have to divide by 3,"},{"Start":"15:24.160 ","End":"15:32.979","Text":"so it\u0027s 1/3, but I also have to divide by the inner derivative."},{"Start":"15:32.979 ","End":"15:35.530","Text":"This is the linear expressions, so we can do that."},{"Start":"15:35.530 ","End":"15:39.040","Text":"I have to divide by minus 2/3,"},{"Start":"15:39.040 ","End":"15:47.920","Text":"so I have to multiply by minus 3/2 instead of dividing by minus 2/3."},{"Start":"15:47.920 ","End":"15:54.070","Text":"I still have a root 13/3,"},{"Start":"15:54.070 ","End":"16:03.560","Text":"and all this has to be taken between 0 and 3."},{"Start":"16:04.680 ","End":"16:09.700","Text":"At the end, I still have to take the integral from"},{"Start":"16:09.700 ","End":"16:16.940","Text":"0 to 2 Pi of this whole thing, d Theta."},{"Start":"16:18.270 ","End":"16:22.600","Text":"Here\u0027s what we get after a bit of simplification."},{"Start":"16:22.600 ","End":"16:25.405","Text":"I\u0027m going to do my usual trick."},{"Start":"16:25.405 ","End":"16:30.505","Text":"I\u0027ll get rid of the minus and I\u0027ll switch the order of these 2."},{"Start":"16:30.505 ","End":"16:32.665","Text":"There I did that,"},{"Start":"16:32.665 ","End":"16:38.530","Text":"and now, if I plug-in 0,"},{"Start":"16:38.530 ","End":"16:44.890","Text":"2 cubed is 8 minus,"},{"Start":"16:44.890 ","End":"16:47.980","Text":"when I plug in 3 for z,"},{"Start":"16:47.980 ","End":"16:52.180","Text":"it just comes out to be 0 because 2/3 of 3 is 2,"},{"Start":"16:52.180 ","End":"16:56.245","Text":"2 minus 2 is 0, so it\u0027s 8 minus 0."},{"Start":"16:56.245 ","End":"17:01.135","Text":"It\u0027s just square root of 13"},{"Start":"17:01.135 ","End":"17:07.495","Text":"times 8 over 6 and then,"},{"Start":"17:07.495 ","End":"17:16.660","Text":"I want the integral of this from 0 to 2 Pi d Theta."},{"Start":"17:16.660 ","End":"17:18.625","Text":"Well, this is a constant."},{"Start":"17:18.625 ","End":"17:22.360","Text":"The integral of 1 is just the difference of 2 Pi minus 0,"},{"Start":"17:22.360 ","End":"17:24.385","Text":"which is 2 Pi."},{"Start":"17:24.385 ","End":"17:30.010","Text":"What I get is 2 Pi root"},{"Start":"17:30.010 ","End":"17:36.460","Text":"13 times 8 over 6,"},{"Start":"17:36.460 ","End":"17:45.445","Text":"and I can cancel 2 into 6 goes 3 times,"},{"Start":"17:45.445 ","End":"17:48.370","Text":"and so remembering where we came from,"},{"Start":"17:48.370 ","End":"17:52.915","Text":"which was the double integral along S_1 of the,"},{"Start":"17:52.915 ","End":"17:55.570","Text":"we can write it whatever it was,"},{"Start":"17:55.570 ","End":"17:57.625","Text":"is equal to this,"},{"Start":"17:57.625 ","End":"18:08.035","Text":"which is 8 over 3 square root of 13 Pi."},{"Start":"18:08.035 ","End":"18:12.175","Text":"Let me highlight this first result."},{"Start":"18:12.175 ","End":"18:16.310","Text":"Then we need the second one and then we\u0027ll add."},{"Start":"18:17.220 ","End":"18:21.325","Text":"Now we\u0027re gonna move on to the next bit, S_2."},{"Start":"18:21.325 ","End":"18:24.760","Text":"Let me just clear the board"},{"Start":"18:24.760 ","End":"18:29.540","Text":"except that I will keep this result and I\u0027ll put it over here."},{"Start":"18:30.150 ","End":"18:34.540","Text":"Now we want to parameterize S_2."},{"Start":"18:34.540 ","End":"18:41.560","Text":"S_2 is a circle of radius 2 in the x,"},{"Start":"18:41.560 ","End":"18:46.840","Text":"y plane, so easiest to use polar coordinates."},{"Start":"18:46.840 ","End":"18:54.220","Text":"For S_2, we can get that r of r and"},{"Start":"18:54.220 ","End":"18:59.635","Text":"Theta is equal to"},{"Start":"18:59.635 ","End":"19:06.340","Text":"r cosine Theta and r sine Theta for x and y."},{"Start":"19:06.340 ","End":"19:12.560","Text":"But z here will be constantly 0."},{"Start":"19:15.150 ","End":"19:25.105","Text":"What we have is that r goes from 0 up to 2,"},{"Start":"19:25.105 ","End":"19:29.380","Text":"and Theta goes from 0."},{"Start":"19:29.380 ","End":"19:32.140","Text":"The full circle is 2 Pi."},{"Start":"19:32.140 ","End":"19:38.290","Text":"It\u0027s like here we had a rectangular region in z and Theta."},{"Start":"19:38.290 ","End":"19:43.570","Text":"Here we have a rectangular region in r and Theta."},{"Start":"19:43.570 ","End":"19:46.015","Text":"Well, I won\u0027t sketch it this time,"},{"Start":"19:46.015 ","End":"19:49.690","Text":"the domain here, let\u0027s call this one D_1."},{"Start":"19:49.690 ","End":"19:56.150","Text":"This domain here, or region is going to be D_2."},{"Start":"19:56.370 ","End":"20:01.940","Text":"Once again, we have to compute the partial derivatives."},{"Start":"20:03.630 ","End":"20:07.270","Text":"I don\u0027t like using this twice, you know what?"},{"Start":"20:07.270 ","End":"20:11.290","Text":"I\u0027ll change it to Rho and Theta now."},{"Start":"20:11.290 ","End":"20:16.990","Text":"With respect to Rho,"},{"Start":"20:16.990 ","End":"20:22.840","Text":"partial derivative is just cosine Theta,"},{"Start":"20:22.840 ","End":"20:29.515","Text":"here sine Theta, and here still 0."},{"Start":"20:29.515 ","End":"20:35.360","Text":"The partial derivative with respect to Theta,"},{"Start":"20:36.240 ","End":"20:39.760","Text":"the cosine gives minus sign,"},{"Start":"20:39.760 ","End":"20:45.030","Text":"minus Rho sine Theta and"},{"Start":"20:45.030 ","End":"20:56.950","Text":"then Rho cosine Theta, and again 0."},{"Start":"20:56.950 ","End":"21:04.870","Text":"If we compute the cross product r Rho cross r Theta,"},{"Start":"21:04.870 ","End":"21:13.100","Text":"I did it at the side and it comes out to be 0, 0, Rho."},{"Start":"21:14.030 ","End":"21:20.005","Text":"I better bring in the formula again, and here it is."},{"Start":"21:20.005 ","End":"21:23.410","Text":"What we get is the double integral,"},{"Start":"21:23.410 ","End":"21:28.370","Text":"this time over the other region D_2."},{"Start":"21:29.340 ","End":"21:36.685","Text":"The f is just square root of x squared plus y squared."},{"Start":"21:36.685 ","End":"21:38.740","Text":"If we do that from here,"},{"Start":"21:38.740 ","End":"21:43.120","Text":"the square root of this squared plus this squared is just Rho,"},{"Start":"21:43.120 ","End":"21:48.205","Text":"because sine squared plus cosine squared is 1 and rho is non-negative."},{"Start":"21:48.205 ","End":"21:53.800","Text":"Then we need the magnitude of this,"},{"Start":"21:53.800 ","End":"21:57.250","Text":"well, the magnitude of this is just Rho."},{"Start":"21:57.250 ","End":"22:00.830","Text":"Then we need dA,"},{"Start":"22:01.620 ","End":"22:08.380","Text":"and so what we get is double integral of"},{"Start":"22:08.380 ","End":"22:14.770","Text":"Rho squared and dA could be d Rho d Theta or d Theta d Rho."},{"Start":"22:14.770 ","End":"22:16.240","Text":"Let\u0027s do it first of all,"},{"Start":"22:16.240 ","End":"22:19.644","Text":"d Rho and then d Theta,"},{"Start":"22:19.644 ","End":"22:25.659","Text":"so this has to be Rho between 0 and 2,"},{"Start":"22:25.659 ","End":"22:30.890","Text":"Theta still between 0 and 2 Pi."},{"Start":"22:31.740 ","End":"22:36.160","Text":"The integral of Rho squared is 1/3,"},{"Start":"22:36.160 ","End":"22:38.635","Text":"which I can bring out to the front."},{"Start":"22:38.635 ","End":"22:44.155","Text":"1/3, the integral from 0 to 2 Pi of"},{"Start":"22:44.155 ","End":"22:51.415","Text":"Rho cubed between 0 and 2 d Theta."},{"Start":"22:51.415 ","End":"22:57.415","Text":"This is just 8 because 2 cubed minus 0 cubed is 8."},{"Start":"22:57.415 ","End":"23:02.590","Text":"What we get is 8/3,"},{"Start":"23:02.590 ","End":"23:06.640","Text":"and then we\u0027re just left with the integral of 1 from 0 to 2 Pi,"},{"Start":"23:06.640 ","End":"23:09.890","Text":"so that\u0027s 2 Pi."},{"Start":"23:11.070 ","End":"23:22.690","Text":"Altogether we get, so it\u0027s 16 over 3 Pi,"},{"Start":"23:22.690 ","End":"23:24.940","Text":"and I\u0027ll highlight it."},{"Start":"23:24.940 ","End":"23:28.915","Text":"Now that\u0027s S_2."},{"Start":"23:28.915 ","End":"23:35.155","Text":"Altogether, what I get is this plus from here,"},{"Start":"23:35.155 ","End":"23:40.870","Text":"16 over 3 Pi."},{"Start":"23:40.870 ","End":"23:42.475","Text":"I\u0027ll write it over here."},{"Start":"23:42.475 ","End":"23:49.585","Text":"I can take 8 Pi over 3 outside the brackets."},{"Start":"23:49.585 ","End":"23:58.010","Text":"What I\u0027m left with here is square root of 13 plus 2."},{"Start":"23:58.560 ","End":"24:06.950","Text":"This is the final answer to the question, and we\u0027re done."}],"ID":9648},{"Watched":false,"Name":"Orientation of Surfaces","Duration":"10m 37s","ChapterTopicVideoID":8785,"CourseChapterTopicPlaylistID":4974,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.240","Text":"In this clip, I\u0027ll be talking about orientation of surfaces,"},{"Start":"00:06.240 ","End":"00:12.360","Text":"and this is something in preparation for surface integrals over vector fields."},{"Start":"00:12.360 ","End":"00:14.565","Text":"What is orientation?"},{"Start":"00:14.565 ","End":"00:18.930","Text":"Well, let\u0027s consider the following picture."},{"Start":"00:18.930 ","End":"00:22.200","Text":"We have here part of a surface,"},{"Start":"00:22.200 ","End":"00:25.210","Text":"and at this point,"},{"Start":"00:25.280 ","End":"00:28.680","Text":"there are 2 normals drawn,"},{"Start":"00:28.680 ","End":"00:32.160","Text":"1 facing upwards, in this case,"},{"Start":"00:32.160 ","End":"00:33.815","Text":"and 1 going downwards."},{"Start":"00:33.815 ","End":"00:39.515","Text":"In general, there are 2 directions that a normal could have."},{"Start":"00:39.515 ","End":"00:44.910","Text":"Now, we often typically restrict a normal to being a unit vector,"},{"Start":"00:44.910 ","End":"00:46.280","Text":"and if we restrict that,"},{"Start":"00:46.280 ","End":"00:50.315","Text":"then there will be exactly 2 normals at every point."},{"Start":"00:50.315 ","End":"00:52.640","Text":"1 being the negative of the other."},{"Start":"00:52.640 ","End":"01:00.540","Text":"In fact, you could say that n_2 the vector is minus n_1 the vector."},{"Start":"01:00.540 ","End":"01:02.360","Text":"This is typically the case,"},{"Start":"01:02.360 ","End":"01:04.190","Text":"and that\u0027s for a single point."},{"Start":"01:04.190 ","End":"01:11.550","Text":"Now, for orientation, I want to assign a normal at every point along the surface,"},{"Start":"01:11.570 ","End":"01:13.860","Text":"make that a unit normal."},{"Start":"01:13.860 ","End":"01:16.130","Text":"A unit normal at every point,"},{"Start":"01:16.130 ","End":"01:17.780","Text":"but in a continuous way."},{"Start":"01:17.780 ","End":"01:20.660","Text":"I don\u0027t want the normal to go 1 way here and right"},{"Start":"01:20.660 ","End":"01:23.600","Text":"alongside it to go down and up in a smooth way,"},{"Start":"01:23.600 ","End":"01:29.650","Text":"and I\u0027ll give you a picture again."},{"Start":"01:29.990 ","End":"01:35.480","Text":"Here\u0027s a picture of essentially the same surface,"},{"Start":"01:35.480 ","End":"01:38.840","Text":"but there\u0027s 2 ways of assigning normals."},{"Start":"01:38.840 ","End":"01:44.630","Text":"I could have all the normals facing this way and all of them facing the other way,"},{"Start":"01:44.630 ","End":"01:47.630","Text":"but I can mix and match."},{"Start":"01:47.630 ","End":"01:49.340","Text":"It has to be continuous or smooth,"},{"Start":"01:49.340 ","End":"01:51.755","Text":"I can\u0027t suddenly changed direction,"},{"Start":"01:51.755 ","End":"01:58.665","Text":"and each of these assignment of unit normals is an orientation."},{"Start":"01:58.665 ","End":"02:02.505","Text":"In general, a surface has 2 orientations."},{"Start":"02:02.505 ","End":"02:06.935","Text":"I\u0027m assuming that the surface has 2 sides,"},{"Start":"02:06.935 ","End":"02:09.050","Text":"you might think that\u0027s a strange thing to say."},{"Start":"02:09.050 ","End":"02:11.270","Text":"Of course, the surface has 2 sides."},{"Start":"02:11.270 ","End":"02:15.350","Text":"But you probably all heard of something called the Mobius strip."},{"Start":"02:15.350 ","End":"02:23.575","Text":"What it looks like is like a strip of paper that\u0027s being joined with a 1/2 twist."},{"Start":"02:23.575 ","End":"02:30.454","Text":"It\u0027s very famous and it\u0027s called the Mobius strip,"},{"Start":"02:30.454 ","End":"02:33.520","Text":"named after someone called Mobius."},{"Start":"02:33.520 ","End":"02:39.110","Text":"Sometimes written with an oe here instead of the o with the umlaut,"},{"Start":"02:39.630 ","End":"02:42.715","Text":"and it only has 1 side."},{"Start":"02:42.715 ","End":"02:47.000","Text":"You can\u0027t put normal at this point,"},{"Start":"02:47.000 ","End":"02:49.010","Text":"and then at this point, this point, this point,"},{"Start":"02:49.010 ","End":"02:51.395","Text":"you keep going round because of the twist,"},{"Start":"02:51.395 ","End":"02:56.870","Text":"you end up back here with the normal going in the other direction."},{"Start":"02:56.870 ","End":"03:00.440","Text":"We won\u0027t be dealing with pathologies like this,"},{"Start":"03:00.440 ","End":"03:03.840","Text":"our surfaces will have 2 sides."},{"Start":"03:04.430 ","End":"03:10.490","Text":"When we do the surface integrals over vector fields,"},{"Start":"03:10.490 ","End":"03:12.410","Text":"we won\u0027t just be given a surface,"},{"Start":"03:12.410 ","End":"03:15.650","Text":"we\u0027ll be given a surface with a specific orientation."},{"Start":"03:15.650 ","End":"03:18.365","Text":"There\u0027s just going to be 2 orientations."},{"Start":"03:18.365 ","End":"03:23.090","Text":"We\u0027ll say the surface with maybe arrows facing upwards,"},{"Start":"03:23.090 ","End":"03:27.810","Text":"or if it\u0027s a closed surface and I\u0027ll get to that in a moment,"},{"Start":"03:27.810 ","End":"03:29.810","Text":"we\u0027ll say facing outwards or inwards."},{"Start":"03:29.810 ","End":"03:34.800","Text":"In fact, let me get to that concept of closed surface."},{"Start":"03:35.080 ","End":"03:43.670","Text":"Loosely speaking, a closed surface like a sphere wraps around a solid body in 3D."},{"Start":"03:43.670 ","End":"03:46.220","Text":"I\u0027m not going to get into a very formal definition,"},{"Start":"03:46.220 ","End":"03:48.680","Text":"but you can just think of it closed,"},{"Start":"03:48.680 ","End":"03:52.235","Text":"it wraps around something."},{"Start":"03:52.235 ","End":"03:54.390","Text":"It\u0027s like a shell."},{"Start":"03:55.000 ","End":"04:02.105","Text":"With closed surfaces there\u0027s a convention that we prefer 1 orientation over the other."},{"Start":"04:02.105 ","End":"04:07.845","Text":"1 obvious orientation is all the unit normals facing outward,"},{"Start":"04:07.845 ","End":"04:11.885","Text":"and the other orientation is to have them all facing inwards."},{"Start":"04:11.885 ","End":"04:14.780","Text":"We say that the facing outward is the positive and the"},{"Start":"04:14.780 ","End":"04:18.480","Text":"facing inward is the negative orientation,"},{"Start":"04:18.710 ","End":"04:22.090","Text":"that\u0027s just a name."},{"Start":"04:22.090 ","End":"04:24.200","Text":"But in any given case,"},{"Start":"04:24.200 ","End":"04:26.450","Text":"when we have a surface with an orientation,"},{"Start":"04:26.450 ","End":"04:30.200","Text":"if it\u0027s not the 1 that we want and the equations give us 1,"},{"Start":"04:30.200 ","End":"04:35.465","Text":"we just can put a minus in front of everything and get the opposite orientation,"},{"Start":"04:35.465 ","End":"04:38.405","Text":"so we can always go back and forth."},{"Start":"04:38.405 ","End":"04:41.720","Text":"There was another situation in which we have"},{"Start":"04:41.720 ","End":"04:46.200","Text":"a preferred positive orientation versus negative,"},{"Start":"04:46.200 ","End":"04:53.300","Text":"and that is when we have a surface that is given by z in terms of x and y,"},{"Start":"04:53.300 ","End":"04:56.215","Text":"or 1 variable in terms of the other."},{"Start":"04:56.215 ","End":"05:00.310","Text":"Here\u0027s the picture, you\u0027ll have to scroll."},{"Start":"05:00.730 ","End":"05:06.560","Text":"Now, when we have a surface that\u0027s given as a function,"},{"Start":"05:06.560 ","End":"05:12.135","Text":"say z equals g of x and y,"},{"Start":"05:12.135 ","End":"05:18.290","Text":"but similar result will hold if any 1 is a function of the other 2,"},{"Start":"05:18.290 ","End":"05:22.205","Text":"I\u0027m just taking z as an example of 1 of 3."},{"Start":"05:22.205 ","End":"05:26.880","Text":"Then let\u0027s say we have a point, I don\u0027t know here,"},{"Start":"05:26.880 ","End":"05:30.135","Text":"then there will be 2 normals,"},{"Start":"05:30.135 ","End":"05:36.285","Text":"1 going this way and 1 going the other way."},{"Start":"05:36.285 ","End":"05:42.240","Text":"The 1 that has an upward component, this 1,"},{"Start":"05:42.240 ","End":"05:48.350","Text":"there will always be 1 that\u0027s above the horizontal plane through this point,"},{"Start":"05:48.350 ","End":"05:54.200","Text":"this will be the positive orientation,"},{"Start":"05:54.200 ","End":"05:57.800","Text":"and this will be the negative orientation."},{"Start":"05:57.800 ","End":"06:00.380","Text":"These things can be made more precise,"},{"Start":"06:00.380 ","End":"06:04.110","Text":"I\u0027m just giving you a general overview."},{"Start":"06:04.780 ","End":"06:07.310","Text":"I copied something from"},{"Start":"06:07.310 ","End":"06:11.255","Text":"a previous clip here and I\u0027ll be getting back to this in a moment."},{"Start":"06:11.255 ","End":"06:17.790","Text":"This was a parametric surface where we had r,"},{"Start":"06:17.790 ","End":"06:21.315","Text":"the surface given in terms of 2 parameters, u and v,"},{"Start":"06:21.315 ","End":"06:24.260","Text":"and I mentioned at that time that if you"},{"Start":"06:24.260 ","End":"06:27.785","Text":"take the cross-product of the 2 partial derivatives,"},{"Start":"06:27.785 ","End":"06:34.070","Text":"we get something that is a normal vector to the surface."},{"Start":"06:34.070 ","End":"06:36.050","Text":"It may not be a unit normal,"},{"Start":"06:36.050 ","End":"06:41.080","Text":"but it\u0027s certainly orthogonal or perpendicular to the surface."},{"Start":"06:41.080 ","End":"06:44.670","Text":"If we want a unit normal,"},{"Start":"06:44.670 ","End":"06:47.285","Text":"so we can always get a unit"},{"Start":"06:47.285 ","End":"06:52.260","Text":"normal vector by just"},{"Start":"06:52.260 ","End":"06:57.500","Text":"dividing any vector by its magnitude gives us a unit vector,"},{"Start":"06:57.500 ","End":"06:59.555","Text":"so I would take this,"},{"Start":"06:59.555 ","End":"07:01.250","Text":"and I\u0027ll just call it for short,"},{"Start":"07:01.250 ","End":"07:11.025","Text":"ru cross rv and divide it by its magnitude of the same thing,"},{"Start":"07:11.025 ","End":"07:16.390","Text":"of ru cross rv."},{"Start":"07:16.520 ","End":"07:20.390","Text":"This is in general how we could get a unit normal."},{"Start":"07:20.390 ","End":"07:23.195","Text":"If we just want a normal, then we can just take the cross product."},{"Start":"07:23.195 ","End":"07:31.825","Text":"Now, what I was going to say is about this surface where z is a function of x and y."},{"Start":"07:31.825 ","End":"07:37.800","Text":"Again, I copied something from a previous clip."},{"Start":"07:38.230 ","End":"07:43.520","Text":"We can always think of this z equals g of x,"},{"Start":"07:43.520 ","End":"07:48.260","Text":"y as parametric because all I have to do is use x,"},{"Start":"07:48.260 ","End":"07:52.490","Text":"y as parameters instead of letters u and v,"},{"Start":"07:52.490 ","End":"07:54.830","Text":"and I\u0027ve got r in terms of x,"},{"Start":"07:54.830 ","End":"07:56.720","Text":"y is just x, y,"},{"Start":"07:56.720 ","End":"07:59.995","Text":"as it is, and z is g of x, y."},{"Start":"07:59.995 ","End":"08:03.080","Text":"Now if I take the 2 partial derivatives,"},{"Start":"08:03.080 ","End":"08:05.220","Text":"this is what I get with respect to x,"},{"Start":"08:05.220 ","End":"08:07.475","Text":"this is what I get with respect to y."},{"Start":"08:07.475 ","End":"08:09.080","Text":"On the cross product,"},{"Start":"08:09.080 ","End":"08:14.100","Text":"the result of the computation is what\u0027s written here."},{"Start":"08:14.100 ","End":"08:21.235","Text":"Notice that there\u0027s a positive z component."},{"Start":"08:21.235 ","End":"08:26.675","Text":"It turns out that in the case of this parametrization,"},{"Start":"08:26.675 ","End":"08:35.620","Text":"this cross product always gives the positive orientation."},{"Start":"08:37.910 ","End":"08:45.110","Text":"We also computed the magnitude of this, which was this."},{"Start":"08:45.110 ","End":"08:54.110","Text":"In fact, if we want unit normal is what I meant to say,"},{"Start":"08:54.110 ","End":"08:57.490","Text":"we can always take the unit normal as"},{"Start":"08:57.490 ","End":"09:04.880","Text":"this gx with a minus,"},{"Start":"09:04.880 ","End":"09:11.045","Text":"minus gy, 1 over"},{"Start":"09:11.045 ","End":"09:21.270","Text":"the square root of gx squared plus gy squared plus 1."},{"Start":"09:21.270 ","End":"09:24.525","Text":"The denominator is positive,"},{"Start":"09:24.525 ","End":"09:26.890","Text":"so even after division,"},{"Start":"09:26.890 ","End":"09:31.269","Text":"this last component is going to be bigger than 0,"},{"Start":"09:31.269 ","End":"09:35.590","Text":"so this unit normal is going to have an upward component,"},{"Start":"09:35.590 ","End":"09:41.000","Text":"or the 1 that faces somewhat upwards is the positive 1."},{"Start":"09:41.220 ","End":"09:47.835","Text":"That\u0027s just a general introduction to"},{"Start":"09:47.835 ","End":"09:56.000","Text":"the 2 orientations and that we won\u0027t be dealing with crazy cases like the Mobius strip."},{"Start":"09:56.000 ","End":"09:58.080","Text":"Surfaces will have 2 sides,"},{"Start":"09:58.080 ","End":"10:00.005","Text":"and if it\u0027s a closed surface,"},{"Start":"10:00.005 ","End":"10:05.030","Text":"then the outward is positive and the inward is negative."},{"Start":"10:05.030 ","End":"10:09.020","Text":"If it\u0027s 1 variable in terms of the other then the upward"},{"Start":"10:09.020 ","End":"10:13.385","Text":"is positive and the downward 1 is negative."},{"Start":"10:13.385 ","End":"10:17.030","Text":"Then we also have the formula that the cross-product of"},{"Start":"10:17.030 ","End":"10:22.935","Text":"the 2 partial derivatives in the parametric case is a normal."},{"Start":"10:22.935 ","End":"10:25.010","Text":"If we want unit normals,"},{"Start":"10:25.010 ","End":"10:26.795","Text":"we can just divide by the magnitude."},{"Start":"10:26.795 ","End":"10:28.970","Text":"That\u0027s a summary so far."},{"Start":"10:28.970 ","End":"10:38.070","Text":"We\u0027ll take a break now and then I\u0027ll talk about surface integrals over vector fields."}],"ID":9649},{"Watched":false,"Name":"Surface Integrals over Vector Fields","Duration":"8m 55s","ChapterTopicVideoID":8786,"CourseChapterTopicPlaylistID":4974,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.400","Text":"Now it\u0027s time to discuss surface integrals over vector fields."},{"Start":"00:05.400 ","End":"00:10.900","Text":"In this case, we\u0027re going to talk about oriented surfaces."},{"Start":"00:11.090 ","End":"00:14.660","Text":"Actually may have forgotten to mention the term oriented surface."},{"Start":"00:14.660 ","End":"00:17.835","Text":"Let me just remind you when we talked about orientation,"},{"Start":"00:17.835 ","End":"00:20.070","Text":"and this is from the previous clip,"},{"Start":"00:20.070 ","End":"00:23.070","Text":"we said that the surface generally has two sides,"},{"Start":"00:23.070 ","End":"00:26.490","Text":"although there are crazy exceptions like the Mobius strip."},{"Start":"00:26.490 ","End":"00:32.240","Text":"As such, at any given point there will be 2 unit normal vectors,"},{"Start":"00:32.240 ","End":"00:36.620","Text":"1 facing each way 1 is the negative of the other."},{"Start":"00:36.620 ","End":"00:38.690","Text":"When we have a surface,"},{"Start":"00:38.690 ","End":"00:42.575","Text":"there\u0027s really two possible choices of orientations,"},{"Start":"00:42.575 ","End":"00:45.060","Text":"because we have to choose the unit"},{"Start":"00:45.060 ","End":"00:48.830","Text":"normal vector at each point but it has to be done in a continuous way."},{"Start":"00:48.830 ","End":"00:52.715","Text":"They\u0027re either all pointing this way or they\u0027re all pointing the opposite way."},{"Start":"00:52.715 ","End":"00:59.310","Text":"A surface with a choice of orientation is called an oriented surface."},{"Start":"00:59.310 ","End":"01:02.960","Text":"A given surface, if it\u0027s well-behaved,"},{"Start":"01:02.960 ","End":"01:08.450","Text":"will give rise to two possible oriented surfaces and in any given problem,"},{"Start":"01:08.450 ","End":"01:12.095","Text":"it will be clear which is the orientation that we want."},{"Start":"01:12.095 ","End":"01:14.690","Text":"When we take some of the formulas we may have"},{"Start":"01:14.690 ","End":"01:19.685","Text":"to throw in a minus if it\u0027s not the one that we want."},{"Start":"01:19.685 ","End":"01:25.430","Text":"First, I\u0027ll show you how we write the surface integral over an oriented surface."},{"Start":"01:25.430 ","End":"01:32.850","Text":"Again, we have a surface S and we have a vector field this time."},{"Start":"01:33.050 ","End":"01:36.770","Text":"If you\u0027ve forgotten what a vector field is in 3D,"},{"Start":"01:36.770 ","End":"01:39.020","Text":"then go and review the material."},{"Start":"01:39.020 ","End":"01:47.390","Text":"We write it as Fds and a dot here."},{"Start":"01:47.390 ","End":"01:51.530","Text":"It\u0027s a dot-product of F with something called ds,"},{"Start":"01:51.530 ","End":"01:54.830","Text":"and this is vector ds."},{"Start":"01:54.830 ","End":"01:57.530","Text":"Now I\u0027m going to define it."},{"Start":"01:57.530 ","End":"01:59.815","Text":"This is just a notation."},{"Start":"01:59.815 ","End":"02:03.560","Text":"S is a surface with orientation,"},{"Start":"02:03.560 ","End":"02:06.470","Text":"which means what we have normal unit vectors."},{"Start":"02:06.470 ","End":"02:13.715","Text":"The definition is the regular surface integral of a scalar function,"},{"Start":"02:13.715 ","End":"02:22.685","Text":"which is F.n and this is the regular ds like we previously learned."},{"Start":"02:22.685 ","End":"02:23.960","Text":"Notice this is a vector,"},{"Start":"02:23.960 ","End":"02:25.085","Text":"this is a vector,"},{"Start":"02:25.085 ","End":"02:27.740","Text":"the dot-product is a scalar."},{"Start":"02:27.740 ","End":"02:34.325","Text":"This is the definition and I\u0027ve highlighted it there."},{"Start":"02:34.325 ","End":"02:36.755","Text":"Now I\u0027ll explain it."},{"Start":"02:36.755 ","End":"02:41.540","Text":"Basically we\u0027ll be dealing with two main kinds of surfaces."},{"Start":"02:41.540 ","End":"02:44.735","Text":"One is parametric surface where we have,"},{"Start":"02:44.735 ","End":"02:50.045","Text":"let\u0027s say r(u, v)."},{"Start":"02:50.045 ","End":"02:57.335","Text":"We were given what this equals in as a function of u and v is xuv, yuv zuv."},{"Start":"02:57.335 ","End":"03:05.555","Text":"The other possibility is that will be given 1 variable as a function of the other two,"},{"Start":"03:05.555 ","End":"03:08.285","Text":"z equals g(x and y)."},{"Start":"03:08.285 ","End":"03:10.880","Text":"Although there\u0027s really 3 possibilities, why I just write them."},{"Start":"03:10.880 ","End":"03:14.745","Text":"I could have y equals g(x, z),"},{"Start":"03:14.745 ","End":"03:22.730","Text":"and I could also have x equals some function of y and z."},{"Start":"03:22.730 ","End":"03:29.840","Text":"Actually each of these has two possibilities for orientation."},{"Start":"03:29.840 ","End":"03:32.960","Text":"One way of finding a normal vector for"},{"Start":"03:32.960 ","End":"03:37.010","Text":"this one I want you to remember this is normal vector would"},{"Start":"03:37.010 ","End":"03:44.610","Text":"be ru cross v. That\u0027s a normal vector."},{"Start":"03:44.610 ","End":"03:47.270","Text":"Of course, to get anything to be a unit vector,"},{"Start":"03:47.270 ","End":"03:52.370","Text":"we would divide it by its magnitude of the same thing,"},{"Start":"03:52.370 ","End":"03:59.915","Text":"ru cross v. Basically if you just remember that ru cross rv is a normal vector,"},{"Start":"03:59.915 ","End":"04:06.080","Text":"and then we would take our normal vector n to be this."},{"Start":"04:06.080 ","End":"04:09.080","Text":"But it could be the plus or minus."},{"Start":"04:09.080 ","End":"04:12.170","Text":"You have to look at what the question states."},{"Start":"04:12.170 ","End":"04:18.980","Text":"The were formulas for here also but I\u0027m not expecting you to remember them."},{"Start":"04:18.980 ","End":"04:24.530","Text":"I\u0027ll just give you an example for the first case where z was a function of x and y,"},{"Start":"04:24.530 ","End":"04:30.650","Text":"we got a normal vector as g with respect to x,"},{"Start":"04:30.650 ","End":"04:33.635","Text":"g with respect to y,"},{"Start":"04:33.635 ","End":"04:37.895","Text":"was it minus, it was minus, and then 1."},{"Start":"04:37.895 ","End":"04:40.730","Text":"Of course again, this could be plus or minus."},{"Start":"04:40.730 ","End":"04:44.150","Text":"But I don\u0027t expect you to remember this."},{"Start":"04:44.150 ","End":"04:46.730","Text":"I\u0027ll show you an easy way of deriving it."},{"Start":"04:46.730 ","End":"04:54.170","Text":"Then we would want a unit normal vector we would divide it by the norm of same thing."},{"Start":"04:54.170 ","End":"04:55.760","Text":"This is typically what we do."},{"Start":"04:55.760 ","End":"04:57.275","Text":"We get a normal vector,"},{"Start":"04:57.275 ","End":"05:01.150","Text":"we divide it by its magnitude we get a unit normal vector."},{"Start":"05:01.150 ","End":"05:04.685","Text":"But I want to even emphasize that in any given case,"},{"Start":"05:04.685 ","End":"05:06.890","Text":"we\u0027re going to decide to keep it as is,"},{"Start":"05:06.890 ","End":"05:11.830","Text":"or possibly reverse the sign according to the way the question is phrased."},{"Start":"05:11.830 ","End":"05:16.820","Text":"Just forgot to write that this is a unit normal vector."},{"Start":"05:16.820 ","End":"05:22.595","Text":"But as I said, don\u0027t remember this."},{"Start":"05:22.595 ","End":"05:25.520","Text":"The only thing I\u0027d expect you to remember is this bit"},{"Start":"05:25.520 ","End":"05:28.280","Text":"here that when we have a parametric surface,"},{"Start":"05:28.280 ","End":"05:32.030","Text":"that the cross-product of the two partial derivatives"},{"Start":"05:32.030 ","End":"05:35.780","Text":"is normal and then dividing it by its magnitude is the usual thing,"},{"Start":"05:35.780 ","End":"05:37.880","Text":"and then we also have to remember the plus or minus."},{"Start":"05:37.880 ","End":"05:42.775","Text":"As for this, I want to show you how to derive it you may not have it on a formula sheet."},{"Start":"05:42.775 ","End":"05:45.710","Text":"I\u0027m going to take the first case and in one of"},{"Start":"05:45.710 ","End":"05:49.475","Text":"the examples we\u0027ll see another one of these two other cases."},{"Start":"05:49.475 ","End":"05:52.685","Text":"If I have z as a function of x and y,"},{"Start":"05:52.685 ","End":"05:57.050","Text":"I could always define a function f(x,"},{"Start":"05:57.050 ","End":"06:05.225","Text":"y,) and z=z minus g(x, y,)."},{"Start":"06:05.225 ","End":"06:08.465","Text":"Now I have a function of 3 variables."},{"Start":"06:08.465 ","End":"06:11.630","Text":"If I let this equal 0,"},{"Start":"06:11.630 ","End":"06:15.290","Text":"then on the one-hand, this part here,"},{"Start":"06:15.290 ","End":"06:18.770","Text":"this equality is the same as this function here."},{"Start":"06:18.770 ","End":"06:19.820","Text":"On the other hand,"},{"Start":"06:19.820 ","End":"06:21.635","Text":"if I looked at the first and the last,"},{"Start":"06:21.635 ","End":"06:26.540","Text":"we see that it\u0027s also a level surface of the function"},{"Start":"06:26.540 ","End":"06:31.520","Text":"f. I hope you remember what level surfaces are,"},{"Start":"06:31.520 ","End":"06:37.710","Text":"they\u0027re just the 3D analogy of level curves."},{"Start":"06:38.620 ","End":"06:41.070","Text":"I\u0027m presuming you\u0027ve covered this."},{"Start":"06:41.070 ","End":"06:44.030","Text":"In any event, let me state the result."},{"Start":"06:44.030 ","End":"06:48.890","Text":"That is that the gradient of a function is perpendicular"},{"Start":"06:48.890 ","End":"06:53.975","Text":"to level curves in 2D or level surfaces in 3D,"},{"Start":"06:53.975 ","End":"06:59.290","Text":"and so in our case,"},{"Start":"07:07.270 ","End":"07:09.800","Text":"well instead of the word perpendicular,"},{"Start":"07:09.800 ","End":"07:13.055","Text":"we can say orthogonal or normal."},{"Start":"07:13.055 ","End":"07:21.960","Text":"Grad f is a normal vector to the surface."},{"Start":"07:22.960 ","End":"07:26.795","Text":"Let\u0027s spell it out. What is grad f?"},{"Start":"07:26.795 ","End":"07:31.140","Text":"Well, for this function, grad f,"},{"Start":"07:31.360 ","End":"07:35.990","Text":"in general, it\u0027s the derivative with respect to x,"},{"Start":"07:35.990 ","End":"07:37.925","Text":"derivative with respect to y,"},{"Start":"07:37.925 ","End":"07:41.150","Text":"derivative with respect to z."},{"Start":"07:41.150 ","End":"07:43.460","Text":"This is equal to, well,"},{"Start":"07:43.460 ","End":"07:46.250","Text":"let\u0027s see that with respect to x,"},{"Start":"07:46.250 ","End":"07:50.905","Text":"we have minus g with respect to x."},{"Start":"07:50.905 ","End":"07:54.915","Text":"With respect to y there\u0027s no y here, so only from here."},{"Start":"07:54.915 ","End":"07:59.120","Text":"It\u0027s minus g with respect to y and with respect to z,"},{"Start":"07:59.120 ","End":"08:02.610","Text":"this is a constant and the derivative of this is 1."},{"Start":"08:03.050 ","End":"08:07.820","Text":"We have this, we know that this isn\u0027t normal then, as usual,"},{"Start":"08:07.820 ","End":"08:12.305","Text":"to get a unit normally just divide by the magnitude of the same thing."},{"Start":"08:12.305 ","End":"08:15.460","Text":"You don\u0027t have to remember this bit,"},{"Start":"08:15.460 ","End":"08:19.050","Text":"you can always re-derive it and similarly for"},{"Start":"08:19.050 ","End":"08:23.869","Text":"the other cases I would write a function y minus g(x,"},{"Start":"08:23.869 ","End":"08:27.170","Text":"z), or x minus g(y, z)."},{"Start":"08:27.170 ","End":"08:28.655","Text":"You can always derive it,"},{"Start":"08:28.655 ","End":"08:32.105","Text":"but if you prefer just to remember it, That\u0027s okay too."},{"Start":"08:32.105 ","End":"08:40.565","Text":"We also tied in the concept of gradient to the concept of normal to a surface."},{"Start":"08:40.565 ","End":"08:42.740","Text":"We got to the same result earlier,"},{"Start":"08:42.740 ","End":"08:47.390","Text":"but you can get to this vector using the gradient more quickly."},{"Start":"08:47.390 ","End":"08:50.450","Text":"I think it\u0027s time to take a break."},{"Start":"08:50.450 ","End":"08:55.050","Text":"After the break, we\u0027ll do some examples."}],"ID":9650},{"Watched":false,"Name":"Surface Integrals over Vector Fields - Example","Duration":"23m 32s","ChapterTopicVideoID":9769,"CourseChapterTopicPlaylistID":4974,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.290 ","End":"00:02.520","Text":"Back from the break."},{"Start":"00:02.520 ","End":"00:05.355","Text":"What we want now is an example."},{"Start":"00:05.355 ","End":"00:11.670","Text":"I want an example of a surface integral of the form as"},{"Start":"00:11.670 ","End":"00:17.970","Text":"above: Vector field F.ds vector."},{"Start":"00:17.970 ","End":"00:22.020","Text":"I have to give you 2 or 3 things."},{"Start":"00:22.020 ","End":"00:24.690","Text":"I have to tell you what the function F is."},{"Start":"00:24.690 ","End":"00:26.580","Text":"Then I have to describe the surface,"},{"Start":"00:26.580 ","End":"00:29.085","Text":"but I also have to give you an orientation."},{"Start":"00:29.085 ","End":"00:33.330","Text":"Let me tell you what the vector field is to the function of x, y,"},{"Start":"00:33.330 ","End":"00:38.550","Text":"z. I want it to be 0, y minus z."},{"Start":"00:38.550 ","End":"00:40.620","Text":"These are 3 components."},{"Start":"00:40.620 ","End":"00:44.720","Text":"It\u0027s defined everywhere in 3 space,"},{"Start":"00:44.720 ","End":"00:46.625","Text":"so no problem there."},{"Start":"00:46.625 ","End":"00:52.820","Text":"Next, I have to give you what S is and I\u0027ll describe it with a picture."},{"Start":"00:52.820 ","End":"00:58.625","Text":"Our surface S is actually made up of 2 parts because I want a closed surface,"},{"Start":"00:58.625 ","End":"01:04.255","Text":"it\u0027s made up of S_1, together with S_2."},{"Start":"01:04.255 ","End":"01:09.530","Text":"Remember we had a concept of piecewise smooth surface."},{"Start":"01:09.530 ","End":"01:12.530","Text":"Well, this is smooth."},{"Start":"01:12.530 ","End":"01:16.040","Text":"It\u0027s going to be a paraboloid and this is smooth."},{"Start":"01:16.040 ","End":"01:18.890","Text":"It\u0027s part of a plane, it\u0027s a disk."},{"Start":"01:18.890 ","End":"01:21.170","Text":"We just take the 2 bits separately."},{"Start":"01:21.170 ","End":"01:23.240","Text":"I have to describe specifically,"},{"Start":"01:23.240 ","End":"01:26.530","Text":"S_1 is the paraboloid,"},{"Start":"01:26.530 ","End":"01:32.375","Text":"y equals x squared plus z squared."},{"Start":"01:32.375 ","End":"01:34.760","Text":"Y is given as a function of x and z."},{"Start":"01:34.760 ","End":"01:37.760","Text":"Remember I told you we could expect 1 of the other 2 cases."},{"Start":"01:37.760 ","End":"01:41.985","Text":"Well, here\u0027s an example where we\u0027re using this form."},{"Start":"01:41.985 ","End":"01:45.590","Text":"It\u0027s a paraboloid and it\u0027s centered on the y-axis,"},{"Start":"01:45.590 ","End":"01:47.345","Text":"that\u0027s the axis of symmetry."},{"Start":"01:47.345 ","End":"01:49.685","Text":"But after tell you up to where."},{"Start":"01:49.685 ","End":"01:57.615","Text":"This is when y is between 0 and 1."},{"Start":"01:57.615 ","End":"02:02.450","Text":"If I would label this then here y equals 1 where it cuts,"},{"Start":"02:02.450 ","End":"02:10.850","Text":"and S_2 is just where this plane y equals 1 cuts the paraboloid."},{"Start":"02:10.850 ","End":"02:13.280","Text":"If I put y equals 1 here,"},{"Start":"02:13.280 ","End":"02:16.159","Text":"what I\u0027ll get is a disk,"},{"Start":"02:16.159 ","End":"02:20.810","Text":"x squared plus z squared equals 1 is the circle,"},{"Start":"02:20.810 ","End":"02:24.995","Text":"but I want the entire disk so less than or equal to 1."},{"Start":"02:24.995 ","End":"02:27.410","Text":"But I\u0027m in 3-dimensional space,"},{"Start":"02:27.410 ","End":"02:29.230","Text":"so I can just forget about y."},{"Start":"02:29.230 ","End":"02:31.050","Text":"How long this disk?"},{"Start":"02:31.050 ","End":"02:38.890","Text":"Y equals 1, and this is the restriction on x and z to be inside the circle."},{"Start":"02:39.650 ","End":"02:43.880","Text":"I just have to still tell you what is the orientation,"},{"Start":"02:43.880 ","End":"02:47.570","Text":"and I\u0027m going to write positive orientation."},{"Start":"02:47.570 ","End":"02:51.170","Text":"Remember what this means in case of a closed surface,"},{"Start":"02:51.170 ","End":"02:54.365","Text":"it means that we\u0027re always pointing outwards."},{"Start":"02:54.365 ","End":"02:56.930","Text":"I\u0027ll just give you an idea like here,"},{"Start":"02:56.930 ","End":"02:58.790","Text":"it would be going this way."},{"Start":"02:58.790 ","End":"03:00.830","Text":"Here, might be going this way."},{"Start":"03:00.830 ","End":"03:04.145","Text":"Here, it would be going along, here."},{"Start":"03:04.145 ","End":"03:07.760","Text":"Along the disk, it would always be constant and in"},{"Start":"03:07.760 ","End":"03:14.265","Text":"this direction here.1 thing that will be clear,"},{"Start":"03:14.265 ","End":"03:17.180","Text":"well we\u0027ll see whether to take the plus or minus."},{"Start":"03:17.180 ","End":"03:21.230","Text":"But the obvious thing about all these arrows on the paraboloid is that they\u0027re"},{"Start":"03:21.230 ","End":"03:25.730","Text":"all going towards the x-z plane."},{"Start":"03:25.730 ","End":"03:28.130","Text":"In other words, the y component will be negative."},{"Start":"03:28.130 ","End":"03:29.795","Text":"We\u0027ll see that in the moment."},{"Start":"03:29.795 ","End":"03:32.675","Text":"On here it will be a constant."},{"Start":"03:32.675 ","End":"03:39.560","Text":"The unit vector will just be the unit vector j or 010."},{"Start":"03:39.560 ","End":"03:41.660","Text":"I\u0027m getting ahead of myself."},{"Start":"03:41.660 ","End":"03:45.955","Text":"Let\u0027s slow down and compute it all."},{"Start":"03:45.955 ","End":"03:49.760","Text":"The overall idea is to compute 2 separate integrals,"},{"Start":"03:49.760 ","End":"03:58.320","Text":"is to say that the integral over S of whatever it was is equal to the integral over S_1;"},{"Start":"03:58.320 ","End":"04:04.895","Text":"The paraboloid part, plus the integral over S_2, the disk part."},{"Start":"04:04.895 ","End":"04:09.785","Text":"Let\u0027s start with, let\u0027s just call this the paraboloid,"},{"Start":"04:09.785 ","End":"04:16.120","Text":"and we\u0027ll call this the disk."},{"Start":"04:16.120 ","End":"04:20.560","Text":"For the paraboloid, for S_1,"},{"Start":"04:21.730 ","End":"04:26.825","Text":"from here we can write a function of x,"},{"Start":"04:26.825 ","End":"04:34.950","Text":"y, and z as y minus x squared minus z squared,"},{"Start":"04:35.900 ","End":"04:40.785","Text":"and we can assign this to equal 0."},{"Start":"04:40.785 ","End":"04:43.520","Text":"Basically, I\u0027m putting this to the other side and making it equal to 0,"},{"Start":"04:43.520 ","End":"04:46.895","Text":"but also calling this a function of 3 variables"},{"Start":"04:46.895 ","End":"04:51.250","Text":"so that what we get is a level curve where f is 0."},{"Start":"04:51.250 ","End":"05:01.289","Text":"The gradient is normal."},{"Start":"05:02.080 ","End":"05:08.340","Text":"Let\u0027s take the gradient of f and see what this is equal to."},{"Start":"05:08.340 ","End":"05:11.975","Text":"We want the derivative with respect to x,"},{"Start":"05:11.975 ","End":"05:14.900","Text":"so that\u0027s minus 2x."},{"Start":"05:14.900 ","End":"05:20.050","Text":"Derivative with respect to y is 1."},{"Start":"05:20.050 ","End":"05:25.335","Text":"Derivative with respect to z is minus 2z."},{"Start":"05:25.335 ","End":"05:28.520","Text":"At this point, we have to decide if this is going in"},{"Start":"05:28.520 ","End":"05:32.180","Text":"the right direction according with our orientation."},{"Start":"05:32.180 ","End":"05:33.995","Text":"Now if you look at this,"},{"Start":"05:33.995 ","End":"05:37.550","Text":"the y coordinate is positive,"},{"Start":"05:37.550 ","End":"05:42.860","Text":"which means that it\u0027s somehow getting closer to the increasing y direction."},{"Start":"05:42.860 ","End":"05:44.090","Text":"This is not good."},{"Start":"05:44.090 ","End":"05:46.400","Text":"This is the opposite of what we want."},{"Start":"05:46.400 ","End":"05:50.545","Text":"We want to take our normal vector to be,"},{"Start":"05:50.545 ","End":"05:52.850","Text":"let me just write plus-minus here."},{"Start":"05:52.850 ","End":"06:01.100","Text":"I\u0027m going to switch direction and get 2x, minus 1, 2z."},{"Start":"06:01.100 ","End":"06:05.570","Text":"Remember we said we have to change sign if we just strictly follow the formulas,"},{"Start":"06:05.570 ","End":"06:08.255","Text":"we may not get the right orientation,"},{"Start":"06:08.255 ","End":"06:12.055","Text":"so this is what we want."},{"Start":"06:12.055 ","End":"06:18.400","Text":"From here we now know our unit normal n will equal,"},{"Start":"06:18.400 ","End":"06:21.800","Text":"this normal here on the other side,"},{"Start":"06:21.800 ","End":"06:26.700","Text":"2x, minus 1, 2z."},{"Start":"06:27.550 ","End":"06:30.890","Text":"I\u0027m going to divide it by the magnitude."},{"Start":"06:30.890 ","End":"06:36.485","Text":"I want to just leave it as the magnitude of grad f,"},{"Start":"06:36.485 ","End":"06:39.260","Text":"I don\u0027t need the minus because it\u0027s magnitude,"},{"Start":"06:39.260 ","End":"06:41.930","Text":"I have a reason for leaving it like this."},{"Start":"06:41.930 ","End":"06:47.535","Text":"I also forgot to label this bit here,"},{"Start":"06:47.535 ","End":"06:51.360","Text":"y is g of x and z,"},{"Start":"06:51.360 ","End":"06:53.175","Text":"that\u0027s the g here."},{"Start":"06:53.175 ","End":"06:58.320","Text":"Over here, notice that it is a minus gx,"},{"Start":"06:58.320 ","End":"07:03.340","Text":"1, and minus gz, it reversed here."},{"Start":"07:03.950 ","End":"07:08.030","Text":"Now, remember we\u0027re working on the paraboloid first,"},{"Start":"07:08.030 ","End":"07:11.045","Text":"I\u0027m going to scroll back up to show you the formula."},{"Start":"07:11.045 ","End":"07:15.515","Text":"This is the formula that I need to get it from"},{"Start":"07:15.515 ","End":"07:24.280","Text":"a vector field integral to a regular surface integral."},{"Start":"07:24.280 ","End":"07:26.610","Text":"If I do that,"},{"Start":"07:26.610 ","End":"07:35.030","Text":"what I\u0027ll get is that the surface integral over S_1 of"},{"Start":"07:35.030 ","End":"07:42.350","Text":"F. ds is equal"},{"Start":"07:42.350 ","End":"07:47.225","Text":"to the double integral over S_1 of irregular surface integral,"},{"Start":"07:47.225 ","End":"07:49.775","Text":"which is F dot n,"},{"Start":"07:49.775 ","End":"07:53.785","Text":"that\u0027s the scalar, dS."},{"Start":"07:53.785 ","End":"07:57.300","Text":"What this is equal to,"},{"Start":"07:57.300 ","End":"08:00.240","Text":"let\u0027s just scroll some more,"},{"Start":"08:00.240 ","End":"08:08.430","Text":"it\u0027s equal to the double integral over S_1."},{"Start":"08:08.430 ","End":"08:13.845","Text":"Now, F.n Here is F, here\u0027s n,"},{"Start":"08:13.845 ","End":"08:22.360","Text":"and the dot-product, I\u0027ll leave the denominator for a moment."},{"Start":"08:22.360 ","End":"08:27.290","Text":"I\u0027ve got 0 times 2x, which is 0."},{"Start":"08:27.440 ","End":"08:31.630","Text":"I\u0027ve got minus y,"},{"Start":"08:32.210 ","End":"08:37.740","Text":"and I\u0027ve got minus 2z squared."},{"Start":"08:37.740 ","End":"08:44.105","Text":"Still I have the denominator,"},{"Start":"08:44.105 ","End":"08:47.900","Text":"and this is dS."},{"Start":"08:47.900 ","End":"08:55.850","Text":"Now, we need another formula to convert from dS to irregular double integral."},{"Start":"08:55.850 ","End":"08:58.550","Text":"I pasted the formula here,"},{"Start":"08:58.550 ","End":"09:01.295","Text":"but we\u0027re going to have to adapt it"},{"Start":"09:01.295 ","End":"09:05.840","Text":"because this was for the case where z was a function of x and y."},{"Start":"09:05.840 ","End":"09:10.325","Text":"What we\u0027re going to have here is we are going to have, instead of this,"},{"Start":"09:10.325 ","End":"09:13.355","Text":"we\u0027re going to have x,"},{"Start":"09:13.355 ","End":"09:16.625","Text":"g of x and z,"},{"Start":"09:16.625 ","End":"09:20.915","Text":"because here y is the function of x and z and then z."},{"Start":"09:20.915 ","End":"09:23.655","Text":"Well, I\u0027ll just rewrite this whole thing."},{"Start":"09:23.655 ","End":"09:28.440","Text":"The other thing is that instead of what\u0027s under here,"},{"Start":"09:28.440 ","End":"09:33.705","Text":"we\u0027ll have gx squared,"},{"Start":"09:33.705 ","End":"09:36.810","Text":"as is, here, we\u0027ll have 1,"},{"Start":"09:36.810 ","End":"09:40.855","Text":"and here we\u0027ll have gz squared."},{"Start":"09:40.855 ","End":"09:45.310","Text":"Just adapting it for the case that y is g of x,"},{"Start":"09:45.310 ","End":"09:47.785","Text":"z, just the obvious changes."},{"Start":"09:47.785 ","End":"09:52.730","Text":"But notice that what\u0027s under the square root,"},{"Start":"09:54.350 ","End":"10:00.945","Text":"this part here is just exactly the magnitude of grad f. That mean,"},{"Start":"10:00.945 ","End":"10:06.775","Text":"if we take the square root of the sum of these squared,"},{"Start":"10:06.775 ","End":"10:09.340","Text":"here I have the gx squared,"},{"Start":"10:09.340 ","End":"10:17.090","Text":"1 squared, and gz squared."},{"Start":"10:17.090 ","End":"10:20.050","Text":"As I was saying, this square root bit,"},{"Start":"10:20.050 ","End":"10:28.255","Text":"this part here is exactly the magnitude of grad f. That\u0027s why I left this here,"},{"Start":"10:28.255 ","End":"10:31.010","Text":"because it\u0027s going to cancel out,"},{"Start":"10:31.580 ","End":"10:35.440","Text":"and so we get a double integral."},{"Start":"10:35.440 ","End":"10:37.000","Text":"We haven\u0027t spoken about D,"},{"Start":"10:37.000 ","End":"10:38.710","Text":"we will in a moment."},{"Start":"10:38.710 ","End":"10:42.460","Text":"Integral over D of,"},{"Start":"10:42.460 ","End":"10:44.140","Text":"now this is just the function,"},{"Start":"10:44.140 ","End":"10:46.165","Text":"this is just what\u0027s written here."},{"Start":"10:46.165 ","End":"10:51.375","Text":"But I need it in terms of y and z,"},{"Start":"10:51.375 ","End":"10:58.695","Text":"so I\u0027m going to replace y by x squared plus z squared."},{"Start":"10:58.695 ","End":"11:00.395","Text":"Let me do that on the side,"},{"Start":"11:00.395 ","End":"11:04.490","Text":"I have minus y minus 2z squared,"},{"Start":"11:04.490 ","End":"11:08.390","Text":"but y is x squared plus z squared."},{"Start":"11:08.390 ","End":"11:11.810","Text":"I\u0027ve got x squared plus z squared,"},{"Start":"11:11.810 ","End":"11:16.655","Text":"all this minus and then minus 2z squared."},{"Start":"11:16.655 ","End":"11:25.000","Text":"What I get is minus x squared minus 3z squared."},{"Start":"11:25.000 ","End":"11:29.060","Text":"Copying that here, I can put the minus in front."},{"Start":"11:29.060 ","End":"11:33.695","Text":"Then I can say x squared plus 3z squared,"},{"Start":"11:33.695 ","End":"11:39.005","Text":"and then 1 over magnitude of"},{"Start":"11:39.005 ","End":"11:45.010","Text":"grad f and then times magnitude of grad f,"},{"Start":"11:45.010 ","End":"11:49.470","Text":"and there is little arrows there, dA,"},{"Start":"11:49.470 ","End":"11:56.680","Text":"and I should emphasize that they canceled this with this."},{"Start":"11:57.270 ","End":"12:01.360","Text":"Now we come to the matter of D. What is D?"},{"Start":"12:01.360 ","End":"12:09.380","Text":"It\u0027s the domain of the paraboloid in the x, z plane."},{"Start":"12:09.380 ","End":"12:11.450","Text":"If you think about it,"},{"Start":"12:11.450 ","End":"12:16.900","Text":"it\u0027s the unit disc,"},{"Start":"12:16.900 ","End":"12:18.940","Text":"in the x, z plane."},{"Start":"12:18.940 ","End":"12:21.295","Text":"Y is less than or equal to 1,"},{"Start":"12:21.295 ","End":"12:24.880","Text":"means that x squared plus z squared is less than or equal to 1."},{"Start":"12:24.880 ","End":"12:27.010","Text":"Of course, it\u0027s bigger or equal to 0."},{"Start":"12:27.010 ","End":"12:30.085","Text":"D is the unit disc,"},{"Start":"12:30.085 ","End":"12:34.510","Text":"x squared plus z squared less than or equal to 1."},{"Start":"12:34.510 ","End":"12:40.645","Text":"It\u0027s 1 squared, so it\u0027s a disc of radius 1 and it\u0027s in the x, z plane."},{"Start":"12:40.645 ","End":"12:43.300","Text":"How are we\u0027re going to do this integral?"},{"Start":"12:43.300 ","End":"12:46.075","Text":"This is the unit disc."},{"Start":"12:46.075 ","End":"12:48.340","Text":"It\u0027s not on the diagram,"},{"Start":"12:48.340 ","End":"12:50.950","Text":"but you could imagine in the x, z plane."},{"Start":"12:50.950 ","End":"12:53.440","Text":"The unit disc, it\u0027s just the projection."},{"Start":"12:53.440 ","End":"12:55.210","Text":"It\u0027s like this disc only in the x,"},{"Start":"12:55.210 ","End":"12:58.284","Text":"z plane and it\u0027s circular."},{"Start":"12:58.284 ","End":"13:02.450","Text":"The natural thing to do would be to use polar coordinates."},{"Start":"13:02.610 ","End":"13:09.700","Text":"In polar coordinates, what we get for this domain is"},{"Start":"13:09.700 ","End":"13:17.280","Text":"that Theta goes full circle from 0 to 2 Pi."},{"Start":"13:17.280 ","End":"13:21.360","Text":"The radius goes from 0 to 1."},{"Start":"13:21.360 ","End":"13:27.635","Text":"Of course, we have the usual conversion formulas,"},{"Start":"13:27.635 ","End":"13:29.620","Text":"but we usually know them with x and y."},{"Start":"13:29.620 ","End":"13:31.000","Text":"But the same thing with x and z,"},{"Start":"13:31.000 ","End":"13:36.475","Text":"we have x equals r cosine Theta z."},{"Start":"13:36.475 ","End":"13:38.050","Text":"We normally have y here,"},{"Start":"13:38.050 ","End":"13:41.575","Text":"is r sine Theta."},{"Start":"13:41.575 ","End":"13:45.820","Text":"For later on, just remember that with polar coordinates,"},{"Start":"13:45.820 ","End":"13:51.475","Text":"the formula for dA is rd Theta dr."},{"Start":"13:51.475 ","End":"13:59.450","Text":"Substituting all this here, we get minus."},{"Start":"13:59.580 ","End":"14:06.850","Text":"The domain D now becomes r Theta in these intervals."},{"Start":"14:06.850 ","End":"14:11.950","Text":"Let\u0027s take Theta from 0 to 2 Pi."},{"Start":"14:11.950 ","End":"14:13.900","Text":"The order doesn\u0027t actually matter,"},{"Start":"14:13.900 ","End":"14:17.485","Text":"r from 0 to 1."},{"Start":"14:17.485 ","End":"14:20.260","Text":"Then what do we need?"},{"Start":"14:20.260 ","End":"14:25.150","Text":"We need x squared plus 3z squared."},{"Start":"14:25.150 ","End":"14:35.305","Text":"We need r squared cosine squared Theta,"},{"Start":"14:35.305 ","End":"14:43.105","Text":"plus 3r squared sine squared Theta,"},{"Start":"14:43.105 ","End":"14:46.885","Text":"and then dA is"},{"Start":"14:46.885 ","End":"14:54.700","Text":"rd Theta dr."},{"Start":"14:54.700 ","End":"14:57.830","Text":"I need some more space here."},{"Start":"14:58.890 ","End":"15:02.305","Text":"I can do some simplification here."},{"Start":"15:02.305 ","End":"15:07.420","Text":"What I can say is that r has nothing to do with Theta."},{"Start":"15:07.420 ","End":"15:09.640","Text":"I can take it in front of the first integral."},{"Start":"15:09.640 ","End":"15:12.835","Text":"I have r squared everywhere and an r here."},{"Start":"15:12.835 ","End":"15:18.610","Text":"I can actually take r cubed outside and then just get an integral"},{"Start":"15:18.610 ","End":"15:24.730","Text":"from 0 to 2 Pi of cosine squared plus 3 sine squared."},{"Start":"15:24.730 ","End":"15:32.530","Text":"Now, cosine squared plus 3 sine squared is actually 1 plus 2 sine"},{"Start":"15:32.530 ","End":"15:36.535","Text":"squared Theta because cosine"},{"Start":"15:36.535 ","End":"15:40.930","Text":"squared plus sine squared is 1 and I still have 2 more sine squared,"},{"Start":"15:40.930 ","End":"15:43.150","Text":"and this is d Theta,"},{"Start":"15:43.150 ","End":"15:53.500","Text":"and then all of this is dr. Let me just tell you the answer to this integral."},{"Start":"15:53.500 ","End":"15:56.905","Text":"I don\u0027t want to waste away so much time with just doing integrals,"},{"Start":"15:56.905 ","End":"16:05.875","Text":"comes out to be 2 Theta minus a 1/2 sine 2 Theta,"},{"Start":"16:05.875 ","End":"16:09.985","Text":"but we need to evaluate it between 0 and 2 Pi."},{"Start":"16:09.985 ","End":"16:14.155","Text":"Now the sine, if I put in Theta 0 or 2 Pi is the same thing."},{"Start":"16:14.155 ","End":"16:15.835","Text":"This part doesn\u0027t matter."},{"Start":"16:15.835 ","End":"16:23.135","Text":"We just need the 2 Theta from 0 to 2 Pi and that comes out to be 4 Pi."},{"Start":"16:23.135 ","End":"16:30.360","Text":"This whole thing here is 4 Pi and the integral,"},{"Start":"16:30.360 ","End":"16:32.530","Text":"this is a constant."},{"Start":"16:33.290 ","End":"16:35.760","Text":"I can bring this in front,"},{"Start":"16:35.760 ","End":"16:44.665","Text":"so I\u0027ve got minus 4 Pi and then I have the integral of r cubed,"},{"Start":"16:44.665 ","End":"16:49.730","Text":"which is just r^4th over 4."},{"Start":"16:51.360 ","End":"16:55.280","Text":"I forgot to write 0 to 1."},{"Start":"16:55.800 ","End":"17:00.100","Text":"When 0, it just gives 0 and 1 it\u0027s a 1/4."},{"Start":"17:00.100 ","End":"17:04.420","Text":"It basically comes out to be, let\u0027s see,"},{"Start":"17:04.420 ","End":"17:10.235","Text":"minus 4 Pi times 1/4."},{"Start":"17:10.235 ","End":"17:14.965","Text":"The final answer is minus Pi."},{"Start":"17:14.965 ","End":"17:20.455","Text":"This is not the final answer."},{"Start":"17:20.455 ","End":"17:24.490","Text":"This is just for the S_1,"},{"Start":"17:24.490 ","End":"17:31.780","Text":"but we still need to do the S_2 that\u0027s still remaining."},{"Start":"17:34.860 ","End":"17:38.425","Text":"I\u0027ll record the fact that for this,"},{"Start":"17:38.425 ","End":"17:41.500","Text":"we got the result of"},{"Start":"17:41.500 ","End":"17:49.910","Text":"minus Pi and now we can start working on F_2."},{"Start":"17:50.250 ","End":"17:56.500","Text":"For the other integral on S_2,"},{"Start":"17:56.500 ","End":"17:59.170","Text":"we can just work straight off the definition."},{"Start":"17:59.170 ","End":"18:07.795","Text":"What we want is the doubling to grow across as 2 of the vector field,"},{"Start":"18:07.795 ","End":"18:16.750","Text":"which is 0, y minus z dot a unit normal vector."},{"Start":"18:16.750 ","End":"18:25.795","Text":"Now, this is easy because in this particular case which is part of the plane, y equals 1."},{"Start":"18:25.795 ","End":"18:30.040","Text":"This is completely parallel to the x,"},{"Start":"18:30.040 ","End":"18:34.015","Text":"z plane and it\u0027s perpendicular to the y-axis."},{"Start":"18:34.015 ","End":"18:38.545","Text":"We can just take a unit normal vector in the direction of the y-axis,"},{"Start":"18:38.545 ","End":"18:42.040","Text":"which is of course 0, 1,"},{"Start":"18:42.040 ","End":"18:45.490","Text":"0, or minus that."},{"Start":"18:45.490 ","End":"18:47.020","Text":"If you think a moment,"},{"Start":"18:47.020 ","End":"18:50.185","Text":"we said the positive direction is the outer direction,"},{"Start":"18:50.185 ","End":"18:52.270","Text":"so it\u0027s in the direction of the y-axis."},{"Start":"18:52.270 ","End":"18:59.650","Text":"We don\u0027t need to make anything negative and that is dS."},{"Start":"18:59.650 ","End":"19:04.495","Text":"If you\u0027re not fully convinced of why this is the normal,"},{"Start":"19:04.495 ","End":"19:10.990","Text":"you could also think of it as this is a level curve, y equals 1."},{"Start":"19:10.990 ","End":"19:14.530","Text":"If we take f of x, y,"},{"Start":"19:14.530 ","End":"19:21.565","Text":"and z just to be equal to y and this is the level curve at the value 1,"},{"Start":"19:21.565 ","End":"19:26.170","Text":"and if we take the derivatives with respect to x,"},{"Start":"19:26.170 ","End":"19:30.700","Text":"y, and z, we\u0027ll get 0, 1, and 0."},{"Start":"19:30.700 ","End":"19:33.205","Text":"That\u0027s another way to look at it."},{"Start":"19:33.205 ","End":"19:39.310","Text":"Anyway, now, if we do the dot product,"},{"Start":"19:39.310 ","End":"19:43.615","Text":"everything\u0027s 0 except for the middle 1, which is y."},{"Start":"19:43.615 ","End":"19:48.100","Text":"We have the double integral"},{"Start":"19:48.100 ","End":"19:54.790","Text":"across S_2 of just ydS,"},{"Start":"19:54.790 ","End":"20:00.355","Text":"but y is equal to 1 on this surface."},{"Start":"20:00.355 ","End":"20:03.770","Text":"I can cross the y out."},{"Start":"20:04.080 ","End":"20:08.980","Text":"Let me clear a bit of space here and here,"},{"Start":"20:08.980 ","End":"20:12.170","Text":"and I\u0027ll push this up here."},{"Start":"20:12.180 ","End":"20:17.920","Text":"I copied a formula that we had before,"},{"Start":"20:17.920 ","End":"20:25.840","Text":"which gave the conversion from an integral over S to a regular double integral over"},{"Start":"20:25.840 ","End":"20:34.165","Text":"a domain D in the case where y is given as a function of x and z."},{"Start":"20:34.165 ","End":"20:35.905","Text":"In other words, in our case,"},{"Start":"20:35.905 ","End":"20:41.320","Text":"this will be our g of x and z,"},{"Start":"20:41.320 ","End":"20:43.930","Text":"which is just the constant function."},{"Start":"20:43.930 ","End":"20:45.865","Text":"If that\u0027s the case,"},{"Start":"20:45.865 ","End":"20:48.415","Text":"then this becomes very simple."},{"Start":"20:48.415 ","End":"20:56.694","Text":"We get the double integral over D and we\u0027ll see you in a moment what the D is,"},{"Start":"20:56.694 ","End":"21:01.045","Text":"of, all this is just 1."},{"Start":"21:01.045 ","End":"21:04.400","Text":"I\u0027ll write it just to emphasize it."},{"Start":"21:05.120 ","End":"21:13.570","Text":"We have gx is 0 and gz is 0."},{"Start":"21:13.570 ","End":"21:17.545","Text":"All we\u0027re left with here is also the square root of 1."},{"Start":"21:17.545 ","End":"21:21.950","Text":"We end up with the integral of 1 dA"},{"Start":"21:23.100 ","End":"21:30.685","Text":"and that is equal to the area of D. We just haven\u0027t said yet what D is,"},{"Start":"21:30.685 ","End":"21:32.680","Text":"this is the area of D. Now,"},{"Start":"21:32.680 ","End":"21:41.520","Text":"D is going to be the same D as in the S_1 case."},{"Start":"21:41.520 ","End":"21:45.890","Text":"It\u0027s just the unit circle in the x, z plane."},{"Start":"21:45.890 ","End":"21:50.110","Text":"This is of course S_2 in our case."},{"Start":"21:50.110 ","End":"21:54.520","Text":"D is the unit disc and we don\u0027t need to do"},{"Start":"21:54.520 ","End":"22:02.240","Text":"any calculus and double integrals because we know the area of D is a unit disc."},{"Start":"22:03.600 ","End":"22:07.300","Text":"Write that down and hope that you remember"},{"Start":"22:07.300 ","End":"22:13.855","Text":"the area of a disc or circle is the Pi r squared."},{"Start":"22:13.855 ","End":"22:21.670","Text":"This thing is just equal to Pi."},{"Start":"22:21.670 ","End":"22:23.890","Text":"I\u0027ll just highlight that."},{"Start":"22:23.890 ","End":"22:25.675","Text":"Now, I\u0027ve got the second half,"},{"Start":"22:25.675 ","End":"22:31.340","Text":"the S_2 and this is plus Pi."},{"Start":"22:32.070 ","End":"22:37.690","Text":"Altogether, what we\u0027ll have to do is say what is minus Pi plus Pi?"},{"Start":"22:37.690 ","End":"22:39.850","Text":"I\u0027ll write it over here and we\u0027ll do it in red."},{"Start":"22:39.850 ","End":"22:48.700","Text":"I have minus Pi plus Pi and the answer is 0, and that\u0027s it."},{"Start":"22:48.700 ","End":"22:51.070","Text":"That\u0027s the end of this example."},{"Start":"22:51.070 ","End":"22:52.510","Text":"That\u0027s basically it."},{"Start":"22:52.510 ","End":"23:01.825","Text":"I\u0027ll just end on a note of terminology that this integral over a surface of"},{"Start":"23:01.825 ","End":"23:11.605","Text":"a vector field with an oriented surface is also called"},{"Start":"23:11.605 ","End":"23:20.200","Text":"the flux of the vector field F"},{"Start":"23:20.200 ","End":"23:29.710","Text":"across the oriented surface S. This term is mostly used in physics."},{"Start":"23:29.710 ","End":"23:32.360","Text":"Yeah, that\u0027s it."}],"ID":9651},{"Watched":false,"Name":"Useful Formula for Computing Surface Integrals of Vector Fields","Duration":"11m 30s","ChapterTopicVideoID":8790,"CourseChapterTopicPlaylistID":4974,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.040","Text":"In this clip, I want to introduce a useful formula."},{"Start":"00:05.040 ","End":"00:09.300","Text":"We\u0027re continuing with surface integrals of vector fields and it\u0027s a formula that is"},{"Start":"00:09.300 ","End":"00:13.380","Text":"used so often that I\u0027d like to bring it to you here."},{"Start":"00:13.380 ","End":"00:19.850","Text":"I\u0027m going to get it by just piecing together various stuff we\u0027ve already done."},{"Start":"00:19.850 ","End":"00:26.160","Text":"Here, I copy-pasted the basic definition of the type 2 surface integral when"},{"Start":"00:26.160 ","End":"00:32.295","Text":"it\u0027s the vector ds and it\u0027s written in the form of a type 1 surface integral."},{"Start":"00:32.295 ","End":"00:34.200","Text":"After I take the dot product,"},{"Start":"00:34.200 ","End":"00:37.050","Text":"what I get is a scalar function F,"},{"Start":"00:37.050 ","End":"00:42.120","Text":"ds, and I want to break this down still more."},{"Start":"00:42.120 ","End":"00:46.460","Text":"Here, I copy-pasted from a previous clip that if"},{"Start":"00:46.460 ","End":"00:50.960","Text":"the surface is given as z equals g of x and y,"},{"Start":"00:50.960 ","End":"00:52.715","Text":"that\u0027s our surface S,"},{"Start":"00:52.715 ","End":"00:57.110","Text":"where x and y belong to some domain D in the x,"},{"Start":"00:57.110 ","End":"01:02.045","Text":"y plane and D is the projection of S onto the xy plane,"},{"Start":"01:02.045 ","End":"01:07.820","Text":"then we can write the surface integral as a regular double integral"},{"Start":"01:07.820 ","End":"01:14.105","Text":"over the domain D. Sometimes I use the letter R for region either row,"},{"Start":"01:14.105 ","End":"01:16.985","Text":"then we get this formula."},{"Start":"01:16.985 ","End":"01:22.790","Text":"I guess I really should have said here that F is a function of 3 variables,"},{"Start":"01:22.790 ","End":"01:25.010","Text":"F of x, y, and z."},{"Start":"01:25.010 ","End":"01:26.510","Text":"After I do F dot n,"},{"Start":"01:26.510 ","End":"01:31.560","Text":"I get a scalar function and then I compute it this way."},{"Start":"01:31.560 ","End":"01:36.500","Text":"Now, I still have to tie these in together and I\u0027ve got this concept of the normal."},{"Start":"01:36.500 ","End":"01:38.540","Text":"Now, what is this normal?"},{"Start":"01:38.540 ","End":"01:43.520","Text":"Suppose we take a point on the surface, this one,"},{"Start":"01:43.520 ","End":"01:48.785","Text":"and we want to find a normal vector which is perpendicular,"},{"Start":"01:48.785 ","End":"01:51.025","Text":"it might look like this."},{"Start":"01:51.025 ","End":"01:54.560","Text":"But often, we want a unit normal vector."},{"Start":"01:54.560 ","End":"01:56.885","Text":"In fact, in this formula,"},{"Start":"01:56.885 ","End":"02:03.025","Text":"the n in the formula here is a unit normal."},{"Start":"02:03.025 ","End":"02:05.945","Text":"But even with unit normals,"},{"Start":"02:05.945 ","End":"02:07.930","Text":"there are 2 of them."},{"Start":"02:07.930 ","End":"02:13.410","Text":"This could be one direction and I could also take"},{"Start":"02:13.410 ","End":"02:19.190","Text":"one and do in dotted lines in the opposite direction."},{"Start":"02:19.190 ","End":"02:25.470","Text":"If this one is n,"},{"Start":"02:25.470 ","End":"02:33.175","Text":"the opposite direction would be minus n. We also said that"},{"Start":"02:33.175 ","End":"02:37.925","Text":"the one going upwards has a positive orientation"},{"Start":"02:37.925 ","End":"02:43.320","Text":"and the one going downwards is considered to have the negative orientation."},{"Start":"02:43.360 ","End":"02:45.560","Text":"In any given case,"},{"Start":"02:45.560 ","End":"02:47.000","Text":"when we define a surface,"},{"Start":"02:47.000 ","End":"02:48.650","Text":"we always give an orientation,"},{"Start":"02:48.650 ","End":"02:53.960","Text":"so we\u0027ll know whether this is up or down."},{"Start":"02:53.960 ","End":"02:57.100","Text":"When I say up, I mean it has an upward component."},{"Start":"02:57.100 ","End":"02:59.240","Text":"It\u0027s not totally vertical,"},{"Start":"02:59.240 ","End":"03:04.915","Text":"but it has a positive z component and downwards is the negative z component."},{"Start":"03:04.915 ","End":"03:11.165","Text":"Now, we also had a formula that defined a normal to such a surface."},{"Start":"03:11.165 ","End":"03:15.890","Text":"1 way of doing it in the angular bracket notation would be to"},{"Start":"03:15.890 ","End":"03:22.040","Text":"take the vector minus g partial derivative with respect to x,"},{"Start":"03:22.040 ","End":"03:26.380","Text":"minus g with respect to y and 1."},{"Start":"03:26.380 ","End":"03:31.225","Text":"This would be a normal in the positive orientation."},{"Start":"03:31.225 ","End":"03:35.160","Text":"Now, unit normal, lets call that"},{"Start":"03:35.160 ","End":"03:41.770","Text":"n. If you take a vector and divide by its magnitude,"},{"Start":"03:41.770 ","End":"03:50.220","Text":"we would get the vector minus g_x minus g_y 1,"},{"Start":"03:50.220 ","End":"03:56.770","Text":"and we divide this by the magnitude of just copying the same thing,"},{"Start":"03:56.770 ","End":"04:00.110","Text":"minus g_x minus g_y,"},{"Start":"04:00.570 ","End":"04:06.255","Text":"1, close brackets, magnitude."},{"Start":"04:06.255 ","End":"04:12.320","Text":"In fact this end is the same as this one,"},{"Start":"04:12.320 ","End":"04:15.625","Text":"which is the positive unit normal."},{"Start":"04:15.625 ","End":"04:20.300","Text":"I mean the unit normal in the positive orientation."},{"Start":"04:20.300 ","End":"04:23.795","Text":"I want to piece some stuff together now,"},{"Start":"04:23.795 ","End":"04:26.615","Text":"but I need some more space."},{"Start":"04:26.615 ","End":"04:31.730","Text":"Then I\u0027m going to copy this and say that this integral over"},{"Start":"04:31.730 ","End":"04:37.590","Text":"S of F dot n ds."},{"Start":"04:37.590 ","End":"04:40.760","Text":"The questions are often stated in this for memoir."},{"Start":"04:40.760 ","End":"04:43.640","Text":"I\u0027ve seen this more commonly given than this,"},{"Start":"04:43.640 ","End":"04:45.020","Text":"so this is what\u0027s important."},{"Start":"04:45.020 ","End":"04:48.800","Text":"But if you have this, you convert it to this and then to this,"},{"Start":"04:48.800 ","End":"04:54.675","Text":"and this is going to equal using this formula;"},{"Start":"04:54.675 ","End":"05:01.100","Text":"the double integral over the region or domain D or R,"},{"Start":"05:01.100 ","End":"05:03.800","Text":"whichever, the same thing,"},{"Start":"05:03.800 ","End":"05:05.660","Text":"but not quite the same thing."},{"Start":"05:05.660 ","End":"05:08.750","Text":"I\u0027ll tell you in a minute why not quite the same."},{"Start":"05:08.750 ","End":"05:14.675","Text":"Instead of ds, by this formula,"},{"Start":"05:14.675 ","End":"05:20.780","Text":"I can just write here g_x squared plus g_y squared."},{"Start":"05:20.780 ","End":"05:25.160","Text":"I mean partial derivatives plus 1 dA."},{"Start":"05:25.160 ","End":"05:28.460","Text":"But when I said not quite,"},{"Start":"05:28.460 ","End":"05:30.410","Text":"I mean that just like here,"},{"Start":"05:30.410 ","End":"05:33.425","Text":"we replace z by g of xy."},{"Start":"05:33.425 ","End":"05:35.945","Text":"I didn\u0027t write the x, y, z in,"},{"Start":"05:35.945 ","End":"05:45.610","Text":"but I\u0027ll just replace z by g of xy."},{"Start":"05:45.610 ","End":"05:47.490","Text":"It is a function of x and y,"},{"Start":"05:47.490 ","End":"05:49.320","Text":"and z doesn\u0027t appear."},{"Start":"05:49.320 ","End":"05:52.305","Text":"Now, there is a shortcut here."},{"Start":"05:52.305 ","End":"05:57.020","Text":"Let\u0027s assume at first that we\u0027re taking the positive orientation,"},{"Start":"05:57.020 ","End":"06:05.465","Text":"then I can replace the unit vector and by what\u0027s written here."},{"Start":"06:05.465 ","End":"06:10.550","Text":"But notice that this here is just the"},{"Start":"06:10.550 ","End":"06:16.835","Text":"same as the magnitude of minus g_x minus g_y,"},{"Start":"06:16.835 ","End":"06:21.500","Text":"1, because the magnitude is square root of this squared,"},{"Start":"06:21.500 ","End":"06:23.180","Text":"plus this squared, plus this squared."},{"Start":"06:23.180 ","End":"06:26.645","Text":"When I square them, the minuses disappear,"},{"Start":"06:26.645 ","End":"06:27.860","Text":"and this is what I get."},{"Start":"06:27.860 ","End":"06:32.765","Text":"If I take this over this and multiply it by this,"},{"Start":"06:32.765 ","End":"06:42.660","Text":"then what we get is the double integral over D of F dot with"},{"Start":"06:42.660 ","End":"06:48.060","Text":"the vector minus g_x, minus g_y,"},{"Start":"06:48.060 ","End":"06:54.450","Text":"1 dA."},{"Start":"06:54.450 ","End":"07:03.410","Text":"As before, I\u0027ll just replace any occurrence of z by the function g of x and y."},{"Start":"07:03.410 ","End":"07:07.760","Text":"As I say, this is for the case of the positive orientation,"},{"Start":"07:07.760 ","End":"07:16.380","Text":"that means that when I have my normal vector facing upwards."},{"Start":"07:16.380 ","End":"07:21.425","Text":"It doesn\u0027t have to be directly upwards as long as it has an upward component."},{"Start":"07:21.425 ","End":"07:25.430","Text":"For the negative orientation,"},{"Start":"07:25.430 ","End":"07:29.810","Text":"I would just replace this by its negative."},{"Start":"07:29.810 ","End":"07:35.765","Text":"Let me summarize now and I\u0027ll write down that formula,"},{"Start":"07:35.765 ","End":"07:45.900","Text":"that the double integral over an oriented surface S of a vector of x, y, and z,"},{"Start":"07:45.900 ","End":"07:51.290","Text":"dot a normal vector in the correct orientation,"},{"Start":"07:51.290 ","End":"07:57.860","Text":"ds will equal and it\u0027s going to be 1 of 2 things."},{"Start":"07:57.860 ","End":"08:05.569","Text":"It\u0027s going to equal either the double integral over D"},{"Start":"08:05.569 ","End":"08:15.100","Text":"of F dot vector minus g_x, minus g_y,"},{"Start":"08:15.100 ","End":"08:22.050","Text":"1 dA for the positive oriented."},{"Start":"08:22.050 ","End":"08:27.010","Text":"I copy-pasted. Now I\u0027m going to change the signs, so here."},{"Start":"08:27.010 ","End":"08:32.450","Text":"This is the formula to use when we have a positive z component of the"},{"Start":"08:32.450 ","End":"08:38.780","Text":"normal and this one is a downward facing a negative z in the normal."},{"Start":"08:38.780 ","End":"08:42.410","Text":"You can tell by the way the problem is stated."},{"Start":"08:42.410 ","End":"08:44.710","Text":"It\u0027s often given explicitly."},{"Start":"08:44.710 ","End":"08:49.280","Text":"Also, don\u0027t forget that when we convert to this form,"},{"Start":"08:49.280 ","End":"08:53.330","Text":"if we see any z leftover,"},{"Start":"08:53.330 ","End":"08:56.910","Text":"we\u0027ve got to replace it by g of xy."},{"Start":"08:57.820 ","End":"09:02.975","Text":"Now, I thought this was going to be the formula I was going to give."},{"Start":"09:02.975 ","End":"09:09.800","Text":"Then I remembered there\u0027s 2 more because this applies to when z is a function of x and y,"},{"Start":"09:09.800 ","End":"09:12.920","Text":"but sometimes we have the other situation that x"},{"Start":"09:12.920 ","End":"09:17.245","Text":"or y are given as functions of the other 2."},{"Start":"09:17.245 ","End":"09:23.360","Text":"I\u0027m going to write down the other 2 formulas."},{"Start":"09:23.360 ","End":"09:27.380","Text":"I started with a copy-paste and I erased, but I don\u0027t need."},{"Start":"09:27.380 ","End":"09:32.435","Text":"Now the second pair will be,"},{"Start":"09:32.435 ","End":"09:39.385","Text":"I\u0027ll take it for when x is a function of y and z."},{"Start":"09:39.385 ","End":"09:43.390","Text":"In this case, if it\u0027s the dx,"},{"Start":"09:43.390 ","End":"09:48.140","Text":"then here\u0027s 1 and here\u0027s the minus 1."},{"Start":"09:48.650 ","End":"10:00.470","Text":"It will be when y equals g of x and z. I\u0027ll get here a 1 in the y position and a minus 1."},{"Start":"10:00.470 ","End":"10:02.810","Text":"The other 2 here,"},{"Start":"10:02.810 ","End":"10:05.045","Text":"there\u0027ll be the negative."},{"Start":"10:05.045 ","End":"10:11.915","Text":"It\u0027s minus g with respect to y minus g with respect to z."},{"Start":"10:11.915 ","End":"10:14.150","Text":"Here everything\u0027s, just the reverse;"},{"Start":"10:14.150 ","End":"10:16.325","Text":"g with respect to y,"},{"Start":"10:16.325 ","End":"10:18.290","Text":"g with respect to z."},{"Start":"10:18.290 ","End":"10:20.330","Text":"Sets of upwards and downwards, what do you say?"},{"Start":"10:20.330 ","End":"10:24.310","Text":"Maybe a positive x component and a minus x component,"},{"Start":"10:24.310 ","End":"10:31.480","Text":"a negative facing in the direction of the positive x-axis on the negative x-axis,"},{"Start":"10:31.480 ","End":"10:33.300","Text":"however you want to look at it."},{"Start":"10:33.300 ","End":"10:36.020","Text":"The last formula when y,"},{"Start":"10:36.020 ","End":"10:43.220","Text":"so here g_x, the negative here,"},{"Start":"10:43.220 ","End":"10:49.400","Text":"negative g_z here, positive g_x, positive g_z,"},{"Start":"10:49.400 ","End":"10:58.030","Text":"and this is when it has a positive y component and a negative y component for the normal."},{"Start":"10:59.780 ","End":"11:04.400","Text":"As before, we shouldn\u0027t forget when we substitute x, y, and z,"},{"Start":"11:04.400 ","End":"11:06.980","Text":"we should replace y by g of x and z,"},{"Start":"11:06.980 ","End":"11:10.040","Text":"just as we did with z and with x here."},{"Start":"11:10.040 ","End":"11:12.695","Text":"There\u0027s really the 6 formulas,"},{"Start":"11:12.695 ","End":"11:16.080","Text":"but they\u0027re all essentially the same."},{"Start":"11:17.120 ","End":"11:25.490","Text":"There\u0027s plenty of examples in the solved exercises that follow the tutorial,"},{"Start":"11:25.490 ","End":"11:30.480","Text":"so that\u0027s it here. We\u0027re done."}],"ID":9652},{"Watched":false,"Name":"Exercise 1","Duration":"9m 38s","ChapterTopicVideoID":8799,"CourseChapterTopicPlaylistID":4974,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.025","Text":"In this exercise, we have to compute this type 1 surface integral."},{"Start":"00:05.025 ","End":"00:11.280","Text":"S is part of this plane that\u0027s above the following rectangle."},{"Start":"00:11.280 ","End":"00:13.860","Text":"All will be explained."},{"Start":"00:13.860 ","End":"00:17.760","Text":"I brought in a general picture that might help us."},{"Start":"00:17.760 ","End":"00:25.095","Text":"Only in our case, we have the region R and the surface above it S,"},{"Start":"00:25.095 ","End":"00:31.475","Text":"and what we can do is bring in a general formula."},{"Start":"00:31.475 ","End":"00:34.520","Text":"For example, this, in general,"},{"Start":"00:34.520 ","End":"00:37.045","Text":"will be a surface g equals,"},{"Start":"00:37.045 ","End":"00:39.625","Text":"sorry, z equals g of x,y."},{"Start":"00:39.625 ","End":"00:45.000","Text":"In our case, this will be g of x,y."},{"Start":"00:45.000 ","End":"00:48.830","Text":"The other thing we have is a function that we want to integrate,"},{"Start":"00:48.830 ","End":"00:50.870","Text":"some function of x, y,"},{"Start":"00:50.870 ","End":"00:54.240","Text":"and z, and this will be here,"},{"Start":"00:54.240 ","End":"00:59.150","Text":"what we\u0027ll call f. We\u0027ll also have a region."},{"Start":"00:59.150 ","End":"01:02.285","Text":"I\u0027ll explain this notation in a minute."},{"Start":"01:02.285 ","End":"01:04.895","Text":"When we have all that,"},{"Start":"01:04.895 ","End":"01:09.980","Text":"the region actually is the projection of the surface onto the x-y plane."},{"Start":"01:09.980 ","End":"01:15.970","Text":"Then there\u0027s the formula for the integral and that says that in general,"},{"Start":"01:15.970 ","End":"01:21.540","Text":"the surface integral of this function,"},{"Start":"01:21.540 ","End":"01:23.810","Text":"f of x, y, z,"},{"Start":"01:23.810 ","End":"01:30.230","Text":"dS is equal to the regular double integral"},{"Start":"01:30.230 ","End":"01:36.750","Text":"over the region R of the following if here it\u0027s f of x,"},{"Start":"01:36.750 ","End":"01:39.815","Text":"y, z, I\u0027ve just abbreviated it,"},{"Start":"01:39.815 ","End":"01:41.660","Text":"but in general will be f of x,"},{"Start":"01:41.660 ","End":"01:46.859","Text":"y and instead of z we\u0027ll put g of x, y,"},{"Start":"01:48.260 ","End":"01:51.900","Text":"and instead of dA, sorry,"},{"Start":"01:51.900 ","End":"01:55.035","Text":"instead of dS, we have a peculiar expression,"},{"Start":"01:55.035 ","End":"02:00.620","Text":"dA in this expression is the derivative of g with respect to x squared,"},{"Start":"02:00.620 ","End":"02:04.715","Text":"the derivative of g with respect to y squared plus 1."},{"Start":"02:04.715 ","End":"02:10.500","Text":"However, you should have written here f of x, y, and z."},{"Start":"02:10.500 ","End":"02:13.020","Text":"Let\u0027s see what happens in our case."},{"Start":"02:13.020 ","End":"02:15.590","Text":"First of all, I\u0027ll show you what this notation means in"},{"Start":"02:15.590 ","End":"02:19.865","Text":"case you\u0027re not familiar with set theory and Cartesian product."},{"Start":"02:19.865 ","End":"02:23.540","Text":"This here just means that x is from the interval 0, 3,"},{"Start":"02:23.540 ","End":"02:29.450","Text":"which means that x is between 0 and 3 and that y is"},{"Start":"02:29.450 ","End":"02:36.610","Text":"between 0 and 2 and we just get this rectangle,"},{"Start":"02:36.950 ","End":"02:41.850","Text":"and this will be R and I\u0027ll shade it,"},{"Start":"02:41.850 ","End":"02:44.280","Text":"and then we\u0027ll just have to apply this formula."},{"Start":"02:44.280 ","End":"02:46.290","Text":"I\u0027ll just move this out of the way,"},{"Start":"02:46.290 ","End":"02:48.199","Text":"and applying the formula,"},{"Start":"02:48.199 ","End":"02:51.350","Text":"we get that the double integral over S,"},{"Start":"02:51.350 ","End":"02:52.370","Text":"I\u0027m just copying this,"},{"Start":"02:52.370 ","End":"02:54.120","Text":"f of x, y, z,"},{"Start":"02:54.120 ","End":"02:58.075","Text":"which is x squared yZdS,"},{"Start":"02:58.075 ","End":"03:04.895","Text":"is equal to a regular double integral over the region R. In this case,"},{"Start":"03:04.895 ","End":"03:10.890","Text":"our specific region of f,"},{"Start":"03:11.400 ","End":"03:14.260","Text":"which is just this function."},{"Start":"03:14.260 ","End":"03:17.170","Text":"What we do is just take x and y,"},{"Start":"03:17.170 ","End":"03:19.915","Text":"leave them as is, x squared y."},{"Start":"03:19.915 ","End":"03:23.575","Text":"But instead of z, we put in g of x, y,"},{"Start":"03:23.575 ","End":"03:26.435","Text":"which is this expression here,"},{"Start":"03:26.435 ","End":"03:31.455","Text":"1 plus 2x plus 3y."},{"Start":"03:31.455 ","End":"03:37.300","Text":"Then we have to multiply by this peculiar expression."},{"Start":"03:37.300 ","End":"03:40.150","Text":"Perhaps I should have written this on a new line."},{"Start":"03:40.150 ","End":"03:44.455","Text":"Let\u0027s see what\u0027s this peculiar expression under the square root."},{"Start":"03:44.455 ","End":"03:47.650","Text":"Well, we need the partial derivatives with"},{"Start":"03:47.650 ","End":"03:51.280","Text":"respect to x and with respect to y of g, which is this."},{"Start":"03:51.280 ","End":"03:53.245","Text":"I can write those here, even,"},{"Start":"03:53.245 ","End":"03:56.530","Text":"g with respect to x is just a constant."},{"Start":"03:56.530 ","End":"04:03.185","Text":"It\u0027s just 2, and g with respect to y is just the constant 3."},{"Start":"04:03.185 ","End":"04:04.880","Text":"If I put those in here,"},{"Start":"04:04.880 ","End":"04:09.540","Text":"I\u0027ve got the square root of gx squared is 2 squared"},{"Start":"04:09.950 ","End":"04:18.910","Text":"and g with respect to y is 3 squared plus 1 and then dA."},{"Start":"04:19.940 ","End":"04:22.695","Text":"Let\u0027s see what this is equal to."},{"Start":"04:22.695 ","End":"04:24.320","Text":"Well, this part here,"},{"Start":"04:24.320 ","End":"04:27.425","Text":"4 plus 9 plus 1 is 14."},{"Start":"04:27.425 ","End":"04:30.619","Text":"This is the square root of 14. It\u0027s a constant."},{"Start":"04:30.619 ","End":"04:35.214","Text":"I\u0027m just going to take the square root of 14 out in front."},{"Start":"04:35.214 ","End":"04:38.660","Text":"Now, the other thing I\u0027m going to do is instead of"},{"Start":"04:38.660 ","End":"04:42.545","Text":"writing the double integral over a region,"},{"Start":"04:42.545 ","End":"04:48.295","Text":"I\u0027ll break it up into an iterated integral dx dy, or dy dx."},{"Start":"04:48.295 ","End":"04:50.120","Text":"That doesn\u0027t really matter."},{"Start":"04:50.120 ","End":"04:51.650","Text":"A rectangle can go both ways."},{"Start":"04:51.650 ","End":"04:53.525","Text":"I\u0027ll take it as vertical slices."},{"Start":"04:53.525 ","End":"05:01.610","Text":"I\u0027ll take the outer integral as x from 0-3 and the inner integral,"},{"Start":"05:01.610 ","End":"05:07.680","Text":"I\u0027ll take y goes from 0-2,"},{"Start":"05:07.680 ","End":"05:14.585","Text":"and then we\u0027ll take this bit and then we\u0027ll put dy dx, at the end."},{"Start":"05:14.585 ","End":"05:18.455","Text":"Now this bit I can expand and get x squared"},{"Start":"05:18.455 ","End":"05:26.265","Text":"y plus 2x times this is 2x cubed y,"},{"Start":"05:26.265 ","End":"05:32.475","Text":"and then 3y times this is 3x squared y squared,"},{"Start":"05:32.475 ","End":"05:39.430","Text":"and dA becomes dy dx."},{"Start":"05:40.550 ","End":"05:44.265","Text":"This, mostly just technical now."},{"Start":"05:44.265 ","End":"05:46.665","Text":"We have a double integral,"},{"Start":"05:46.665 ","End":"05:51.390","Text":"the iterated, dy dx,"},{"Start":"05:51.390 ","End":"05:55.145","Text":"and as usual, we\u0027ll do the inner integral first."},{"Start":"05:55.145 ","End":"05:58.805","Text":"I mean, the dy integral, I\u0027ll highlight it."},{"Start":"05:58.805 ","End":"06:01.539","Text":"This is the one we\u0027re going to do first."},{"Start":"06:01.539 ","End":"06:06.440","Text":"I\u0027m going to clear some space because we don\u0027t need the pictures anymore."},{"Start":"06:06.440 ","End":"06:08.990","Text":"Let\u0027s see. You know what,"},{"Start":"06:08.990 ","End":"06:11.255","Text":"I\u0027ll do this one at the side."},{"Start":"06:11.255 ","End":"06:12.950","Text":"Somewhere down there."},{"Start":"06:12.950 ","End":"06:14.645","Text":"I don\u0027t want to interrupt the flow."},{"Start":"06:14.645 ","End":"06:16.490","Text":"This one will be,"},{"Start":"06:16.490 ","End":"06:22.170","Text":"let\u0027s see if I integrate this with respect to y. I\u0027ll get from x squared y,"},{"Start":"06:22.170 ","End":"06:28.980","Text":"I\u0027ll get 1/2x squared y squared because y gives me y squared over 2."},{"Start":"06:28.980 ","End":"06:33.810","Text":"Next, I\u0027ll get, from here,"},{"Start":"06:33.810 ","End":"06:39.075","Text":"2y will give me y squared."},{"Start":"06:39.075 ","End":"06:42.270","Text":"I\u0027ll get x cubed,"},{"Start":"06:42.270 ","End":"06:46.740","Text":"y squared with respect to y, of course."},{"Start":"06:46.740 ","End":"06:51.000","Text":"Here, from the 3y squared,"},{"Start":"06:51.000 ","End":"06:52.950","Text":"I\u0027ll get y cubed,"},{"Start":"06:52.950 ","End":"06:55.510","Text":"I\u0027ll get x squared, y cubed."},{"Start":"06:55.510 ","End":"06:58.085","Text":"That\u0027s the integral of this with respect to y."},{"Start":"06:58.085 ","End":"07:02.815","Text":"Then I have to evaluate this from 0-2."},{"Start":"07:02.815 ","End":"07:06.585","Text":"Of course, that\u0027s y goes from 0-2,"},{"Start":"07:06.585 ","End":"07:08.550","Text":"lest you get confused,"},{"Start":"07:08.550 ","End":"07:10.815","Text":"and this is equal 2."},{"Start":"07:10.815 ","End":"07:13.185","Text":"Now if I plug in 0,"},{"Start":"07:13.185 ","End":"07:16.010","Text":"I\u0027m not going to get anything when y is 0."},{"Start":"07:16.010 ","End":"07:19.175","Text":"I just need to plug in the 2."},{"Start":"07:19.175 ","End":"07:23.415","Text":"What I get is y is 2,"},{"Start":"07:23.415 ","End":"07:24.990","Text":"2 squared is 4,"},{"Start":"07:24.990 ","End":"07:26.850","Text":"4 over 2 is 2."},{"Start":"07:26.850 ","End":"07:29.400","Text":"That\u0027s 2x squared."},{"Start":"07:29.400 ","End":"07:32.940","Text":"Next one, y is 2."},{"Start":"07:32.940 ","End":"07:40.695","Text":"That\u0027s 4x cubed and then here when y is 2,"},{"Start":"07:40.695 ","End":"07:45.555","Text":"I\u0027m going to get 2 cubed is 8x squared."},{"Start":"07:45.555 ","End":"07:49.860","Text":"Of course, I can collect the 2 and the 8"},{"Start":"07:49.860 ","End":"07:57.710","Text":"together and get just 10x squared plus 4x cubed."},{"Start":"07:57.710 ","End":"08:02.500","Text":"Now I\u0027m going to take this and go back to where we came from,"},{"Start":"08:02.500 ","End":"08:06.610","Text":"and so we get the square root of 14,"},{"Start":"08:06.610 ","End":"08:11.600","Text":"the integral as x goes from 0-3,"},{"Start":"08:11.600 ","End":"08:16.575","Text":"all this bit came out to 10x"},{"Start":"08:16.575 ","End":"08:24.940","Text":"squared plus 4x cubed, and that\u0027s dx."},{"Start":"08:26.600 ","End":"08:31.720","Text":"Let\u0027s see; this is equal to square root of 14."},{"Start":"08:31.720 ","End":"08:34.780","Text":"Now the integral of this would be"},{"Start":"08:34.780 ","End":"08:44.780","Text":"10 over 3x cubed plus 4x to the 4th over 4 is just x to the 4th,"},{"Start":"08:44.780 ","End":"08:49.380","Text":"and I want to take this from 0-3."},{"Start":"08:50.780 ","End":"08:53.310","Text":"When I put in 0,"},{"Start":"08:53.310 ","End":"08:54.770","Text":"I\u0027m not going to get anything,"},{"Start":"08:54.770 ","End":"08:58.050","Text":"I\u0027m just going to get the bit with 3."},{"Start":"08:58.050 ","End":"09:04.590","Text":"I\u0027ve got square root of 14 times,"},{"Start":"09:04.590 ","End":"09:07.770","Text":"let\u0027s see, if x is 3,"},{"Start":"09:07.770 ","End":"09:11.895","Text":"3 cubed over 3 is like 3 squared, is 9,"},{"Start":"09:11.895 ","End":"09:14.235","Text":"that\u0027s going to be 90,"},{"Start":"09:14.235 ","End":"09:19.990","Text":"and 3 to the 4th is 81."},{"Start":"09:20.870 ","End":"09:25.935","Text":"Our final answer will be 90 plus 81,"},{"Start":"09:25.935 ","End":"09:32.670","Text":"is 171 times the square root of 14."},{"Start":"09:32.670 ","End":"09:38.190","Text":"I\u0027m just going to highlight this and declare that we are done."}],"ID":9653},{"Watched":false,"Name":"Exercise 2","Duration":"8m 32s","ChapterTopicVideoID":8800,"CourseChapterTopicPlaylistID":4974,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.965","Text":"In this exercise, we have to compute a type 1 surface integral."},{"Start":"00:04.965 ","End":"00:06.705","Text":"This is the 1."},{"Start":"00:06.705 ","End":"00:13.650","Text":"S is defined to be the surface where y equals x squared plus 4z."},{"Start":"00:13.650 ","End":"00:16.260","Text":"We\u0027re given restrictions on x and z."},{"Start":"00:16.260 ","End":"00:21.810","Text":"I\u0027ve sketched that x goes from 0 to 2,"},{"Start":"00:21.810 ","End":"00:24.165","Text":"and z also goes from 0 to 2."},{"Start":"00:24.165 ","End":"00:25.890","Text":"We\u0027ll give this region a name."},{"Start":"00:25.890 ","End":"00:28.455","Text":"We usually choose ROD."},{"Start":"00:28.455 ","End":"00:31.170","Text":"Note that in this exercise,"},{"Start":"00:31.170 ","End":"00:34.245","Text":"it\u0027s y which is the function of the other 2."},{"Start":"00:34.245 ","End":"00:35.730","Text":"This is some function."},{"Start":"00:35.730 ","End":"00:39.730","Text":"This one, we\u0027ll call it g of x and z."},{"Start":"00:39.800 ","End":"00:45.150","Text":"As always, the function to be integrated, that\u0027s the one I call f."},{"Start":"00:45.150 ","End":"00:51.255","Text":"So this x here is f of x, y, and z."},{"Start":"00:51.255 ","End":"00:55.860","Text":"There\u0027s 3 different formulas."},{"Start":"00:55.860 ","End":"00:58.245","Text":"One, when y is a function of x and z."},{"Start":"00:58.245 ","End":"01:01.220","Text":"The usual one is when we have z as a function of x and y,"},{"Start":"01:01.220 ","End":"01:03.290","Text":"and so on for the third one."},{"Start":"01:03.290 ","End":"01:11.440","Text":"The formula to be used in this case when y is the odd one out is as follows,"},{"Start":"01:11.690 ","End":"01:25.980","Text":"the line integral over the surface S of this function of x, y, and z dS"},{"Start":"01:25.980 ","End":"01:32.530","Text":"is equal to the regular double integral over the region R."},{"Start":"01:32.530 ","End":"01:36.640","Text":"R is the region where g is defined on"},{"Start":"01:36.640 ","End":"01:41.590","Text":"and it\u0027s also the projection of S onto the whatever plane it is,"},{"Start":"01:41.590 ","End":"01:44.360","Text":"the x, z plane in this case."},{"Start":"01:44.820 ","End":"01:50.110","Text":"What we do is we take f and we keep x as is,"},{"Start":"01:50.110 ","End":"01:56.755","Text":"but y in this case is replaced by g of the other 2 variables,"},{"Start":"01:56.755 ","End":"01:59.825","Text":"and z also as is."},{"Start":"01:59.825 ","End":"02:08.180","Text":"The dS is replaced by the square root of something like we saw in the other case, and dA."},{"Start":"02:08.220 ","End":"02:13.495","Text":"The square root is the sum of the squares of the partial derivatives,"},{"Start":"02:13.495 ","End":"02:16.120","Text":"at least in the x place,"},{"Start":"02:16.120 ","End":"02:17.320","Text":"I get g x squared,"},{"Start":"02:17.320 ","End":"02:18.430","Text":"and in the z place,"},{"Start":"02:18.430 ","End":"02:21.940","Text":"I get z squared."},{"Start":"02:21.940 ","End":"02:26.400","Text":"y is the odd one out so I just put 1 there."},{"Start":"02:26.400 ","End":"02:30.070","Text":"Of course, you could put the 1 at the beginning or the end."},{"Start":"02:30.480 ","End":"02:33.175","Text":"That\u0027s how it works in general."},{"Start":"02:33.175 ","End":"02:36.140","Text":"Let\u0027s see what happens in our case."},{"Start":"02:36.140 ","End":"02:39.090","Text":"I wanted to say, of course there\u0027s 3 such formulas."},{"Start":"02:39.090 ","End":"02:43.440","Text":"One, when we have y is a function of x and z,"},{"Start":"02:43.440 ","End":"02:45.990","Text":"and similarly for each of the other 2 variables."},{"Start":"02:45.990 ","End":"02:46.980","Text":"They\u0027re all similar."},{"Start":"02:46.980 ","End":"02:49.225","Text":"They follow the same pattern."},{"Start":"02:49.225 ","End":"02:52.580","Text":"In our case, R is a nice rectangular region."},{"Start":"02:52.580 ","End":"02:56.615","Text":"I can write it as a double iterated integral."},{"Start":"02:56.615 ","End":"03:00.650","Text":"I could do it as x and then z, or vice versa."},{"Start":"03:00.650 ","End":"03:04.609","Text":"I\u0027ll do it as x goes from 0 to 2,"},{"Start":"03:04.609 ","End":"03:07.460","Text":"z goes from 0 to 2."},{"Start":"03:07.460 ","End":"03:11.619","Text":"Here, I put f, which is the function just x."},{"Start":"03:11.619 ","End":"03:15.690","Text":"It didn\u0027t have any y in it,"},{"Start":"03:15.690 ","End":"03:18.140","Text":"but if I had a y somewhere here,"},{"Start":"03:18.140 ","End":"03:24.600","Text":"I would have to replace y by x squared plus 4z by g of x, z,"},{"Start":"03:24.600 ","End":"03:26.960","Text":"but it just didn\u0027t have any y here."},{"Start":"03:26.960 ","End":"03:30.100","Text":"Then I need the square root thing,"},{"Start":"03:30.100 ","End":"03:32.795","Text":"so I\u0027ll need g_x and g_z,"},{"Start":"03:32.795 ","End":"03:36.480","Text":"I can get those from here."},{"Start":"03:40.700 ","End":"03:42.960","Text":"This is the function g."},{"Start":"03:42.960 ","End":"03:53.465","Text":"With respect to x, it\u0027s just 2x, and with respect to z,"},{"Start":"03:53.465 ","End":"03:55.910","Text":"it\u0027s the constant 4."},{"Start":"03:55.910 ","End":"04:07.875","Text":"What I get here is g_x squared plus 1 plus 4 squared."},{"Start":"04:07.875 ","End":"04:14.110","Text":"All this is going to be dz dx instead of dA."},{"Start":"04:15.500 ","End":"04:19.305","Text":"Now, I like to pull things upfront."},{"Start":"04:19.305 ","End":"04:24.270","Text":"What I mean is this expression doesn\u0027t involve z,"},{"Start":"04:24.270 ","End":"04:27.320","Text":"and my inner integral is going to be dz."},{"Start":"04:27.320 ","End":"04:29.720","Text":"To make it simpler,"},{"Start":"04:29.720 ","End":"04:32.930","Text":"I can just pull this in front and say that this is equal"},{"Start":"04:32.930 ","End":"04:37.505","Text":"to the integral as x goes from 0 to 2 of this,"},{"Start":"04:37.505 ","End":"04:40.485","Text":"which is x times the square root."},{"Start":"04:40.485 ","End":"04:42.635","Text":"I can simplify it at the same time."},{"Start":"04:42.635 ","End":"04:48.260","Text":"This is 4x squared plus 1 plus 16, that\u0027s 17."},{"Start":"04:48.260 ","End":"04:50.945","Text":"Then times the integral,"},{"Start":"04:50.945 ","End":"04:57.880","Text":"all I\u0027m left with is 1 dz, and then dx."},{"Start":"04:57.880 ","End":"05:01.760","Text":"As usual, we start from the inside."},{"Start":"05:01.760 ","End":"05:08.865","Text":"I forgot to write that z goes from 0 to 2."},{"Start":"05:08.865 ","End":"05:11.120","Text":"When we have such an integral of 1,"},{"Start":"05:11.120 ","End":"05:14.585","Text":"it\u0027s just the upper limit minus the lower limit, 2 minus 0."},{"Start":"05:14.585 ","End":"05:18.055","Text":"This whole thing comes out to be just 2."},{"Start":"05:18.055 ","End":"05:21.740","Text":"What I end up with now is the integral"},{"Start":"05:21.740 ","End":"05:37.270","Text":"from 0 to 2 of 2x times the square root of 4x squared plus 17, dx."},{"Start":"05:37.270 ","End":"05:42.260","Text":"Now, we could solve this integration using a substitution."},{"Start":"05:42.260 ","End":"05:48.275","Text":"I could substitute 4x squared plus 17 or even the square root of this,"},{"Start":"05:48.275 ","End":"05:52.730","Text":"but I don\u0027t want to pull out the heavy artillery."},{"Start":"05:52.730 ","End":"05:55.175","Text":"There\u0027s a simpler way of doing it."},{"Start":"05:55.175 ","End":"06:00.230","Text":"There\u0027s a lot of these schemes when I use the concept of a box."},{"Start":"06:00.230 ","End":"06:08.290","Text":"If I have the integral of the square root of some function of x, call it box,"},{"Start":"06:08.290 ","End":"06:13.470","Text":"and then I have alongside, box prime,"},{"Start":"06:13.470 ","End":"06:15.690","Text":"let\u0027s say this is dx,"},{"Start":"06:15.690 ","End":"06:23.235","Text":"then this is just equal to box to the power of 3 over 2."},{"Start":"06:23.235 ","End":"06:25.965","Text":"Basically, I\u0027m thinking that this is to the power of 1/2."},{"Start":"06:25.965 ","End":"06:29.885","Text":"I raise the power by 1 and divide by 3 over 2,"},{"Start":"06:29.885 ","End":"06:33.260","Text":"which is like putting a 2/3 here."},{"Start":"06:33.260 ","End":"06:35.220","Text":"We\u0027re doing a definite integral,"},{"Start":"06:35.220 ","End":"06:37.340","Text":"but if you were just needing an indefinite integral,"},{"Start":"06:37.340 ","End":"06:42.080","Text":"you would add a plus C, which in our case we wouldn\u0027t."},{"Start":"06:42.080 ","End":"06:45.360","Text":"In our case, let\u0027s see."},{"Start":"06:45.590 ","End":"06:52.260","Text":"The box is going to be 4x squared plus 17."},{"Start":"06:52.260 ","End":"06:56.610","Text":"The box prime, would be the derivative of this would be 8x."},{"Start":"06:56.610 ","End":"06:58.290","Text":"That\u0027s not what we have."},{"Start":"06:58.290 ","End":"07:01.070","Text":"We have 2x, but you know the usual tricks,"},{"Start":"07:01.070 ","End":"07:03.295","Text":"we multiply and divide."},{"Start":"07:03.295 ","End":"07:05.910","Text":"I fix it and I write an 8 here,"},{"Start":"07:05.910 ","End":"07:07.680","Text":"but then I have to compensate,"},{"Start":"07:07.680 ","End":"07:10.425","Text":"so I write 1/4 in front,"},{"Start":"07:10.425 ","End":"07:12.905","Text":"and then I\u0027m all right because I\u0027m still with 2."},{"Start":"07:12.905 ","End":"07:15.565","Text":"Then I can use this formula."},{"Start":"07:15.565 ","End":"07:21.390","Text":"What I get is, well, the 1/4 has to stay,"},{"Start":"07:21.390 ","End":"07:27.090","Text":"and then I get 2/3 from here."},{"Start":"07:27.090 ","End":"07:31.080","Text":"Then I have to do this between,"},{"Start":"07:31.080 ","End":"07:33.600","Text":"well, let me just write it out first,"},{"Start":"07:33.600 ","End":"07:44.220","Text":"4x squared plus 17 to the power of 3 over 2."},{"Start":"07:44.220 ","End":"07:50.820","Text":"I have to take this between 0 and 2."},{"Start":"07:50.820 ","End":"07:54.795","Text":"The 1/4 times the 2/3 can stay."},{"Start":"07:54.795 ","End":"07:58.440","Text":"In fact, the 2s cancel,"},{"Start":"07:58.440 ","End":"08:00.570","Text":"I\u0027d get 1 over 2 times 3."},{"Start":"08:00.570 ","End":"08:05.925","Text":"I\u0027d get 1/6 and then I plug in the 2."},{"Start":"08:05.925 ","End":"08:08.550","Text":"If x is 2, this is 16 plus 17,"},{"Start":"08:08.550 ","End":"08:13.530","Text":"is 33 to the power of 3 over 2."},{"Start":"08:13.530 ","End":"08:19.840","Text":"I plug in 0, I\u0027ve got 17 to the 3 over 2 and I subtract."},{"Start":"08:19.850 ","End":"08:23.240","Text":"I don\u0027t think there\u0027s much simplification to be done here."},{"Start":"08:23.240 ","End":"08:25.435","Text":"You could do it on the calculator."},{"Start":"08:25.435 ","End":"08:31.390","Text":"I\u0027m just going to highlight it, and that\u0027s finished."}],"ID":9654},{"Watched":false,"Name":"Exercise 3","Duration":"12m 30s","ChapterTopicVideoID":8801,"CourseChapterTopicPlaylistID":4974,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.965","Text":"In this exercise, we have to compute this surface integral."},{"Start":"00:04.965 ","End":"00:07.965","Text":"We\u0027re given that S is part of the plane,"},{"Start":"00:07.965 ","End":"00:10.770","Text":"z is so and so."},{"Start":"00:10.770 ","End":"00:14.729","Text":"That\u0027s inside the cylinder and this is the function,"},{"Start":"00:14.729 ","End":"00:16.620","Text":"a closed curve in the (x,"},{"Start":"00:16.620 ","End":"00:21.150","Text":"y) plane but cylinder means that z extends indefinitely."},{"Start":"00:21.150 ","End":"00:25.230","Text":"I think maybe pictures could help."},{"Start":"00:25.230 ","End":"00:27.750","Text":"Here\u0027s a general picture."},{"Start":"00:27.750 ","End":"00:29.830","Text":"Whenever we have z as a function (x,"},{"Start":"00:29.830 ","End":"00:31.830","Text":"y) over some regional chain,"},{"Start":"00:31.830 ","End":"00:35.475","Text":"I prefer to call this R. In our case,"},{"Start":"00:35.475 ","End":"00:40.620","Text":"R is going to be the unit disk and I\u0027ll draw that separately."},{"Start":"00:40.620 ","End":"00:45.500","Text":"Here it is. The cylinder looked at from above,"},{"Start":"00:45.500 ","End":"00:50.340","Text":"from the z-direction just looks like a circle but inside,"},{"Start":"00:50.340 ","End":"00:54.350","Text":"so we take the interior also and get the whole unit disk and that will be"},{"Start":"00:54.350 ","End":"00:59.135","Text":"our region R and here we\u0027ll have the surface above it."},{"Start":"00:59.135 ","End":"01:05.215","Text":"In general, z is a function of x and y although x is missing here."},{"Start":"01:05.215 ","End":"01:13.550","Text":"The function I usually give the name g. Z which is g(x,"},{"Start":"01:13.550 ","End":"01:17.030","Text":"y) is y plus 3."},{"Start":"01:17.030 ","End":"01:22.915","Text":"X doesn\u0027t appear specifically you could write plus 0_x if you wanted to put x in."},{"Start":"01:22.915 ","End":"01:28.645","Text":"The double integral is over another function (x, y) and z."},{"Start":"01:28.645 ","End":"01:32.309","Text":"Here also 1 of the variables is missing but in general,"},{"Start":"01:32.309 ","End":"01:34.215","Text":"I have f (x,"},{"Start":"01:34.215 ","End":"01:40.395","Text":"y) and z is equal to y_z."},{"Start":"01:40.395 ","End":"01:43.580","Text":"X is missing but it\u0027s still a function of 3 variables."},{"Start":"01:43.580 ","End":"01:47.570","Text":"The theory says that when we\u0027re in such a situation that we have"},{"Start":"01:47.570 ","End":"01:53.630","Text":"this surface that can be projected onto this region or if you look at it the other way,"},{"Start":"01:53.630 ","End":"01:56.149","Text":"you have a function over this region,"},{"Start":"01:56.149 ","End":"01:58.490","Text":"z as a function of x and y,"},{"Start":"01:58.490 ","End":"02:07.535","Text":"then the surface integral"},{"Start":"02:07.535 ","End":"02:10.670","Text":"over S of this function (x,"},{"Start":"02:10.670 ","End":"02:20.195","Text":"y, z) dS is equal to a regular double of integral over R or in this case here,"},{"Start":"02:20.195 ","End":"02:26.410","Text":"double integral over R of f. It\u0027s also of (x,"},{"Start":"02:26.410 ","End":"02:28.130","Text":"y) but not of z."},{"Start":"02:28.130 ","End":"02:31.830","Text":"Z are replaced by g(x, y)."},{"Start":"02:32.450 ","End":"02:38.690","Text":"Then what we do is we replace dS by some peculiar expression."},{"Start":"02:38.690 ","End":"02:41.985","Text":"It\u0027s the square root of something, dA."},{"Start":"02:41.985 ","End":"02:48.140","Text":"Now, the square root it\u0027s just g with respect to x,"},{"Start":"02:48.140 ","End":"02:56.070","Text":"partial derivative squared plus g with respect to y squared plus 1."},{"Start":"02:56.070 ","End":"03:00.200","Text":"That\u0027s the standard formula when z is a function of x and y."},{"Start":"03:00.200 ","End":"03:05.355","Text":"Z is the odd one out because the 3 actual formulas,"},{"Start":"03:05.355 ","End":"03:09.140","Text":"1, when you have y as a function of x and z, and so on."},{"Start":"03:09.140 ","End":"03:11.570","Text":"Using that formula here,"},{"Start":"03:11.570 ","End":"03:14.305","Text":"we get that our integral is,"},{"Start":"03:14.305 ","End":"03:21.330","Text":"the 1 here is double integral over R which is the unit disc."},{"Start":"03:21.330 ","End":"03:24.465","Text":"Disc includes the interior circle usually,"},{"Start":"03:24.465 ","End":"03:28.830","Text":"this means the outside of f( x, y)."},{"Start":"03:28.830 ","End":"03:33.155","Text":"G means we take this but wherever we see z,"},{"Start":"03:33.155 ","End":"03:42.785","Text":"we substitute g. There is no x but I\u0027ve got y and z from here is y plus 3."},{"Start":"03:42.785 ","End":"03:48.680","Text":"It\u0027s y plus 3 and then instead of dS,"},{"Start":"03:48.680 ","End":"03:51.605","Text":"you put this funny square root thing."},{"Start":"03:51.605 ","End":"03:55.365","Text":"Leave it there for a moment and then dA."},{"Start":"03:55.365 ","End":"03:57.890","Text":"What is the square root?"},{"Start":"03:57.890 ","End":"04:02.775","Text":"G with respect to x, g is here."},{"Start":"04:02.775 ","End":"04:07.679","Text":"G doesn\u0027t depend on x so g with respect to x is 0,"},{"Start":"04:07.679 ","End":"04:11.415","Text":"g with respect to y, that is 1."},{"Start":"04:11.415 ","End":"04:16.210","Text":"That\u0027s 1 squared and here we have plus 1."},{"Start":"04:18.980 ","End":"04:23.570","Text":"This bit is the square root of 2 and I\u0027ll take it out front in a moment."},{"Start":"04:23.570 ","End":"04:26.135","Text":"In fact, I\u0027ll do that now. I\u0027ll say that\u0027s the square root of"},{"Start":"04:26.135 ","End":"04:29.795","Text":"2 times the double integral."},{"Start":"04:29.795 ","End":"04:35.285","Text":"But now I want to write y, y plus 3."},{"Start":"04:35.285 ","End":"04:37.970","Text":"I\u0027m going to do it in polar coordinates."},{"Start":"04:37.970 ","End":"04:41.645","Text":"In principle, you could do it with vertical slices,"},{"Start":"04:41.645 ","End":"04:44.510","Text":"horizontal slices, type 1, type 2 region."},{"Start":"04:44.510 ","End":"04:47.375","Text":"Since it has circular symmetry around the origin,"},{"Start":"04:47.375 ","End":"04:50.065","Text":"best to go with polar here."},{"Start":"04:50.065 ","End":"04:52.730","Text":"I\u0027ll remind you of the formulas in a moment."},{"Start":"04:52.730 ","End":"04:57.305","Text":"Just remember that what we take is that for each point,"},{"Start":"04:57.305 ","End":"05:05.525","Text":"the angle from the positive x-axis is the Theta and r is the distance from the origin."},{"Start":"05:05.525 ","End":"05:11.545","Text":"We can see that if we start from here and go all the way around and end up here,"},{"Start":"05:11.545 ","End":"05:14.040","Text":"Theta goes from 0 to 2 Pi."},{"Start":"05:14.040 ","End":"05:15.630","Text":"R, distance from the origin,"},{"Start":"05:15.630 ","End":"05:18.420","Text":"goes from 0 to 1."},{"Start":"05:18.420 ","End":"05:24.995","Text":"What we get for the limits of integration is that Theta from 0 to 2 Pi,"},{"Start":"05:24.995 ","End":"05:28.790","Text":"360 degrees, r from 0 to 1."},{"Start":"05:28.790 ","End":"05:31.315","Text":"Now we need to convert."},{"Start":"05:31.315 ","End":"05:33.930","Text":"Maybe I\u0027ll write the formulas over here."},{"Start":"05:33.930 ","End":"05:36.835","Text":"The formulas for converting to r Theta,"},{"Start":"05:36.835 ","End":"05:39.275","Text":"x equals r cosine Theta,"},{"Start":"05:39.275 ","End":"05:48.300","Text":"y equals r sine Theta and dA is rdrd Theta."},{"Start":"05:48.300 ","End":"05:51.045","Text":"We get y is"},{"Start":"05:51.045 ","End":"05:59.310","Text":"r sine Theta and then r sine Theta plus 3."},{"Start":"05:59.310 ","End":"06:05.349","Text":"Then dA, rdrd Theta."},{"Start":"06:05.470 ","End":"06:08.915","Text":"Let\u0027s see what this equals."},{"Start":"06:08.915 ","End":"06:13.625","Text":"Square root of 2 double integral."},{"Start":"06:13.625 ","End":"06:17.580","Text":"Collecting the rs and opening brackets."},{"Start":"06:17.580 ","End":"06:27.315","Text":"The first term is r cubed sine squared Theta and the second"},{"Start":"06:27.315 ","End":"06:30.960","Text":"will just be 3r squared sine"},{"Start":"06:30.960 ","End":"06:40.215","Text":"Theta drd Theta."},{"Start":"06:40.215 ","End":"06:44.400","Text":"The limits 0-1 for r,"},{"Start":"06:44.400 ","End":"06:48.325","Text":"0-2 Pi for Theta."},{"Start":"06:48.325 ","End":"06:54.850","Text":"We start from the inside and work our way outwards so the first o is this."},{"Start":"06:54.850 ","End":"06:57.975","Text":"I think I\u0027ll do this at the side."},{"Start":"06:57.975 ","End":"07:00.960","Text":"I have some room down here."},{"Start":"07:00.960 ","End":"07:03.770","Text":"What I have is 2 pieces,"},{"Start":"07:03.770 ","End":"07:07.600","Text":"I can actually break it up into 2 integrals because of this plus."},{"Start":"07:07.600 ","End":"07:15.665","Text":"I can say that I have the first bit and I can take Theta outside."},{"Start":"07:15.665 ","End":"07:19.310","Text":"It\u0027s tough with Theta in front of the brackets so I have sine"},{"Start":"07:19.310 ","End":"07:25.380","Text":"squared of Theta and then the integral"},{"Start":"07:25.380 ","End":"07:32.210","Text":"from 0-1 of r cubed dr plus the second piece where I\u0027ll take"},{"Start":"07:32.210 ","End":"07:41.280","Text":"the 3 sine Theta in front of the brackets and then I"},{"Start":"07:41.280 ","End":"07:45.855","Text":"have just the integral of r squared from 0-1"},{"Start":"07:45.855 ","End":"07:51.230","Text":"dr. That\u0027s the bit that I\u0027ve highlighted here."},{"Start":"07:51.230 ","End":"07:54.560","Text":"This integral, r to the fourth over 4,"},{"Start":"07:54.560 ","End":"07:56.780","Text":"so this gives me 1/4."},{"Start":"07:56.780 ","End":"08:03.450","Text":"From here I get 1/4 sine squared Theta and the second bit,"},{"Start":"08:03.450 ","End":"08:08.680","Text":"r cubed over 3 but from 0-1, it\u0027s just 1/3."},{"Start":"08:09.980 ","End":"08:17.190","Text":"It\u0027s 3 times 1/3 sine Theta."},{"Start":"08:17.190 ","End":"08:22.755","Text":"Of course, 3 with 1/3 cancels, it\u0027s just 1."},{"Start":"08:22.755 ","End":"08:33.545","Text":"Going back here, we have that this equals the square root of 2 integral from 0-2 Pi"},{"Start":"08:33.545 ","End":"08:37.370","Text":"of 1/4 sine squared"},{"Start":"08:37.370 ","End":"08:45.525","Text":"Theta plus sine Theta d Theta."},{"Start":"08:45.525 ","End":"08:49.375","Text":"I know how to do the integral of sine is minus cosine."},{"Start":"08:49.375 ","End":"08:51.115","Text":"What do I do with the sine squared?"},{"Start":"08:51.115 ","End":"08:54.620","Text":"Let me do this bit also as a side exercise."},{"Start":"08:54.620 ","End":"09:01.840","Text":"I have the integral of sine squared Theta d Theta."},{"Start":"09:01.840 ","End":"09:08.015","Text":"I\u0027ll use trigonometric identities because sine squared Theta is"},{"Start":"09:08.015 ","End":"09:15.070","Text":"just 1/2 of 1 minus cosine 2 Theta using 1 of the trigonometrical identities."},{"Start":"09:15.070 ","End":"09:18.025","Text":"Then we got rid of the exponent."},{"Start":"09:18.025 ","End":"09:26.910","Text":"We can say now that this is equal to the integral of 1/2 is just 1/2 of Theta."},{"Start":"09:26.910 ","End":"09:36.335","Text":"The integral of cosine"},{"Start":"09:36.335 ","End":"09:43.825","Text":"2 Theta is sine of 2 Theta but not quite,"},{"Start":"09:43.825 ","End":"09:48.185","Text":"times 1/2 because the internal derivative is 2."},{"Start":"09:48.185 ","End":"09:53.525","Text":"But we also have this 1/2 here so it\u0027s 1/2 times 1/2 sine 2 Theta."},{"Start":"09:53.525 ","End":"09:56.690","Text":"I\u0027ll put a plus c here but we don\u0027t need it."},{"Start":"09:56.690 ","End":"10:03.815","Text":"Going back here, I did just the sine squared part over here but I also have 1/4."},{"Start":"10:03.815 ","End":"10:10.299","Text":"This integral is 1/4 1/2 Theta"},{"Start":"10:10.299 ","End":"10:16.095","Text":"minus 1/4 sine 2 Theta."},{"Start":"10:16.095 ","End":"10:20.040","Text":"For sine, I have to put minus cosine Theta."},{"Start":"10:20.040 ","End":"10:22.265","Text":"That\u0027s the integral."},{"Start":"10:22.265 ","End":"10:25.210","Text":"I have to take this"},{"Start":"10:26.060 ","End":"10:34.569","Text":"between 0 and 2 Pi and then there\u0027s a square root of 2 also."},{"Start":"10:35.030 ","End":"10:38.500","Text":"What do we get? Square root of 2."},{"Start":"10:38.500 ","End":"10:41.090","Text":"Sometimes I like to break it up into"},{"Start":"10:41.090 ","End":"10:44.930","Text":"separate pieces and take each piece between 0 and 2 Pi."},{"Start":"10:44.930 ","End":"10:46.760","Text":"For example, in the first one,"},{"Start":"10:46.760 ","End":"10:52.875","Text":"I could write 1/8 Theta and evaluate that between 0 and 2 Pi."},{"Start":"10:52.875 ","End":"10:57.855","Text":"Then for the next bit I could take 1/16,"},{"Start":"10:57.855 ","End":"10:59.790","Text":"that\u0027s 1/4 times 1/4,"},{"Start":"10:59.790 ","End":"11:04.935","Text":"and then sine 2 Theta and evaluate that from 0 to 2 Pi."},{"Start":"11:04.935 ","End":"11:10.750","Text":"Then I could also subtract cosine Theta from 0 to 2 Pi."},{"Start":"11:10.750 ","End":"11:16.120","Text":"Sometimes I like to do it this way just for a change."},{"Start":"11:16.700 ","End":"11:22.255","Text":"What I get is the square root of 2 will stay."},{"Start":"11:22.255 ","End":"11:29.280","Text":"1/8 Theta from 0 to Pi,"},{"Start":"11:29.280 ","End":"11:31.440","Text":"at 0, it\u0027s nothing because it ain\u0027t."},{"Start":"11:31.440 ","End":"11:36.600","Text":"2 Pi, it\u0027s just 2 Pi over 8 which is Pi over 4."},{"Start":"11:36.600 ","End":"11:38.565","Text":"Pi over 4 is this bit."},{"Start":"11:38.565 ","End":"11:42.939","Text":"Now, Sine 2 Theta,"},{"Start":"11:42.939 ","End":"11:48.510","Text":"when Theta is 0, it\u0027s sine of 0 or I get sine of 4 Pi."},{"Start":"11:48.510 ","End":"11:49.955","Text":"Either way, it\u0027s 0."},{"Start":"11:49.955 ","End":"11:52.355","Text":"Multiples of Pi sine is 0."},{"Start":"11:52.355 ","End":"11:55.955","Text":"The 1/16 won\u0027t change the fact that this is 0."},{"Start":"11:55.955 ","End":"11:58.365","Text":"Then I have to subtract."},{"Start":"11:58.365 ","End":"12:01.540","Text":"Cosine Theta, at 2 Pi and at 0,"},{"Start":"12:01.540 ","End":"12:02.600","Text":"they\u0027re the same value."},{"Start":"12:02.600 ","End":"12:04.340","Text":"I don\u0027t even have to know what that value is."},{"Start":"12:04.340 ","End":"12:06.205","Text":"It happens to be equal to 1."},{"Start":"12:06.205 ","End":"12:08.190","Text":"But at 0 and 2 Pi,"},{"Start":"12:08.190 ","End":"12:13.600","Text":"it\u0027s equal to 1 in both cases so it\u0027s also just 0."},{"Start":"12:13.790 ","End":"12:19.155","Text":"What I end up is if I combine these 2,"},{"Start":"12:19.155 ","End":"12:30.010","Text":"is square root of 2 times Pi over 4 and that\u0027s the final answer."}],"ID":9655},{"Watched":false,"Name":"Exercise 4","Duration":"11m 27s","ChapterTopicVideoID":8802,"CourseChapterTopicPlaylistID":4974,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.370","Text":"In this exercise, we have to compute this surface integral,"},{"Start":"00:05.370 ","End":"00:10.545","Text":"and S is given to be the hemisphere."},{"Start":"00:10.545 ","End":"00:14.130","Text":"This equation is the equation of a whole sphere,"},{"Start":"00:14.130 ","End":"00:17.190","Text":"but the z bigger equal to 0"},{"Start":"00:17.190 ","End":"00:22.290","Text":"makes it just half a sphere, the 1 that\u0027s above the x, y plane."},{"Start":"00:22.290 ","End":"00:25.439","Text":"The radius is 2 because 4 is 2 squared."},{"Start":"00:25.439 ","End":"00:30.059","Text":"Now, if I look from the point of view of the z-axis"},{"Start":"00:30.059 ","End":"00:33.150","Text":"or what\u0027s the projection of this onto the x, y plane,"},{"Start":"00:33.150 ","End":"00:37.485","Text":"it will just be the disk of radius 2."},{"Start":"00:37.485 ","End":"00:45.710","Text":"Here\u0027s a picture of it and I\u0027ll label this R or d for regions or domains."},{"Start":"00:45.710 ","End":"00:49.685","Text":"Now, to actually get z as a function of x and y,"},{"Start":"00:49.685 ","End":"00:54.650","Text":"we can do because the hemisphere is actually a function."},{"Start":"00:54.650 ","End":"00:56.240","Text":"The whole sphere is not."},{"Start":"00:56.240 ","End":"00:59.780","Text":"For example, not for example, I\u0027ll show you how we do that."},{"Start":"00:59.780 ","End":"01:09.045","Text":"We could say that z squared is equal to 4 minus x squared minus y squared."},{"Start":"01:09.045 ","End":"01:13.550","Text":"Then z would normally be plus or minus the square root of this,"},{"Start":"01:13.550 ","End":"01:16.310","Text":"but because of the negative z,"},{"Start":"01:16.310 ","End":"01:20.630","Text":"we have the z equals only the plus square root,"},{"Start":"01:20.630 ","End":"01:24.215","Text":"the 4 minus x squared minus y squared."},{"Start":"01:24.215 ","End":"01:27.980","Text":"Notice that the domain is exactly this because"},{"Start":"01:27.980 ","End":"01:32.510","Text":"the domain is where what\u0027s under the square root sign is bigger or equal to 0."},{"Start":"01:32.510 ","End":"01:37.805","Text":"If I say that 4 minus x squared minus y squared is bigger or equal to 0,"},{"Start":"01:37.805 ","End":"01:43.100","Text":"this exactly gives me that x squared plus y squared is less than or equal to 4,"},{"Start":"01:43.100 ","End":"01:46.909","Text":"which is the circle together with its interior,"},{"Start":"01:46.909 ","End":"01:49.705","Text":"meaning the disk as shaded."},{"Start":"01:49.705 ","End":"01:51.845","Text":"Now we have a function,"},{"Start":"01:51.845 ","End":"01:55.200","Text":"and when we have z as a function of x and y,"},{"Start":"01:55.200 ","End":"01:59.340","Text":"I like to call that g of x, y."},{"Start":"01:59.340 ","End":"02:03.225","Text":"The function to be integrated,"},{"Start":"02:03.225 ","End":"02:06.815","Text":"this 1 here, I usually reserve the letter f for that."},{"Start":"02:06.815 ","End":"02:09.380","Text":"This is f of x, y, and z."},{"Start":"02:09.380 ","End":"02:11.960","Text":"Now we have a standard situation of"},{"Start":"02:11.960 ","End":"02:17.820","Text":"the surface projected onto the x, y plane, region R."},{"Start":"02:17.820 ","End":"02:22.040","Text":"We have the function g describing the surface"},{"Start":"02:22.040 ","End":"02:24.620","Text":"and we have the function to be integrated."},{"Start":"02:24.620 ","End":"02:33.840","Text":"We use the formula that the double integral over S of f of x, y, z."},{"Start":"02:33.840 ","End":"02:41.825","Text":"DS is equal to plain double integral over the region"},{"Start":"02:41.825 ","End":"02:46.430","Text":"of a function of 2 variables which I can get by"},{"Start":"02:46.430 ","End":"02:53.455","Text":"replacing the last 1 z by g of x, y."},{"Start":"02:53.455 ","End":"02:58.370","Text":"The dS comes out to be something a bit peculiar."},{"Start":"02:58.370 ","End":"03:01.520","Text":"It\u0027s the square root of something dA,"},{"Start":"03:01.520 ","End":"03:05.450","Text":"and that something is we take the function g,"},{"Start":"03:05.450 ","End":"03:09.140","Text":"take its partial derivative with respect to x squared plus the"},{"Start":"03:09.140 ","End":"03:13.780","Text":"partial derivative with respect to y squared and then plus 1."},{"Start":"03:13.780 ","End":"03:14.880","Text":"That\u0027s the formula."},{"Start":"03:14.880 ","End":"03:21.150","Text":"Before we continue, we would have gx and gy."},{"Start":"03:21.150 ","End":"03:23.190","Text":"Perhaps we\u0027ll do that to the side here."},{"Start":"03:23.190 ","End":"03:24.300","Text":"This is g."},{"Start":"03:24.300 ","End":"03:27.260","Text":"Now I want to differentiate with respect to x."},{"Start":"03:27.260 ","End":"03:33.445","Text":"So g with respect to x of x, y is equal to?"},{"Start":"03:33.445 ","End":"03:35.520","Text":"I have a square root."},{"Start":"03:35.520 ","End":"03:41.960","Text":"The derivative of square root is 1 over twice the square root, that\u0027s for starters."},{"Start":"03:41.960 ","End":"03:43.115","Text":"Let me just copy that,"},{"Start":"03:43.115 ","End":"03:47.490","Text":"4 minus x squared minus y squared."},{"Start":"03:47.630 ","End":"03:53.045","Text":"Then we have to multiply by the inner derivative and it\u0027s with respect to x."},{"Start":"03:53.045 ","End":"03:57.620","Text":"The inner derivative of this is minus 2x."},{"Start":"03:57.620 ","End":"04:03.400","Text":"Actually, the 2 can cancel so it makes it a bit simpler."},{"Start":"04:03.400 ","End":"04:09.470","Text":"Gy is really similar, same denominator,"},{"Start":"04:09.470 ","End":"04:12.780","Text":"4 minus x squared minus y squared,"},{"Start":"04:12.780 ","End":"04:14.970","Text":"but instead of x, I get a y."},{"Start":"04:14.970 ","End":"04:18.000","Text":"I\u0027ve already canceled the 2 that didn\u0027t bother to."},{"Start":"04:18.000 ","End":"04:20.580","Text":"Now, plugging that into here"},{"Start":"04:20.580 ","End":"04:32.980","Text":"and also plugging the value of f, which is this, what we get is,"},{"Start":"04:32.980 ","End":"04:37.850","Text":"you know what? I\u0027m thinking that I could rewrite this."},{"Start":"04:37.850 ","End":"04:41.540","Text":"I could take the x squared plus y"},{"Start":"04:41.540 ","End":"04:47.075","Text":"squared outside the brackets times z out the brackets."},{"Start":"04:47.075 ","End":"04:49.770","Text":"I think this will be a bit simpler."},{"Start":"04:49.840 ","End":"04:57.660","Text":"That we will get the double integral over the disk R."},{"Start":"04:57.660 ","End":"05:03.270","Text":"Just copying this but replacing z by g."},{"Start":"05:03.270 ","End":"05:07.919","Text":"We have x squared plus y squared."},{"Start":"05:07.919 ","End":"05:13.055","Text":"Then we have g of x, y, which is this,"},{"Start":"05:13.055 ","End":"05:18.340","Text":"the square root of 4 minus x squared minus y squared."},{"Start":"05:18.340 ","End":"05:20.030","Text":"Now we\u0027re up to here,"},{"Start":"05:20.030 ","End":"05:26.795","Text":"and now the square root of gx squared."},{"Start":"05:26.795 ","End":"05:31.100","Text":"What I\u0027ll get is minus x, when I square,"},{"Start":"05:31.100 ","End":"05:37.190","Text":"it is x squared and the denominator is just without the square root."},{"Start":"05:37.190 ","End":"05:41.330","Text":"Now, 4 minus x squared minus y squared."},{"Start":"05:41.330 ","End":"05:43.910","Text":"That\u0027s this bit squared."},{"Start":"05:43.910 ","End":"05:51.165","Text":"Now gy squared, same thing just with a y squared on top."},{"Start":"05:51.165 ","End":"05:57.120","Text":"It\u0027s also 4 minus x squared minus y squared."},{"Start":"05:57.120 ","End":"06:03.975","Text":"Then there\u0027s a plus 1 from the formula, and this is dA."},{"Start":"06:03.975 ","End":"06:09.110","Text":"The bit under the square root here I\u0027d like to just do it the side somewhere here."},{"Start":"06:09.110 ","End":"06:17.475","Text":"I\u0027ll make a common denominator and put everything over 4 minus x squared minus y squared."},{"Start":"06:17.475 ","End":"06:20.115","Text":"Then from here, I\u0027ll get x squared."},{"Start":"06:20.115 ","End":"06:22.800","Text":"From here, I\u0027ll get y squared,"},{"Start":"06:22.800 ","End":"06:27.170","Text":"and from 1, I\u0027ll just copy the numerator and"},{"Start":"06:27.170 ","End":"06:30.290","Text":"denominator and say it\u0027s like 4 minus x squared"},{"Start":"06:30.290 ","End":"06:33.590","Text":"minus y squared over 4 minus x squared minus y squared."},{"Start":"06:33.590 ","End":"06:35.225","Text":"This bit over this bit is 1."},{"Start":"06:35.225 ","End":"06:38.840","Text":"Look, the x squared cancels with"},{"Start":"06:38.840 ","End":"06:45.765","Text":"the minus x squared and the y squared with the minus y squared,"},{"Start":"06:45.765 ","End":"06:49.390","Text":"and all I\u0027m left with is 4."},{"Start":"06:49.400 ","End":"06:52.710","Text":"So getting back to here,"},{"Start":"06:52.710 ","End":"06:59.070","Text":"this is equal to x squared plus y squared."},{"Start":"06:59.070 ","End":"07:04.780","Text":"Then square root of 4 minus x squared minus y squared."},{"Start":"07:04.780 ","End":"07:07.560","Text":"Oops, I forgot to write the integral."},{"Start":"07:08.150 ","End":"07:12.500","Text":"What I want here is the square root of this thing."},{"Start":"07:12.500 ","End":"07:14.870","Text":"I can do that for numerator and denominator separately."},{"Start":"07:14.870 ","End":"07:18.175","Text":"The square root of 4 is 2."},{"Start":"07:18.175 ","End":"07:21.590","Text":"Then the square root of the denominator is just square root"},{"Start":"07:21.590 ","End":"07:25.235","Text":"of 4 minus x squared minus y squared."},{"Start":"07:25.235 ","End":"07:28.195","Text":"Again, we get lucky,"},{"Start":"07:28.195 ","End":"07:30.240","Text":"if you want to call it that."},{"Start":"07:30.240 ","End":"07:36.010","Text":"This cancels with this, and I forgotten the dA."},{"Start":"07:38.970 ","End":"07:44.980","Text":"This is an excellent example for using polar coordinates."},{"Start":"07:44.980 ","End":"07:49.800","Text":"Not only is our region circle around the origin,"},{"Start":"07:49.800 ","End":"07:54.000","Text":"but x squared plus y squared is 1 of those expressions that\u0027s easy to convert."},{"Start":"07:54.000 ","End":"08:01.665","Text":"Let me bring in the formulas for the polar conversion."},{"Start":"08:01.665 ","End":"08:03.915","Text":"There are several things,"},{"Start":"08:03.915 ","End":"08:08.490","Text":"x equals R cosine Theta."},{"Start":"08:08.490 ","End":"08:10.875","Text":"Convert everything to R and Theta."},{"Start":"08:10.875 ","End":"08:14.560","Text":"y is R sine Theta."},{"Start":"08:14.560 ","End":"08:22.350","Text":"We have to replace dA by RdRd Theta."},{"Start":"08:22.350 ","End":"08:26.375","Text":"There\u0027s a fourth equation that we often throw in because it\u0027s so useful,"},{"Start":"08:26.375 ","End":"08:29.030","Text":"that if we see x squared plus y squared,"},{"Start":"08:29.030 ","End":"08:32.400","Text":"we can replace that with R squared."},{"Start":"08:33.470 ","End":"08:38.030","Text":"That\u0027s as far as converting algebraically,"},{"Start":"08:38.030 ","End":"08:40.340","Text":"but we also have to convert the region."},{"Start":"08:40.340 ","End":"08:43.580","Text":"I have to write this region in polar terms."},{"Start":"08:43.580 ","End":"08:50.300","Text":"Well, we know how to do this to a circle centered at the origin."},{"Start":"08:50.300 ","End":"08:53.395","Text":"I\u0027ll just go ahead and write it."},{"Start":"08:53.395 ","End":"08:55.230","Text":"Whole circle,"},{"Start":"08:55.230 ","End":"08:57.505","Text":"the angle Theta goes from 0 to 2Pi."},{"Start":"08:57.505 ","End":"09:01.415","Text":"I start off with the integral from 0 to 2Pi,"},{"Start":"09:01.415 ","End":"09:04.660","Text":"and that will go with the d Theta at the end."},{"Start":"09:04.660 ","End":"09:09.320","Text":"R goes at each point for any Theta."},{"Start":"09:09.320 ","End":"09:13.595","Text":"R goes from here to here, from 0 to 2."},{"Start":"09:13.595 ","End":"09:17.090","Text":"So that\u0027s from 0 to 2, then it\u0027ll be a dR."},{"Start":"09:17.090 ","End":"09:21.245","Text":"What are we left with?"},{"Start":"09:21.245 ","End":"09:22.880","Text":"We have the 2 here."},{"Start":"09:22.880 ","End":"09:27.540","Text":"As a constant, I\u0027ll just pull that out in front."},{"Start":"09:27.540 ","End":"09:30.420","Text":"All I\u0027m left with is x squared plus y squared,"},{"Start":"09:30.420 ","End":"09:32.509","Text":"and according to this formula,"},{"Start":"09:32.509 ","End":"09:34.700","Text":"that is equal to R squared."},{"Start":"09:34.700 ","End":"09:37.190","Text":"I\u0027m left with R squared,"},{"Start":"09:37.190 ","End":"09:45.270","Text":"and then the DA here, that is from this formula, RdRd Theta."},{"Start":"09:48.500 ","End":"09:51.120","Text":"I\u0027ll do this first."},{"Start":"09:51.120 ","End":"09:56.185","Text":"I\u0027ll do this at the side somewhere where I have space up here."},{"Start":"09:56.185 ","End":"09:58.625","Text":"We have the integral from 0 to 2."},{"Start":"09:58.625 ","End":"10:02.510","Text":"R squared R is of course R cubed dR,"},{"Start":"10:02.510 ","End":"10:15.735","Text":"and that gives us R^4 over 4 from 0 to 2,"},{"Start":"10:15.735 ","End":"10:18.660","Text":"and that will equal."},{"Start":"10:18.660 ","End":"10:27.315","Text":"If I plug in 2, 2^4 over 4 minus 0^4 over 4."},{"Start":"10:27.315 ","End":"10:35.820","Text":"This is just 16 over 4 which is 4 minus 0."},{"Start":"10:35.820 ","End":"10:37.710","Text":"So this is 4."},{"Start":"10:37.710 ","End":"10:46.085","Text":"If this is 4, I can pull it out to the front and combine it with the 2 to give 8."},{"Start":"10:46.085 ","End":"10:53.565","Text":"So we get 8 times the integral of just the Theta 1."},{"Start":"10:53.565 ","End":"10:57.140","Text":"0 to 2Pi, there\u0027s nothing left."},{"Start":"10:57.140 ","End":"11:01.500","Text":"I just put it as 1 d Theta."},{"Start":"11:01.630 ","End":"11:06.650","Text":"When you have the integral of 1, just the rule of thumb,"},{"Start":"11:06.650 ","End":"11:12.440","Text":"I remember, just subtract the upper minus the lower 2Pi minus 0 is 2Pi."},{"Start":"11:12.440 ","End":"11:14.300","Text":"So it\u0027s 8 times 2Pi,"},{"Start":"11:14.300 ","End":"11:21.880","Text":"and I guess 16Pi is simplest we can go."},{"Start":"11:21.880 ","End":"11:27.180","Text":"I\u0027ll highlight it because that\u0027s our final answer, and we\u0027re done."}],"ID":9656},{"Watched":false,"Name":"Exercise 5","Duration":"20m 16s","ChapterTopicVideoID":8803,"CourseChapterTopicPlaylistID":4974,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.260","Text":"In this exercise, we have to compute a surface"},{"Start":"00:04.260 ","End":"00:12.000","Text":"integral where S is defined parametrically in terms of 2 parameters."},{"Start":"00:12.000 ","End":"00:13.920","Text":"A surface requires 2 parameters,"},{"Start":"00:13.920 ","End":"00:17.895","Text":"u and v, and it\u0027s given as follows."},{"Start":"00:17.895 ","End":"00:21.675","Text":"We\u0027re given that u and v,"},{"Start":"00:21.675 ","End":"00:27.290","Text":"the domain for them or the region that they\u0027re defined on is given by these inequalities."},{"Start":"00:27.290 ","End":"00:30.665","Text":"It\u0027s actually a rectangle and I\u0027ll sketch it in a moment."},{"Start":"00:30.665 ","End":"00:33.905","Text":"Actually we are going to do that right away."},{"Start":"00:33.905 ","End":"00:41.045","Text":"I have a u axis and a v axis and u between 1 and 2,"},{"Start":"00:41.045 ","End":"00:44.525","Text":"maybe this 1 and this is 2,"},{"Start":"00:44.525 ","End":"00:48.695","Text":"and then v between 0 and Pi over 2."},{"Start":"00:48.695 ","End":"00:51.360","Text":"Pi over 2 is about 1.5,"},{"Start":"00:51.360 ","End":"00:53.535","Text":"so maybe somewhere here."},{"Start":"00:53.535 ","End":"01:00.700","Text":"I would get the following rectangle here."},{"Start":"01:01.460 ","End":"01:03.810","Text":"This is Pi over 2,"},{"Start":"01:03.810 ","End":"01:05.985","Text":"1 and 2, and I\u0027ll shade it."},{"Start":"01:05.985 ","End":"01:13.435","Text":"Then I\u0027ll label it R and this is not the same as S. S is"},{"Start":"01:13.435 ","End":"01:21.530","Text":"the surface you get if we apply this 1 here,"},{"Start":"01:21.530 ","End":"01:24.230","Text":"with all values of u and v in here,"},{"Start":"01:24.230 ","End":"01:28.669","Text":"I get different values of R in 3D."},{"Start":"01:28.669 ","End":"01:30.245","Text":"Anyway, I\u0027m not going to sketch it."},{"Start":"01:30.245 ","End":"01:35.130","Text":"We don\u0027t need a sketch to do this problem."},{"Start":"01:35.130 ","End":"01:39.180","Text":"Okay, so this function, x, y, z,"},{"Start":"01:39.180 ","End":"01:43.640","Text":"I call that f. So we have f of x, y,"},{"Start":"01:43.640 ","End":"01:47.120","Text":"and z equaling just x, y,"},{"Start":"01:47.120 ","End":"01:52.925","Text":"z. I also give names to each of these 3 components,"},{"Start":"01:52.925 ","End":"01:57.800","Text":"but just call this bit x of u and"},{"Start":"01:57.800 ","End":"02:04.265","Text":"v. This bit here will be y of u and v,"},{"Start":"02:04.265 ","End":"02:12.620","Text":"and this here will be z of u and v. The formula we use for"},{"Start":"02:12.620 ","End":"02:21.485","Text":"the parametric when the surface is given by the 2 parameters is as follows."},{"Start":"02:21.485 ","End":"02:24.760","Text":"The surface integral of f,"},{"Start":"02:24.760 ","End":"02:27.440","Text":"well, f of x, y, z,"},{"Start":"02:27.440 ","End":"02:34.010","Text":"dS becomes the double integral of r."},{"Start":"02:34.010 ","End":"02:40.610","Text":"That\u0027s going to be a uv integral of f,"},{"Start":"02:40.610 ","End":"02:42.620","Text":"but not quite in this form."},{"Start":"02:42.620 ","End":"02:45.560","Text":"What I do is, I replace x, y,"},{"Start":"02:45.560 ","End":"02:49.310","Text":"and z by the functions of u and v,"},{"Start":"02:49.310 ","End":"02:51.020","Text":"so x of u and v,"},{"Start":"02:51.020 ","End":"02:52.865","Text":"y of u and v,"},{"Start":"02:52.865 ","End":"03:02.390","Text":"and z of u and v. Then we replace dS by"},{"Start":"03:02.390 ","End":"03:08.525","Text":"the magnitude of the vector r with respect to u"},{"Start":"03:08.525 ","End":"03:15.995","Text":"cross r with respect to v. This is the cross product of 2 vectors."},{"Start":"03:15.995 ","End":"03:17.630","Text":"Write it with an arrow."},{"Start":"03:17.630 ","End":"03:21.485","Text":"Bold is like putting an arrow over it."},{"Start":"03:21.485 ","End":"03:30.115","Text":"Finally, dA, where A is du dv or dv du."},{"Start":"03:30.115 ","End":"03:34.205","Text":"I\u0027d like to start with computing this."},{"Start":"03:34.205 ","End":"03:39.120","Text":"I just know that we forgot the other bar here."},{"Start":"03:39.120 ","End":"03:43.590","Text":"This is the magnitude of a vector."},{"Start":"03:43.590 ","End":"03:46.925","Text":"Let\u0027s spend some time computing this,"},{"Start":"03:46.925 ","End":"03:49.175","Text":"and then we\u0027ll return here."},{"Start":"03:49.175 ","End":"03:51.780","Text":"I\u0027ll do this at the side."},{"Start":"03:53.210 ","End":"03:56.370","Text":"Start by copying r,"},{"Start":"03:56.370 ","End":"04:00.310","Text":"which is u cosine v,"},{"Start":"04:01.760 ","End":"04:10.510","Text":"i plus u sine v, j plus 3uk."},{"Start":"04:12.620 ","End":"04:17.435","Text":"Now, let\u0027s do the derivative with respect to u,"},{"Start":"04:17.435 ","End":"04:21.870","Text":"so r with respect to u is equal"},{"Start":"04:21.870 ","End":"04:27.780","Text":"to v is a constant,"},{"Start":"04:27.780 ","End":"04:29.955","Text":"so we just get here"},{"Start":"04:29.955 ","End":"04:38.040","Text":"cosine v i cross."},{"Start":"04:38.040 ","End":"04:41.320","Text":"Here we\u0027ll get sine vj,"},{"Start":"04:43.430 ","End":"04:48.840","Text":"and here with respect to u, just 3k."},{"Start":"04:48.840 ","End":"04:51.670","Text":"Now with respect to v,"},{"Start":"04:51.740 ","End":"04:54.510","Text":"u is the constant."},{"Start":"04:54.510 ","End":"04:57.930","Text":"Derivative of cosine is minus sine,"},{"Start":"04:57.930 ","End":"05:05.805","Text":"and so I\u0027ve got minus u sine v times i."},{"Start":"05:05.805 ","End":"05:08.390","Text":"Then derivative of sine is cosine."},{"Start":"05:08.390 ","End":"05:15.350","Text":"So it\u0027s u cosine vj."},{"Start":"05:15.350 ","End":"05:17.895","Text":"As far as v goes through u,"},{"Start":"05:17.895 ","End":"05:21.120","Text":"just give me nothing good, just plus 0."},{"Start":"05:21.120 ","End":"05:24.585","Text":"Well, I\u0027ll just write it to show I haven\u0027t forgotten."},{"Start":"05:24.585 ","End":"05:29.055","Text":"Now the cross-product, so"},{"Start":"05:29.055 ","End":"05:37.040","Text":"ru cross with rv."},{"Start":"05:37.040 ","End":"05:39.335","Text":"There are several formulas."},{"Start":"05:39.335 ","End":"05:43.400","Text":"Let me assume that you do know what a determinant is."},{"Start":"05:43.400 ","End":"05:48.135","Text":"There is a non-determinant formula which we could look up."},{"Start":"05:48.135 ","End":"05:56.485","Text":"1 way with determinant is just to write on the top row i, j,"},{"Start":"05:56.485 ","End":"06:01.650","Text":"and k. Then in the next row,"},{"Start":"06:01.650 ","End":"06:06.030","Text":"you write the components of i, j,"},{"Start":"06:06.030 ","End":"06:07.665","Text":"and k. In this case,"},{"Start":"06:07.665 ","End":"06:13.020","Text":"cosine v. I think I need to make it larger."},{"Start":"06:13.020 ","End":"06:20.490","Text":"Cosine v, sine v and then 3,"},{"Start":"06:20.490 ","End":"06:27.555","Text":"and here minus u sine v,"},{"Start":"06:27.555 ","End":"06:34.660","Text":"u cosine v and 0."},{"Start":"06:35.030 ","End":"06:40.350","Text":"I\u0027ll expand this using the co-factor method,"},{"Start":"06:40.350 ","End":"06:44.085","Text":"expanded by the top row."},{"Start":"06:44.085 ","End":"06:54.840","Text":"The i part will be the determinant of this 2 by 2 matrix here."},{"Start":"06:55.550 ","End":"06:59.595","Text":"Let\u0027s see, I hope I gave enough room."},{"Start":"06:59.595 ","End":"07:06.320","Text":"The i part, I get sine v times 0 minus"},{"Start":"07:06.320 ","End":"07:15.690","Text":"3u cosine v. So it\u0027s just minus 3u cosine v,"},{"Start":"07:15.690 ","End":"07:17.475","Text":"and that\u0027s the i part."},{"Start":"07:17.475 ","End":"07:26.085","Text":"For the j, like I cross out the row and column with the j and I have this and this."},{"Start":"07:26.085 ","End":"07:28.535","Text":"I have to take the determinant of this,"},{"Start":"07:28.535 ","End":"07:30.995","Text":"but there\u0027s also a matter of a sine."},{"Start":"07:30.995 ","End":"07:33.095","Text":"It goes plus, minus, plus,"},{"Start":"07:33.095 ","End":"07:35.980","Text":"so I need a minus for this 1."},{"Start":"07:35.980 ","End":"07:39.830","Text":"Then I do this times this minus this times this,"},{"Start":"07:39.830 ","End":"07:44.130","Text":"cosine v times 0 is 0."},{"Start":"07:44.240 ","End":"07:50.280","Text":"Minus minus, so it\u0027s plus 3u"},{"Start":"07:50.280 ","End":"07:57.585","Text":"sine v. That\u0027s the j."},{"Start":"07:57.585 ","End":"08:06.245","Text":"For the k, that\u0027s going to be a plus just what\u0027s left is this square here,"},{"Start":"08:06.245 ","End":"08:09.325","Text":"this times this minus this times this."},{"Start":"08:09.325 ","End":"08:16.775","Text":"I get cosine v times u cosine v is u cosine"},{"Start":"08:16.775 ","End":"08:25.970","Text":"squared v minus this times this,"},{"Start":"08:25.970 ","End":"08:27.530","Text":"but this is a negative,"},{"Start":"08:27.530 ","End":"08:29.360","Text":"so that becomes plus,"},{"Start":"08:29.360 ","End":"08:36.370","Text":"and it\u0027s going to be u sine squared v. K."},{"Start":"08:39.240 ","End":"08:43.315","Text":"That is the cross product."},{"Start":"08:43.315 ","End":"08:48.620","Text":"In case you really lost two determinants I\u0027ll tell you what?"},{"Start":"08:49.410 ","End":"08:52.870","Text":"I\u0027ll write down for you a more general,"},{"Start":"08:52.870 ","End":"08:54.400","Text":"if you don\u0027t know determinants."},{"Start":"08:54.400 ","End":"08:56.515","Text":"If I have two vectors,"},{"Start":"08:56.515 ","End":"09:04.720","Text":"the first one is a_1i plus a_2j plus a_3k,"},{"Start":"09:06.570 ","End":"09:13.165","Text":"and I want to cross product this to cross multiply with b_1i"},{"Start":"09:13.165 ","End":"09:20.205","Text":"plus b_2j plus b_3"},{"Start":"09:20.205 ","End":"09:25.960","Text":"k. Then this is equal to,"},{"Start":"09:28.380 ","End":"09:31.300","Text":"in the first position,"},{"Start":"09:31.300 ","End":"09:33.190","Text":"I use 2 and 3."},{"Start":"09:33.190 ","End":"09:38.950","Text":"I got a_2, b_3 minus a_3,"},{"Start":"09:38.950 ","End":"09:44.800","Text":"b_2 i, and then I have minus."},{"Start":"09:44.800 ","End":"09:48.850","Text":"Then for this one I need 1 and 3,"},{"Start":"09:48.850 ","End":"09:52.870","Text":"so it\u0027s a_1 b_3"},{"Start":"09:52.870 ","End":"09:59.470","Text":"minus a_3b_1 j."},{"Start":"09:59.470 ","End":"10:02.560","Text":"Then plus, and the third position,"},{"Start":"10:02.560 ","End":"10:12.565","Text":"a_1 b_2 minus a_2 b_1,"},{"Start":"10:12.565 ","End":"10:19.870","Text":"and that\u0027s k. This is a formula for those who don\u0027t like determinants."},{"Start":"10:19.870 ","End":"10:27.744","Text":"You should get the same result if we do it that way where the a vector is this one."},{"Start":"10:27.744 ","End":"10:30.730","Text":"This would be my a in this case."},{"Start":"10:30.730 ","End":"10:35.240","Text":"This would be the b in this formula."},{"Start":"10:36.930 ","End":"10:39.250","Text":"Well, we\u0027re not done yet."},{"Start":"10:39.250 ","End":"10:40.660","Text":"We\u0027ve done the cross product."},{"Start":"10:40.660 ","End":"10:44.560","Text":"We now we need the magnitude,"},{"Start":"10:44.560 ","End":"10:48.160","Text":"sometimes it\u0027s called the norm of this vector."},{"Start":"10:48.160 ","End":"10:51.490","Text":"That\u0027s r_u cross r_v."},{"Start":"10:51.490 ","End":"10:59.770","Text":"Now I need the magnitude of r_u cross r_v,"},{"Start":"10:59.770 ","End":"11:04.165","Text":"which means taking the square root of"},{"Start":"11:04.165 ","End":"11:10.345","Text":"the sum of the squares of each of the components."},{"Start":"11:10.345 ","End":"11:13.580","Text":"I need this squared."},{"Start":"11:15.360 ","End":"11:24.640","Text":"I\u0027ll just write it as 3u cosine v squared."},{"Start":"11:24.640 ","End":"11:27.655","Text":"Then the second one squared."},{"Start":"11:27.655 ","End":"11:38.335","Text":"This is minus 3u sine v squared."},{"Start":"11:38.335 ","End":"11:42.745","Text":"Then the last one squared."},{"Start":"11:42.745 ","End":"11:47.530","Text":"But look a cosine squared plus sine squared is 1."},{"Start":"11:47.530 ","End":"11:50.770","Text":"This thing is just u."},{"Start":"11:50.770 ","End":"11:57.220","Text":"This last bit is equal to u because of the cosine squared plus sine squared."},{"Start":"11:57.220 ","End":"12:03.340","Text":"This is just u squared and then square root of this."},{"Start":"12:03.340 ","End":"12:05.710","Text":"Let\u0027s see what we get."},{"Start":"12:05.710 ","End":"12:07.825","Text":"If I can simplify this."},{"Start":"12:07.825 ","End":"12:16.750","Text":"Well, yeah, we could simplify this because here I get 9u squared times cosine squared,"},{"Start":"12:16.750 ","End":"12:20.410","Text":"and here I have 9u squared times sine squared."},{"Start":"12:20.410 ","End":"12:23.590","Text":"Because cosine squared plus sine squared is one,"},{"Start":"12:23.590 ","End":"12:29.905","Text":"I\u0027ll get the square root of 9u squared,"},{"Start":"12:29.905 ","End":"12:35.680","Text":"like I said, times cosine squared plus sine squared which is one, plus u squared."},{"Start":"12:35.680 ","End":"12:43.630","Text":"It comes out to be the square root of 10 times u."},{"Start":"12:43.630 ","End":"12:46.630","Text":"Well, should be the absolute value of u."},{"Start":"12:46.630 ","End":"12:51.220","Text":"But because u is positive."},{"Start":"12:51.220 ","End":"12:53.305","Text":"I mean it\u0027s between 1 and 2,"},{"Start":"12:53.305 ","End":"12:55.585","Text":"so I don\u0027t need the absolute value."},{"Start":"12:55.585 ","End":"12:58.390","Text":"All this together is square root of 10u."},{"Start":"12:58.390 ","End":"13:01.190","Text":"Let me go back up now."},{"Start":"13:01.430 ","End":"13:09.700","Text":"We figured out that this bit is the square root of 10u."},{"Start":"13:09.870 ","End":"13:15.470","Text":"That includes the norm, the magnitude."},{"Start":"13:16.980 ","End":"13:20.110","Text":"Now I\u0027m going to start computing this."},{"Start":"13:20.110 ","End":"13:22.630","Text":"Now the double integral over R,"},{"Start":"13:22.630 ","End":"13:26.665","Text":"I can break it up separately into the integral."},{"Start":"13:26.665 ","End":"13:32.035","Text":"Let\u0027s take first of all the u from 1 to 2,"},{"Start":"13:32.035 ","End":"13:42.280","Text":"and then the integral of v from zero to Pi over 2."},{"Start":"13:42.280 ","End":"13:47.705","Text":"Then I have to compute f, which is xyz."},{"Start":"13:47.705 ","End":"13:55.270","Text":"But replacing xyz maybe I\u0027ll scroll further backup so we can see something."},{"Start":"13:55.270 ","End":"14:00.039","Text":"Xyz is just, this is x."},{"Start":"14:00.039 ","End":"14:03.790","Text":"It\u0027s u cosine v,"},{"Start":"14:03.790 ","End":"14:12.980","Text":"that\u0027s x of u and v times y. Y is u sine v,"},{"Start":"14:14.670 ","End":"14:19.760","Text":"and z is 3u."},{"Start":"14:19.800 ","End":"14:22.780","Text":"That\u0027s just the x, that\u0027s the y,"},{"Start":"14:22.780 ","End":"14:30.970","Text":"that\u0027s the z from the formula f of x, y, z is xyz."},{"Start":"14:30.970 ","End":"14:33.625","Text":"Then I need this bit,"},{"Start":"14:33.625 ","End":"14:41.125","Text":"which is the square root of 10u and then dA,"},{"Start":"14:41.125 ","End":"14:45.080","Text":"in this case will be dvdu."},{"Start":"14:45.390 ","End":"14:51.490","Text":"The outer one usually matches this one,"},{"Start":"14:51.490 ","End":"14:54.730","Text":"and the inner one is the v_1."},{"Start":"14:54.730 ","End":"15:01.180","Text":"Now let\u0027s simplify this and see what we get."},{"Start":"15:01.180 ","End":"15:05.065","Text":"First of all, constants can be pulled right out in front."},{"Start":"15:05.065 ","End":"15:08.665","Text":"I have a 3 and I have a square root of 10."},{"Start":"15:08.665 ","End":"15:11.965","Text":"I can put that here."},{"Start":"15:11.965 ","End":"15:15.100","Text":"Next, let\u0027s see."},{"Start":"15:15.100 ","End":"15:17.890","Text":"Since the first integral is dv,"},{"Start":"15:17.890 ","End":"15:20.650","Text":"I\u0027m going to collect the u\u0027s together."},{"Start":"15:20.650 ","End":"15:24.444","Text":"I have u from here, from here,"},{"Start":"15:24.444 ","End":"15:29.140","Text":"from here, and from here."},{"Start":"15:29.140 ","End":"15:33.070","Text":"It looks like u^4."},{"Start":"15:33.070 ","End":"15:38.050","Text":"I can pull the u^4 in front of the integral,"},{"Start":"15:38.050 ","End":"15:48.220","Text":"which is for v. This is the u integral from 1 to 2 the v integral from 0 to Pi over 2."},{"Start":"15:48.220 ","End":"15:53.230","Text":"What I\u0027m left with just v is cosine v"},{"Start":"15:53.230 ","End":"16:03.025","Text":"sine vdv and finally a du."},{"Start":"16:03.025 ","End":"16:10.570","Text":"The inner integral with dv is this bit here,"},{"Start":"16:10.570 ","End":"16:13.405","Text":"and this is what we do first."},{"Start":"16:13.405 ","End":"16:16.975","Text":"I\u0027d like to use a trigonometric identity."},{"Start":"16:16.975 ","End":"16:24.580","Text":"I know that the trigonometric identity for 2 cosine v sine v. Let me put a 2 here."},{"Start":"16:24.580 ","End":"16:26.605","Text":"I have to compensate."},{"Start":"16:26.605 ","End":"16:29.575","Text":"I\u0027ll put over 2 here."},{"Start":"16:29.575 ","End":"16:35.030","Text":"Then this will be just this part."},{"Start":"16:35.190 ","End":"16:38.200","Text":"Again I like to do things at the side."},{"Start":"16:38.200 ","End":"16:41.140","Text":"I have got a bit of space here."},{"Start":"16:41.140 ","End":"16:50.859","Text":"The integral from 0 to Pi over 2,"},{"Start":"16:50.859 ","End":"16:59.390","Text":"the 2 cosine v sine v is just sine of 2vdv."},{"Start":"17:01.080 ","End":"17:08.560","Text":"This is equal to the integral of sine is minus cosine."},{"Start":"17:08.560 ","End":"17:12.160","Text":"But I\u0027m also going to have to divide by 2."},{"Start":"17:12.160 ","End":"17:18.385","Text":"That\u0027s going to be minus 1/2 cosine 2v"},{"Start":"17:18.385 ","End":"17:26.200","Text":"evaluated from 0 to Pi/ 2."},{"Start":"17:26.200 ","End":"17:32.005","Text":"Let\u0027s see. If I put Pi/ 2 in,"},{"Start":"17:32.005 ","End":"17:35.470","Text":"then twice Pi/2 is Pi."},{"Start":"17:35.470 ","End":"17:40.510","Text":"Cosine of Pi is minus 1."},{"Start":"17:40.510 ","End":"17:46.240","Text":"Minus 1 times minus 1/2 is plus 1/2."},{"Start":"17:46.240 ","End":"17:49.720","Text":"Now I have to subtract the bit with 0,"},{"Start":"17:49.720 ","End":"17:52.704","Text":"2v is also 0."},{"Start":"17:52.704 ","End":"17:55.825","Text":"Cosine of 0 is 1."},{"Start":"17:55.825 ","End":"17:58.375","Text":"This is minus 1/2."},{"Start":"17:58.375 ","End":"18:03.355","Text":"It\u0027s minus, minus 1/2."},{"Start":"18:03.355 ","End":"18:07.040","Text":"This just comes out to be 1."},{"Start":"18:09.840 ","End":"18:16.135","Text":"This whole highlighted the bit is just 1."},{"Start":"18:16.135 ","End":"18:18.490","Text":"Just like I can throw it out."},{"Start":"18:18.490 ","End":"18:26.725","Text":"Now what I get returning to the main thread is that all this equals,"},{"Start":"18:26.725 ","End":"18:28.425","Text":"it\u0027s a bit crowded."},{"Start":"18:28.425 ","End":"18:30.875","Text":"I\u0027ll put a separator here."},{"Start":"18:30.875 ","End":"18:40.950","Text":"Continuing here, I have 3 root 10 over 2"},{"Start":"18:40.950 ","End":"18:45.990","Text":"times the integral from"},{"Start":"18:45.990 ","End":"18:54.195","Text":"1 to 2 of u^4 du."},{"Start":"18:54.195 ","End":"18:57.749","Text":"This is equal to,"},{"Start":"18:57.749 ","End":"19:07.424","Text":"let\u0027s see, 3 root 10 over 2."},{"Start":"19:07.424 ","End":"19:11.950","Text":"Now this is u^5 over 5,"},{"Start":"19:12.300 ","End":"19:18.130","Text":"which I want to take from 1 to 2."},{"Start":"19:18.130 ","End":"19:24.160","Text":"I\u0027ll get 3 root 10 over 2."},{"Start":"19:24.160 ","End":"19:26.110","Text":"Now if I plug-in 2,"},{"Start":"19:26.110 ","End":"19:33.425","Text":"I\u0027ve got 2^5 is 32 over 5 minus 1 over 5."},{"Start":"19:33.425 ","End":"19:41.370","Text":"That will give me 31 over 5."},{"Start":"19:43.000 ","End":"19:46.265","Text":"Let\u0027s try and simplify a little bit."},{"Start":"19:46.265 ","End":"19:50.855","Text":"This is 3 times 31 is 93."},{"Start":"19:50.855 ","End":"19:57.760","Text":"Then we have root 10 over 2 times"},{"Start":"19:57.760 ","End":"20:04.490","Text":"5 is 10 and you can go a bit further and say root 10 goes into 10 root 10 times."},{"Start":"20:04.490 ","End":"20:10.350","Text":"I could write it as 93 over root 10."},{"Start":"20:10.440 ","End":"20:13.340","Text":"Finally, this was a lot of work,"},{"Start":"20:13.340 ","End":"20:16.320","Text":"but here\u0027s our final answer."}],"ID":9657},{"Watched":false,"Name":"Exercise 6","Duration":"17m 53s","ChapterTopicVideoID":8791,"CourseChapterTopicPlaylistID":4974,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.765","Text":"This exercise is very simple."},{"Start":"00:03.765 ","End":"00:10.750","Text":"Simply phrased, just compute the surface area of a sphere with a given radius."},{"Start":"00:12.950 ","End":"00:17.430","Text":"We want to use surface integrals for"},{"Start":"00:17.430 ","End":"00:21.410","Text":"this and we don\u0027t have anything written as far as equation."},{"Start":"00:21.410 ","End":"00:23.760","Text":"The first thing to do is to write the equation of"},{"Start":"00:23.760 ","End":"00:27.150","Text":"the sphere of radius r. That equation in"},{"Start":"00:27.150 ","End":"00:35.380","Text":"standard is x squared plus y squared plus z squared equals R squared."},{"Start":"00:35.440 ","End":"00:42.980","Text":"Now, I can do the surface area if I have one variable as a function of the other"},{"Start":"00:42.980 ","End":"00:49.580","Text":"2 and so what I\u0027d like to do is extract z as a function of x and y."},{"Start":"00:49.580 ","End":"00:51.050","Text":"It could have been any one of the 3,"},{"Start":"00:51.050 ","End":"00:55.145","Text":"but usually we prefer the z to be the odd one out."},{"Start":"00:55.145 ","End":"01:01.370","Text":"Now the sphere is not really a function of one variable in terms of the others."},{"Start":"01:01.370 ","End":"01:05.075","Text":"If I take z as an upper hemisphere and a lower hemisphere,"},{"Start":"01:05.075 ","End":"01:06.770","Text":"each one of them could be a function."},{"Start":"01:06.770 ","End":"01:09.320","Text":"I\u0027m going to go with the upper hemisphere."},{"Start":"01:09.320 ","End":"01:11.960","Text":"I\u0027m going to say that z equals,"},{"Start":"01:11.960 ","End":"01:18.995","Text":"I take the R squared and subtract from it x squared and y squared,"},{"Start":"01:18.995 ","End":"01:23.840","Text":"minus y squared, and then I take the square root,"},{"Start":"01:23.840 ","End":"01:32.700","Text":"but I\u0027ll take the positive square root and then that will be the upper hemisphere."},{"Start":"01:32.700 ","End":"01:35.085","Text":"The one above the x-y plane."},{"Start":"01:35.085 ","End":"01:39.020","Text":"If I compute the surface area of this,"},{"Start":"01:39.020 ","End":"01:42.570","Text":"I\u0027ll have to then multiply by 2 at the end"},{"Start":"01:42.570 ","End":"01:47.350","Text":"and not to forget to do this otherwise it\u0027ll be off by a factor of 2."},{"Start":"01:47.680 ","End":"01:53.930","Text":"Now, what I want is the domain or the projection of"},{"Start":"01:53.930 ","End":"02:00.250","Text":"this upper hemisphere onto the x-y plane and here\u0027s the sketch."},{"Start":"02:00.250 ","End":"02:07.550","Text":"I claim it\u0027s simply the disc with radius R. I\u0027ve used now the letter capital R,"},{"Start":"02:07.550 ","End":"02:12.635","Text":"so I\u0027ll call this region D. We sometimes use D for domain or other region."},{"Start":"02:12.635 ","End":"02:15.170","Text":"Why is it the whole disk?"},{"Start":"02:15.170 ","End":"02:19.070","Text":"It\u0027s actually defined for whenever what\u0027s under the square root sign is"},{"Start":"02:19.070 ","End":"02:23.045","Text":"non-negative and if you say that this thing is bigger or equal to 0,"},{"Start":"02:23.045 ","End":"02:28.385","Text":"it gives you that x squared plus y squared has to be less than or equal to R squared."},{"Start":"02:28.385 ","End":"02:30.920","Text":"If it was equal to R squared,"},{"Start":"02:30.920 ","End":"02:34.220","Text":"then it\u0027s just the unit circle,"},{"Start":"02:34.220 ","End":"02:37.415","Text":"but less than or equal to gives us the interior as well."},{"Start":"02:37.415 ","End":"02:40.850","Text":"Haven\u0027t drawn it here but you could imagine that this is the base"},{"Start":"02:40.850 ","End":"02:45.090","Text":"of the projection of a hemisphere."},{"Start":"02:45.090 ","End":"02:48.420","Text":"An upper hemisphere that lies above the x-y plane."},{"Start":"02:48.420 ","End":"02:57.510","Text":"We\u0027re getting closer. Now, I want to call this function g of x, y."},{"Start":"02:57.510 ","End":"02:59.870","Text":"Let\u0027s let the g of x,"},{"Start":"02:59.870 ","End":"03:05.270","Text":"y be the function square root of R squared minus x squared minus"},{"Start":"03:05.270 ","End":"03:11.875","Text":"y squared and we do have a formula for the integral."},{"Start":"03:11.875 ","End":"03:22.825","Text":"There\u0027s one vital step I missed and that is that the area of any surface,"},{"Start":"03:22.825 ","End":"03:32.115","Text":"S is the double integral over that surface of the constant function 1."},{"Start":"03:32.115 ","End":"03:34.885","Text":"That\u0027s a function, dS."},{"Start":"03:34.885 ","End":"03:41.720","Text":"Now in our case, we\u0027ll take S as the upper hemisphere and again,"},{"Start":"03:41.720 ","End":"03:44.870","Text":"we have to remember to multiply by 2 at the end,"},{"Start":"03:44.870 ","End":"03:51.020","Text":"and now we can use the formula for computing surface integrals."},{"Start":"03:51.020 ","End":"03:53.675","Text":"This function, although it\u0027s a constant,"},{"Start":"03:53.675 ","End":"03:55.130","Text":"it is a function of x, y,"},{"Start":"03:55.130 ","End":"04:00.585","Text":"and z, and I\u0027ll call that my function f of x, y, and z."},{"Start":"04:00.585 ","End":"04:02.315","Text":"Now we have 2 functions."},{"Start":"04:02.315 ","End":"04:06.200","Text":"The function to be integrated and the function g,"},{"Start":"04:06.200 ","End":"04:09.875","Text":"which describes the surface when z is a function of x, y."},{"Start":"04:09.875 ","End":"04:19.790","Text":"In this case, we get the general formula is that the double integral over S, in general,"},{"Start":"04:19.790 ","End":"04:23.340","Text":"it\u0027s going to be of f ds,"},{"Start":"04:23.340 ","End":"04:25.530","Text":"f of x, y, z. I just didn\u0027t write the x, y,"},{"Start":"04:25.530 ","End":"04:32.240","Text":"z, is equal to the double integral over the domain that is projected onto."},{"Start":"04:32.240 ","End":"04:33.590","Text":"We\u0027ve used the letter R before,"},{"Start":"04:33.590 ","End":"04:40.770","Text":"but we sometimes use D of f. But instead of x,"},{"Start":"04:40.770 ","End":"04:43.265","Text":"y, z, I have x, y,"},{"Start":"04:43.265 ","End":"04:48.055","Text":"and z is replaced by g of x, y."},{"Start":"04:48.055 ","End":"04:50.415","Text":"That\u0027s this one here,"},{"Start":"04:50.415 ","End":"05:00.010","Text":"and dS is replaced by the square root of some funny expression and then finally dA."},{"Start":"05:00.010 ","End":"05:03.560","Text":"This expression, I\u0027ve memorized it,"},{"Start":"05:03.560 ","End":"05:09.110","Text":"but you should have it at hand is g with"},{"Start":"05:09.110 ","End":"05:16.195","Text":"respect to x partial derivative squared plus gy squared plus 1."},{"Start":"05:16.195 ","End":"05:19.610","Text":"Let\u0027s see. We\u0027re going to do some substitution and"},{"Start":"05:19.610 ","End":"05:22.850","Text":"we\u0027re also going to need the partial derivatives."},{"Start":"05:22.850 ","End":"05:25.660","Text":"Let\u0027s see what this comes out to."},{"Start":"05:25.660 ","End":"05:32.265","Text":"This is equal to the double integral over D. Now,"},{"Start":"05:32.265 ","End":"05:34.710","Text":"f is the constant function 1."},{"Start":"05:34.710 ","End":"05:37.860","Text":"You don\u0027t have to do anything here, it\u0027s just 1."},{"Start":"05:37.860 ","End":"05:40.510","Text":"I\u0027ll write that 1 here."},{"Start":"05:40.510 ","End":"05:47.345","Text":"At times the square root of g with respect to x."},{"Start":"05:47.345 ","End":"05:48.680","Text":"Let\u0027s see what that is."},{"Start":"05:48.680 ","End":"05:52.340","Text":"We have a square root so we start off with something"},{"Start":"05:52.340 ","End":"05:59.370","Text":"over twice the square root of the same thing."},{"Start":"05:59.370 ","End":"06:03.005","Text":"R squared minus x squared minus y squared."},{"Start":"06:03.005 ","End":"06:08.120","Text":"Yeah, it\u0027s going to be a little bit messy and because it\u0027s not x,"},{"Start":"06:08.120 ","End":"06:11.480","Text":"it\u0027s a function of x we need the inner derivative,"},{"Start":"06:11.480 ","End":"06:19.895","Text":"which is minus 2x and then similarly for y. Oh yeah,"},{"Start":"06:19.895 ","End":"06:25.890","Text":"and I forgot to say this whole thing is squared and now we need something for y."},{"Start":"06:25.890 ","End":"06:29.225","Text":"Already I know I\u0027m going to have to extend this quite a bit."},{"Start":"06:29.225 ","End":"06:33.350","Text":"I\u0027ll get the same thing basically with just y on the top because"},{"Start":"06:33.350 ","End":"06:38.210","Text":"the inner derivative is going to be y minus 2y."},{"Start":"06:38.210 ","End":"06:40.234","Text":"Same thing on the bottom,"},{"Start":"06:40.234 ","End":"06:46.280","Text":"twice the square root of R squared minus x squared minus y squared squared."},{"Start":"06:46.280 ","End":"06:51.195","Text":"That\u0027s this bit and then plus 1 and dA."},{"Start":"06:51.195 ","End":"06:54.670","Text":"I want to simplify this now."},{"Start":"06:54.670 ","End":"07:00.020","Text":"I\u0027d like to do the simplification at the side not to interrupt the main flow."},{"Start":"07:00.020 ","End":"07:03.485","Text":"Let\u0027s see what\u0027s under the square root sign."},{"Start":"07:03.485 ","End":"07:06.860","Text":"What we have from the first one,"},{"Start":"07:06.860 ","End":"07:12.620","Text":"as I can cancel the 2 and I\u0027ll just"},{"Start":"07:12.620 ","End":"07:18.540","Text":"get x squared over the minus,"},{"Start":"07:18.540 ","End":"07:20.865","Text":"of course becomes a plus,"},{"Start":"07:20.865 ","End":"07:26.945","Text":"and the denominator squared is R squared minus x squared minus y squared."},{"Start":"07:26.945 ","End":"07:31.690","Text":"The next bit after canceling the 2s again and I square it,"},{"Start":"07:31.690 ","End":"07:42.080","Text":"is just plus y squared over R squared minus x squared minus y squared,"},{"Start":"07:42.080 ","End":"07:46.315","Text":"and the last one is plus 1."},{"Start":"07:46.315 ","End":"07:49.770","Text":"Now, the plus 1,"},{"Start":"07:49.770 ","End":"07:52.305","Text":"I want to put a common denominator."},{"Start":"07:52.305 ","End":"07:57.170","Text":"1 I can write as R squared minus x squared"},{"Start":"07:57.170 ","End":"08:02.690","Text":"minus y squared over R squared minus x squared minus y squared."},{"Start":"08:02.690 ","End":"08:04.505","Text":"Same thing top and bottom."},{"Start":"08:04.505 ","End":"08:08.750","Text":"That\u0027s the one and now I can do a common denominator on this"},{"Start":"08:08.750 ","End":"08:13.580","Text":"and say that this is equal to x squared plus y"},{"Start":"08:13.580 ","End":"08:18.560","Text":"squared plus R squared minus x squared minus y"},{"Start":"08:18.560 ","End":"08:25.860","Text":"squared over R squared minus x squared minus y squared."},{"Start":"08:26.080 ","End":"08:30.200","Text":"Stuff cancels again, x squared with x squared,"},{"Start":"08:30.200 ","End":"08:32.945","Text":"y squared with y squared."},{"Start":"08:32.945 ","End":"08:35.870","Text":"When I get back here,"},{"Start":"08:35.870 ","End":"08:44.490","Text":"what I\u0027m left with is the double integral over D of the square root of this."},{"Start":"08:49.820 ","End":"08:55.570","Text":"Square root of R squared is R. It\u0027s positive otherwise we\u0027d take absolute value,"},{"Start":"08:55.570 ","End":"09:01.925","Text":"over the square root of R squared minus x squared minus y squared."},{"Start":"09:01.925 ","End":"09:04.240","Text":"This is what I\u0027m putting instead of this."},{"Start":"09:04.240 ","End":"09:05.420","Text":"R is a constant,"},{"Start":"09:05.420 ","End":"09:07.190","Text":"they\u0027ll take it out to the front."},{"Start":"09:07.190 ","End":"09:10.220","Text":"I just have the integral of"},{"Start":"09:10.220 ","End":"09:19.810","Text":"1 over square root of R squared minus x squared minus y squared dA."},{"Start":"09:19.810 ","End":"09:25.295","Text":"Again I want to remind you that we have to multiply by 2 at the end."},{"Start":"09:25.295 ","End":"09:27.830","Text":"How do we do this?"},{"Start":"09:27.830 ","End":"09:31.820","Text":"If you look at the domain D,"},{"Start":"09:31.820 ","End":"09:34.910","Text":"well, it begs to be done in polar coordinates."},{"Start":"09:34.910 ","End":"09:36.770","Text":"It has this circular symmetry about"},{"Start":"09:36.770 ","End":"09:40.760","Text":"the origin and even has stuff like x squared plus y squared,"},{"Start":"09:40.760 ","End":"09:42.500","Text":"which is very good for polar."},{"Start":"09:42.500 ","End":"09:47.090","Text":"My strong recommendations is that we do this with polar coordinates."},{"Start":"09:47.090 ","End":"09:50.350","Text":"I\u0027ll just write the word Polar."},{"Start":"09:51.210 ","End":"09:53.800","Text":"Hope you remember polar."},{"Start":"09:53.800 ","End":"09:55.720","Text":"Remember, instead of x and y,"},{"Start":"09:55.720 ","End":"09:58.135","Text":"we use r and Theta."},{"Start":"09:58.135 ","End":"10:00.385","Text":"In this case, for this circle,"},{"Start":"10:00.385 ","End":"10:07.315","Text":"Theta goes all the way around from 0 to 2Pi and r goes from"},{"Start":"10:07.315 ","End":"10:14.890","Text":"0 to R. What we get when we substitute?"},{"Start":"10:14.890 ","End":"10:16.300","Text":"Well, need some formulas as well."},{"Start":"10:16.300 ","End":"10:24.040","Text":"Perhaps I\u0027ll just remind you of the polar through anyway,"},{"Start":"10:24.040 ","End":"10:25.735","Text":"I\u0027m going to write this."},{"Start":"10:25.735 ","End":"10:28.120","Text":"Well, actually there are several equations,"},{"Start":"10:28.120 ","End":"10:29.200","Text":"but we don\u0027t need all of them."},{"Start":"10:29.200 ","End":"10:30.520","Text":"I don\u0027t need x and y."},{"Start":"10:30.520 ","End":"10:38.560","Text":"All I need really is to remember is that dA is r dr d Theta."},{"Start":"10:38.560 ","End":"10:41.770","Text":"There is an equation for what x equals and for what y equals."},{"Start":"10:41.770 ","End":"10:43.840","Text":"We won\u0027t need those."},{"Start":"10:43.840 ","End":"10:50.335","Text":"Optional 4th formula, we will use is that x squared plus y squared equals r squared."},{"Start":"10:50.335 ","End":"10:52.315","Text":"I don\u0027t need the other formulas,"},{"Start":"10:52.315 ","End":"10:55.180","Text":"so I just substitute those here."},{"Start":"10:55.180 ","End":"10:57.715","Text":"I\u0027m going to convert the region D,"},{"Start":"10:57.715 ","End":"11:00.730","Text":"like I said, to Theta from 0-2Pi,"},{"Start":"11:00.730 ","End":"11:08.055","Text":"r from 0 to R. What we will get in polar is r,"},{"Start":"11:08.055 ","End":"11:10.170","Text":"the integral, like I said,"},{"Start":"11:10.170 ","End":"11:14.265","Text":"Theta from 0-2Pi, it\u0027s the full circle, 360 degrees."},{"Start":"11:14.265 ","End":"11:18.970","Text":"The radius goes from the center 0 to the perimeter,"},{"Start":"11:18.970 ","End":"11:29.065","Text":"which is R. Then I have here 1 over the square root of r squared."},{"Start":"11:29.065 ","End":"11:32.500","Text":"Now here, x squared plus y squared is r squared,"},{"Start":"11:32.500 ","End":"11:34.435","Text":"so it\u0027s minus r squared,"},{"Start":"11:34.435 ","End":"11:38.710","Text":"and dA is r dr d Theta."},{"Start":"11:38.710 ","End":"11:41.890","Text":"Why don\u0027t I just write it here as"},{"Start":"11:41.890 ","End":"11:52.030","Text":"r dr d Theta like this?"},{"Start":"11:52.030 ","End":"11:57.250","Text":"As always, we start with the inner integral."},{"Start":"11:57.250 ","End":"12:01.730","Text":"Would be this 1, the dr integral."},{"Start":"12:02.480 ","End":"12:07.080","Text":"I\u0027d like to do this integral at the side."},{"Start":"12:07.080 ","End":"12:13.960","Text":"What I have is the integral from 0 to r"},{"Start":"12:13.960 ","End":"12:20.835","Text":"of r dr over"},{"Start":"12:20.835 ","End":"12:26.075","Text":"square root of r squared minus r squared."},{"Start":"12:26.075 ","End":"12:29.440","Text":"I could do this with a substitution,"},{"Start":"12:29.440 ","End":"12:32.470","Text":"but that may be an overkill,"},{"Start":"12:32.470 ","End":"12:34.690","Text":"there is a simpler way."},{"Start":"12:34.690 ","End":"12:41.395","Text":"I\u0027d like this to be like 1 over the square root of something."},{"Start":"12:41.395 ","End":"12:45.955","Text":"Because I know how to do the integral"},{"Start":"12:45.955 ","End":"12:52.585","Text":"of 1 over the square root of something as some function of x,"},{"Start":"12:52.585 ","End":"12:56.470","Text":"provided that I have the derivative of that thing on the top."},{"Start":"12:56.470 ","End":"12:59.845","Text":"That\u0027s it. This is a function of x dx."},{"Start":"12:59.845 ","End":"13:03.950","Text":"Then the integral of this is,"},{"Start":"13:04.170 ","End":"13:09.610","Text":"I claim it\u0027s equal to twice the square root of"},{"Start":"13:09.610 ","End":"13:16.400","Text":"that box plus a constant if you\u0027re doing an indefinite integral."},{"Start":"13:18.600 ","End":"13:21.325","Text":"Well, there\u0027s several ways of looking at it,"},{"Start":"13:21.325 ","End":"13:23.590","Text":"but 1 way is to just differentiate this."},{"Start":"13:23.590 ","End":"13:25.420","Text":"The derivative of the square root of"},{"Start":"13:25.420 ","End":"13:29.140","Text":"something is 1 over twice the square root of something."},{"Start":"13:29.140 ","End":"13:32.350","Text":"Then the inner derivative and the 2s cancel."},{"Start":"13:32.350 ","End":"13:39.800","Text":"Or I could think of it as something to the minus a 1/2."},{"Start":"13:41.610 ","End":"13:44.650","Text":"Well, you could do it using exponents,"},{"Start":"13:44.650 ","End":"13:49.075","Text":"but basically, this is a basic formula and like I said,"},{"Start":"13:49.075 ","End":"13:54.640","Text":"it\u0027s because if I differentiate the square root of something,"},{"Start":"13:54.640 ","End":"14:02.395","Text":"I get 1 over twice the square root of something and then times something prime."},{"Start":"14:02.395 ","End":"14:03.880","Text":"If I didn\u0027t want the 2 here,"},{"Start":"14:03.880 ","End":"14:08.965","Text":"I can just bring the 2 over 2 here and that gives me this formula."},{"Start":"14:08.965 ","End":"14:16.915","Text":"Now, box will be R squared minus r squared,"},{"Start":"14:16.915 ","End":"14:22.510","Text":"R minus r. I need here the derivative of that,"},{"Start":"14:22.510 ","End":"14:25.675","Text":"the inner derivative is minus 2r."},{"Start":"14:25.675 ","End":"14:33.054","Text":"If I put here a minus 2,"},{"Start":"14:33.054 ","End":"14:40.765","Text":"but compensate for it by dividing by minus 2 or that\u0027s minus a 1/2 here,"},{"Start":"14:40.765 ","End":"14:43.375","Text":"the minus 2 and the minus a 1/2 cancel."},{"Start":"14:43.375 ","End":"14:45.580","Text":"That\u0027s okay."},{"Start":"14:45.580 ","End":"14:50.440","Text":"Now this is just box prime, what I have here."},{"Start":"14:50.440 ","End":"14:55.690","Text":"I\u0027ve got this template and so I can now do the integral."},{"Start":"14:55.690 ","End":"15:05.380","Text":"What we get is minus a 1/2 and then times the 2,"},{"Start":"15:05.380 ","End":"15:10.414","Text":"because this is the line I\u0027m reading off the integral,"},{"Start":"15:10.414 ","End":"15:14.100","Text":"times 2 square root of box,"},{"Start":"15:14.100 ","End":"15:18.240","Text":"which was r squared minus r squared and I\u0027m going don\u0027t need"},{"Start":"15:18.240 ","End":"15:23.690","Text":"the C because we\u0027re doing a definite integral and the 2 with the 2 will cancel."},{"Start":"15:23.690 ","End":"15:29.770","Text":"I need to take it between 0 and R. What I\u0027m looking for"},{"Start":"15:29.770 ","End":"15:36.850","Text":"is this from 0 to R. Well,"},{"Start":"15:36.850 ","End":"15:39.220","Text":"what I like to do is if there\u0027s a minus,"},{"Start":"15:39.220 ","End":"15:41.350","Text":"I like to reverse the order."},{"Start":"15:41.350 ","End":"15:46.780","Text":"What I have is the square root of R squared minus r squared."},{"Start":"15:46.780 ","End":"15:50.920","Text":"I get rid of the minus and I reverse these 2,"},{"Start":"15:50.920 ","End":"15:52.569","Text":"the order of subtraction,"},{"Start":"15:52.569 ","End":"15:54.310","Text":"it takes care of it."},{"Start":"15:54.310 ","End":"16:00.280","Text":"I go from R to 0 is what I get."},{"Start":"16:00.280 ","End":"16:02.185","Text":"Then here, let\u0027s see,"},{"Start":"16:02.185 ","End":"16:11.545","Text":"plug in 0 for r. This is the limits for r. When r is 0,"},{"Start":"16:11.545 ","End":"16:17.530","Text":"I get just square root of r squared is r. When r is R,"},{"Start":"16:17.530 ","End":"16:22.585","Text":"I get the square root of R squared minus r squared is just 0 minus 0,"},{"Start":"16:22.585 ","End":"16:27.115","Text":"which is just R. All this thing comes out to be"},{"Start":"16:27.115 ","End":"16:33.310","Text":"R. Now I go back and substitute it back here,"},{"Start":"16:33.310 ","End":"16:42.174","Text":"that this whole thing came out to be just R. I have,"},{"Start":"16:42.174 ","End":"16:45.865","Text":"let\u0027s see, R squared."},{"Start":"16:45.865 ","End":"16:47.875","Text":"This R goes with this R,"},{"Start":"16:47.875 ","End":"16:55.420","Text":"R squared times the integral from 0-2Pi. There\u0027s nothing left."},{"Start":"16:55.420 ","End":"17:00.200","Text":"It\u0027s like just a 1 d Theta."},{"Start":"17:04.350 ","End":"17:09.640","Text":"The integral of 1 is always the upper limit minus the lower limit,"},{"Start":"17:09.640 ","End":"17:12.400","Text":"so that\u0027s 2Pi minus 0 is 2Pi."},{"Start":"17:12.400 ","End":"17:17.230","Text":"This equals 2Pi R squared."},{"Start":"17:17.230 ","End":"17:19.915","Text":"But now I want to remind you we wrote it."},{"Start":"17:19.915 ","End":"17:23.545","Text":"We have to multiply by 2 at the end."},{"Start":"17:23.545 ","End":"17:27.925","Text":"This was the area of the hemisphere."},{"Start":"17:27.925 ","End":"17:33.595","Text":"Our answer is twice this,"},{"Start":"17:33.595 ","End":"17:35.230","Text":"I\u0027m multiplying by 2,"},{"Start":"17:35.230 ","End":"17:40.640","Text":"so it\u0027s 4Pi R squared."},{"Start":"17:40.740 ","End":"17:46.660","Text":"This is a well-known formula for the surface area of a sphere of"},{"Start":"17:46.660 ","End":"17:53.480","Text":"radius R. We did it with the surface integrals and got the right answer. We\u0027re done."}],"ID":9658},{"Watched":false,"Name":"Exercise 7","Duration":"12m 59s","ChapterTopicVideoID":8792,"CourseChapterTopicPlaylistID":4974,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.970","Text":"In this exercise, we\u0027re given a thin sheet,"},{"Start":"00:02.970 ","End":"00:05.970","Text":"I don\u0027t know a sheet metal or something."},{"Start":"00:05.970 ","End":"00:08.850","Text":"We neglect the thickness of it,"},{"Start":"00:08.850 ","End":"00:11.880","Text":"but it has a density,"},{"Start":"00:11.880 ","End":"00:16.290","Text":"which is like the mass per unit area."},{"Start":"00:16.290 ","End":"00:18.690","Text":"In general, it\u0027s a function of x, y,"},{"Start":"00:18.690 ","End":"00:20.400","Text":"and z, but in our particular case,"},{"Start":"00:20.400 ","End":"00:21.780","Text":"it\u0027s going to be constant."},{"Start":"00:21.780 ","End":"00:25.890","Text":"That\u0027s what we call homogeneous material,"},{"Start":"00:25.890 ","End":"00:28.620","Text":"everywhere the same density."},{"Start":"00:28.620 ","End":"00:31.710","Text":"The shape of it is part of the paraboloid,"},{"Start":"00:31.710 ","End":"00:33.330","Text":"this is the paraboloid."},{"Start":"00:33.330 ","End":"00:37.405","Text":"But we want that part that\u0027s below the plane z equals 1."},{"Start":"00:37.405 ","End":"00:39.285","Text":"We have to compute the mass."},{"Start":"00:39.285 ","End":"00:49.114","Text":"The only thing we need from physics is that the S is the surface."},{"Start":"00:49.114 ","End":"00:52.360","Text":"We need to know that the mass,"},{"Start":"00:52.360 ","End":"00:58.955","Text":"let\u0027s say m, is equal to the double integral over the surface."},{"Start":"00:58.955 ","End":"01:02.690","Text":"In general, of delta of x,"},{"Start":"01:02.690 ","End":"01:08.110","Text":"y, and z, dS."},{"Start":"01:08.900 ","End":"01:11.970","Text":"Greek letter D for density."},{"Start":"01:11.970 ","End":"01:14.670","Text":"Now in our case,"},{"Start":"01:14.670 ","End":"01:16.460","Text":"delta is a constant."},{"Start":"01:16.460 ","End":"01:18.230","Text":"This whole thing is not a function of x,"},{"Start":"01:18.230 ","End":"01:20.705","Text":"y, z at some delta naught."},{"Start":"01:20.705 ","End":"01:25.970","Text":"In our case, we get delta"},{"Start":"01:25.970 ","End":"01:33.060","Text":"naught times the double integral over S of just the function 1dS."},{"Start":"01:33.340 ","End":"01:38.195","Text":"This is actually also equal to the area of S, the part here,"},{"Start":"01:38.195 ","End":"01:43.490","Text":"but we\u0027re not going to use that now. How do I do this?"},{"Start":"01:43.490 ","End":"01:46.805","Text":"Well, let\u0027s see if we can describe what S looks like."},{"Start":"01:46.805 ","End":"01:53.150","Text":"If I take this as the z-axis and this as either the x or the y-axis,"},{"Start":"01:53.150 ","End":"01:55.955","Text":"it\u0027s actually going to represent the x-y plane."},{"Start":"01:55.955 ","End":"01:57.904","Text":"If I let x equal 0,"},{"Start":"01:57.904 ","End":"02:00.234","Text":"I\u0027ll get z equals y squared,"},{"Start":"02:00.234 ","End":"02:04.460","Text":"just the regular parabola that goes through 1,"},{"Start":"02:04.460 ","End":"02:07.350","Text":"1 and minus 1, 1."},{"Start":"02:07.350 ","End":"02:12.590","Text":"It would actually look the same if I took this to be the y-axis"},{"Start":"02:12.590 ","End":"02:18.170","Text":"or the x-axis or a sideways view of any cross-section of the x-y plane."},{"Start":"02:18.170 ","End":"02:24.855","Text":"It\u0027s a parabola, but it\u0027s a paraboloid because it\u0027s rotated."},{"Start":"02:24.855 ","End":"02:27.080","Text":"Anyway, that\u0027s just a general idea."},{"Start":"02:27.080 ","End":"02:32.450","Text":"But we only go up to the place where z equals 1."},{"Start":"02:32.450 ","End":"02:35.760","Text":"When z equals 1, we truncate it."},{"Start":"02:35.760 ","End":"02:37.860","Text":"We\u0027ve only got part of it."},{"Start":"02:37.860 ","End":"02:41.200","Text":"That\u0027s part of it up to here."},{"Start":"02:41.980 ","End":"02:52.100","Text":"Better to just shade it and it has a projection on the x-y plane,"},{"Start":"02:52.100 ","End":"02:54.410","Text":"which from the side will look like this,"},{"Start":"02:54.410 ","End":"02:59.240","Text":"but let\u0027s compute what it is in the x-y plane and sketch that."},{"Start":"02:59.240 ","End":"03:04.995","Text":"All we have to do to say that z it\u0027s below the plane z equals 1,"},{"Start":"03:04.995 ","End":"03:07.260","Text":"is to say that this z,"},{"Start":"03:07.260 ","End":"03:14.315","Text":"x squared plus y squared is less than or equal to 1 below or including."},{"Start":"03:14.315 ","End":"03:19.555","Text":"X squared plus y squared less than or equal to 1 is just the unit disk."},{"Start":"03:19.555 ","End":"03:24.635","Text":"This yellow bit is this yellow bit and I\u0027ll label it,"},{"Start":"03:24.635 ","End":"03:26.450","Text":"say off a region."},{"Start":"03:26.450 ","End":"03:29.570","Text":"That\u0027s x squared plus y squared less than or equal to 1, and of course,"},{"Start":"03:29.570 ","End":"03:37.395","Text":"the circle itself as x squared plus y squared equals 1 on the circumference."},{"Start":"03:37.395 ","End":"03:42.880","Text":"Now we want to do the integral to convert"},{"Start":"03:42.880 ","End":"03:47.710","Text":"from a surface integral over the surface to the regular"},{"Start":"03:47.710 ","End":"03:57.640","Text":"double integral in the x-y plane for the case where z is in terms of x and y."},{"Start":"03:57.640 ","End":"04:01.735","Text":"There are other formulas where x is given in terms of y, and z, and so on."},{"Start":"04:01.735 ","End":"04:08.175","Text":"Let\u0027s say this is equal to g of x, y and this,"},{"Start":"04:08.175 ","End":"04:11.410","Text":"the function, is f of x,"},{"Start":"04:11.410 ","End":"04:13.154","Text":"y, and z in general,"},{"Start":"04:13.154 ","End":"04:15.415","Text":"but in our case, it\u0027s a constant."},{"Start":"04:15.415 ","End":"04:24.079","Text":"In general, we have that the double integral over S of f of x, y,"},{"Start":"04:24.079 ","End":"04:29.060","Text":"z. dS is equal to the double integral over d,"},{"Start":"04:29.060 ","End":"04:39.150","Text":"which is the projection of this onto the x-y plane of f without z,"},{"Start":"04:39.150 ","End":"04:42.630","Text":"z is replaced by g of x,"},{"Start":"04:42.630 ","End":"04:45.900","Text":"y as in our case."},{"Start":"04:45.900 ","End":"04:48.609","Text":"This is it. Instead of dS,"},{"Start":"04:48.609 ","End":"04:50.300","Text":"we put some expression,"},{"Start":"04:50.300 ","End":"04:56.160","Text":"I\u0027ll write it in a moment of any expression with square roots, dA."},{"Start":"04:56.160 ","End":"05:00.140","Text":"The expression here is g with respect to x"},{"Start":"05:00.140 ","End":"05:05.800","Text":"squared plus g with respect to y squared plus 1."},{"Start":"05:05.800 ","End":"05:10.025","Text":"We need to know the partial derivatives where we can compute those easily."},{"Start":"05:10.025 ","End":"05:13.640","Text":"In fact, why don\u0027t we just go right ahead and substitute now."},{"Start":"05:13.640 ","End":"05:16.550","Text":"What we get will be,"},{"Start":"05:16.550 ","End":"05:17.885","Text":"I\u0027ll write it over here,"},{"Start":"05:17.885 ","End":"05:22.170","Text":"the double integral over"},{"Start":"05:23.930 ","End":"05:31.655","Text":"R. Often we use the letter D for domains instead of 1/2 a region."},{"Start":"05:31.655 ","End":"05:35.880","Text":"Now, f of anything is just 1."},{"Start":"05:37.910 ","End":"05:46.865","Text":"It\u0027s just 1 times the square root of g with respect to x,"},{"Start":"05:46.865 ","End":"05:51.585","Text":"which is 2x squared,"},{"Start":"05:51.585 ","End":"05:55.335","Text":"g with respect to y is 2y squared,"},{"Start":"05:55.335 ","End":"05:59.395","Text":"plus 1, and dA."},{"Start":"05:59.395 ","End":"06:08.105","Text":"This is just the double integral over S. I still have to take care of the delta naught."},{"Start":"06:08.105 ","End":"06:09.455","Text":"I\u0027ll do that at the end,"},{"Start":"06:09.455 ","End":"06:12.660","Text":"just let\u0027s not forget to multiply by"},{"Start":"06:12.660 ","End":"06:19.155","Text":"that because I\u0027m just taking this bit for the moment."},{"Start":"06:19.155 ","End":"06:23.930","Text":"We have now is, I\u0027ll simplify this,"},{"Start":"06:23.930 ","End":"06:27.560","Text":"but I\u0027ll also convert this to polar coordinates"},{"Start":"06:27.560 ","End":"06:32.295","Text":"because it\u0027s so obviously fitted for polar,"},{"Start":"06:32.295 ","End":"06:37.625","Text":"not only is this region doesn\u0027t have a circular symmetry,"},{"Start":"06:37.625 ","End":"06:41.360","Text":"but we\u0027ll even get expressions with x squared plus y squared,"},{"Start":"06:41.360 ","End":"06:43.450","Text":"which is very good for polar."},{"Start":"06:43.450 ","End":"06:47.780","Text":"I\u0027ll just remind you briefly about polar is that in polar,"},{"Start":"06:47.780 ","End":"06:51.300","Text":"every point has an r and a Theta."},{"Start":"06:52.550 ","End":"06:57.805","Text":"Any point in this disk will have a Theta between 0 and 2Pi,"},{"Start":"06:57.805 ","End":"07:02.160","Text":"the whole circle and an r between 0 and 1."},{"Start":"07:02.510 ","End":"07:12.500","Text":"This becomes the integral where Theta goes from 0-2Pi and r goes from 0-1."},{"Start":"07:12.500 ","End":"07:14.650","Text":"Now we need some formulas."},{"Start":"07:14.650 ","End":"07:17.229","Text":"Now this bit here,"},{"Start":"07:17.229 ","End":"07:19.940","Text":"I could rewrite it,"},{"Start":"07:19.940 ","End":"07:23.620","Text":"4x squared plus 4y squared plus 1."},{"Start":"07:23.620 ","End":"07:30.100","Text":"What\u0027s under the square root is 4x squared plus 4y squared plus 1."},{"Start":"07:30.100 ","End":"07:32.410","Text":"When I convert to polar,"},{"Start":"07:32.410 ","End":"07:34.880","Text":"there\u0027s a whole list of formulas."},{"Start":"07:35.060 ","End":"07:37.420","Text":"To the couple, I won\u0027t be using."},{"Start":"07:37.420 ","End":"07:40.450","Text":"There is an x equals r cosine Theta,"},{"Start":"07:40.450 ","End":"07:41.470","Text":"I don\u0027t need that."},{"Start":"07:41.470 ","End":"07:45.280","Text":"X equals r sine Theta, don\u0027t need that."},{"Start":"07:45.280 ","End":"07:47.940","Text":"dA equals r, dr,"},{"Start":"07:47.940 ","End":"07:50.880","Text":"d Theta, yes, I need that."},{"Start":"07:50.880 ","End":"07:55.740","Text":"There\u0027s a 4th 1 where x squared plus y squared equals r squared,"},{"Start":"07:55.740 ","End":"08:02.984","Text":"very useful because now I can write this as the square root."},{"Start":"08:02.984 ","End":"08:05.640","Text":"Now I have 4x squared plus 4y squared,"},{"Start":"08:05.640 ","End":"08:07.880","Text":"so that\u0027s just 4r squared."},{"Start":"08:07.880 ","End":"08:09.905","Text":"I can take the 4 outside the brackets."},{"Start":"08:09.905 ","End":"08:20.610","Text":"I have 4r squared plus 1 and then the dA is replaced by r, dr, d Theta."},{"Start":"08:20.610 ","End":"08:22.440","Text":"I don\u0027t need pictures anymore,"},{"Start":"08:22.440 ","End":"08:23.805","Text":"now it\u0027s all tactical."},{"Start":"08:23.805 ","End":"08:27.240","Text":"Just let\u0027s remember to multiply"},{"Start":"08:27.240 ","End":"08:31.985","Text":"the delta naught at the end because we\u0027re just doing this part."},{"Start":"08:31.985 ","End":"08:39.080","Text":"Continuing, we start with the innermost integral,"},{"Start":"08:39.080 ","End":"08:42.990","Text":"the dr, this 1 here."},{"Start":"08:43.000 ","End":"08:46.040","Text":"I\u0027ll do this at the side."},{"Start":"08:46.040 ","End":"08:57.760","Text":"I want to compute the integral from 0-1 of the square root of 4r squared plus 1."},{"Start":"08:57.760 ","End":"09:02.470","Text":"Now, instead of writing r, dr,"},{"Start":"09:02.470 ","End":"09:08.390","Text":"I would like to have here the derivative of what\u0027s inside."},{"Start":"09:08.390 ","End":"09:11.470","Text":"I would like to have 8r here."},{"Start":"09:11.470 ","End":"09:13.395","Text":"Now if I write 8r,"},{"Start":"09:13.395 ","End":"09:15.220","Text":"dr instead of r, dr,"},{"Start":"09:15.220 ","End":"09:20.779","Text":"I\u0027ve spoiled it, so I have to compensate by writing 1/8th in front."},{"Start":"09:20.779 ","End":"09:23.795","Text":"Now why do I want to do this?"},{"Start":"09:23.795 ","End":"09:27.540","Text":"Because if this part here,"},{"Start":"09:27.540 ","End":"09:33.750","Text":"I\u0027ll call it box then the 8r is box prime."},{"Start":"09:33.750 ","End":"09:40.685","Text":"There\u0027s a general rule that the integral of square root"},{"Start":"09:40.685 ","End":"09:47.610","Text":"of box times box prime, it\u0027s dx."},{"Start":"09:47.610 ","End":"09:55.470","Text":"Let\u0027s say that they\u0027re both boxes of function of x in general or in this case,"},{"Start":"09:55.470 ","End":"09:59.130","Text":"r. I\u0027ll just leave that out."},{"Start":"09:59.130 ","End":"10:06.530","Text":"Is equal to 2/3 box to the power of 3 over 2."},{"Start":"10:06.530 ","End":"10:08.615","Text":"In general, plus a constant,"},{"Start":"10:08.615 ","End":"10:11.390","Text":"we don\u0027t need the constant because we\u0027re doing a definite integral."},{"Start":"10:11.390 ","End":"10:16.080","Text":"The reason for this is that the square root is just to the power of 1/2."},{"Start":"10:16.080 ","End":"10:17.920","Text":"I raise the power by 1,"},{"Start":"10:17.920 ","End":"10:22.285","Text":"I get 3 over 2 and divide by 3 over 2, it\u0027s 2/3."},{"Start":"10:22.285 ","End":"10:32.450","Text":"Using this template, I can write this as 1/8th times 2/3,"},{"Start":"10:32.450 ","End":"10:36.760","Text":"times the box to the power of 3 over 2,"},{"Start":"10:36.760 ","End":"10:40.570","Text":"which is 4r squared plus 1 to the power of 3 over 2."},{"Start":"10:40.570 ","End":"10:41.715","Text":"I don\u0027t need the constant."},{"Start":"10:41.715 ","End":"10:46.285","Text":"It\u0027s a definite integral and I\u0027m taking it from 0-1."},{"Start":"10:46.285 ","End":"10:50.210","Text":"Now, let\u0027s see what this equals, of course,"},{"Start":"10:50.210 ","End":"10:53.170","Text":"2 into 8 goes 4,"},{"Start":"10:53.300 ","End":"11:00.265","Text":"this bit comes out 1/12th and then I need to subtract something."},{"Start":"11:00.265 ","End":"11:02.499","Text":"When I substitute 1,"},{"Start":"11:02.499 ","End":"11:06.345","Text":"I\u0027ve got 4 plus 1 is 5,"},{"Start":"11:06.345 ","End":"11:09.825","Text":"5 to the power of 3 over 2."},{"Start":"11:09.825 ","End":"11:12.900","Text":"When I substitute 0,"},{"Start":"11:12.900 ","End":"11:16.815","Text":"I\u0027ve got 1 to the power of 3 over 2."},{"Start":"11:16.815 ","End":"11:19.955","Text":"1 to the power of anything is just 1."},{"Start":"11:19.955 ","End":"11:23.960","Text":"That\u0027s this part here."},{"Start":"11:23.960 ","End":"11:29.980","Text":"Continuing here, it is a constant,"},{"Start":"11:29.980 ","End":"11:32.810","Text":"and I can take it out in front of the integral,"},{"Start":"11:32.810 ","End":"11:39.945","Text":"so 1/12th of 5 to the 3 over 2 minus 1"},{"Start":"11:39.945 ","End":"11:49.095","Text":"times the integral from 0-2 Pi of just d Theta or 1d Theta."},{"Start":"11:49.095 ","End":"11:55.660","Text":"Now, the integral of 1 is just the upper minus the lower limit,"},{"Start":"11:55.660 ","End":"11:58.090","Text":"so this whole thing comes out to be 2Pi."},{"Start":"11:58.090 ","End":"12:02.350","Text":"I can note that that\u0027s 2Pi."},{"Start":"12:03.770 ","End":"12:07.275","Text":"If I throw in the Delta naught,"},{"Start":"12:07.275 ","End":"12:11.860","Text":"I get that the mass equals"},{"Start":"12:12.200 ","End":"12:22.865","Text":"1/12th times 5 to the 3 over 2 minus 1 times 2Pi."},{"Start":"12:22.865 ","End":"12:27.865","Text":"I suppose I should really combine the 2 with the 12."},{"Start":"12:27.865 ","End":"12:30.910","Text":"Let\u0027s just write it there and then I\u0027ll put the Pi on top."},{"Start":"12:30.910 ","End":"12:37.840","Text":"It\u0027s Pi over 6 times 5 to the 3 over 2 minus 1."},{"Start":"12:37.840 ","End":"12:40.675","Text":"I\u0027m not going to compute it numerically."},{"Start":"12:40.675 ","End":"12:42.570","Text":"Oh gosh, I almost forgot,"},{"Start":"12:42.570 ","End":"12:44.790","Text":"the Delta naught of course,"},{"Start":"12:44.790 ","End":"12:49.300","Text":"comes in here too, very important."},{"Start":"12:50.150 ","End":"12:52.380","Text":"This is now the answer."},{"Start":"12:52.380 ","End":"12:58.930","Text":"I\u0027ll highlight it and we are done."}],"ID":9659},{"Watched":false,"Name":"Exercise 8","Duration":"18m 59s","ChapterTopicVideoID":8793,"CourseChapterTopicPlaylistID":4974,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.045","Text":"In this exercise, we have to compute,"},{"Start":"00:03.045 ","End":"00:04.770","Text":"as we often do,"},{"Start":"00:04.770 ","End":"00:11.580","Text":"F.ndS over the surface Sn is as usual,"},{"Start":"00:11.580 ","End":"00:15.600","Text":"the outward unit normal on the surface S."},{"Start":"00:15.600 ","End":"00:22.245","Text":"Here we\u0027re given the details that the vector field F is this monster."},{"Start":"00:22.245 ","End":"00:25.980","Text":"3 components, the i component,"},{"Start":"00:25.980 ","End":"00:28.965","Text":"the j component and the k component."},{"Start":"00:28.965 ","End":"00:38.280","Text":"We\u0027re given that S is the open surface defined by this function z of x and y."},{"Start":"00:38.280 ","End":"00:45.994","Text":"It\u0027s a paraboloid and upside down paraboloid and z bigger or equal to 0,"},{"Start":"00:45.994 ","End":"00:48.200","Text":"limits it to the x,"},{"Start":"00:48.200 ","End":"00:50.030","Text":"y plane and above."},{"Start":"00:50.030 ","End":"00:51.995","Text":"This is the sketch I brought,"},{"Start":"00:51.995 ","End":"00:55.040","Text":"from the side it just looks like a parabola."},{"Start":"00:55.040 ","End":"00:57.605","Text":"What I\u0027d like to know is,"},{"Start":"00:57.605 ","End":"01:05.165","Text":"what is this part here which is not included in the sketch?"},{"Start":"01:05.165 ","End":"01:08.250","Text":"Put some dotted lines here."},{"Start":"01:08.500 ","End":"01:13.700","Text":"This is where the surface cuts the x, y plane."},{"Start":"01:13.700 ","End":"01:15.934","Text":"If I let z equals 0,"},{"Start":"01:15.934 ","End":"01:20.285","Text":"what I would get would be that the part below,"},{"Start":"01:20.285 ","End":"01:21.755","Text":"let me call it."},{"Start":"01:21.755 ","End":"01:24.305","Text":"I\u0027ll give it a name D for disk."},{"Start":"01:24.305 ","End":"01:33.275","Text":"It\u0027s a disk of radius 2 would be x squared plus y squared."},{"Start":"01:33.275 ","End":"01:36.980","Text":"Well, the boundaries where it\u0027s equal to 4,"},{"Start":"01:36.980 ","End":"01:41.760","Text":"but we want the interior also so less than or equal to 4."},{"Start":"01:41.760 ","End":"01:44.160","Text":"Since 4 is 2 squared,"},{"Start":"01:44.160 ","End":"01:47.985","Text":"then this is why I put the 2s here."},{"Start":"01:47.985 ","End":"01:52.550","Text":"When x, y is the origin,"},{"Start":"01:52.550 ","End":"01:54.530","Text":"then z is equal to 4,"},{"Start":"01:54.530 ","End":"01:57.410","Text":"4 minus 0 minus 0. this is the right picture."},{"Start":"01:57.410 ","End":"02:02.060","Text":"Then should like to also put in extra picture of just"},{"Start":"02:02.060 ","End":"02:07.920","Text":"this D. Here\u0027s the disk D in the x, y plane."},{"Start":"02:07.920 ","End":"02:12.680","Text":"This surface is S. Now,"},{"Start":"02:12.680 ","End":"02:15.650","Text":"S is an open surface,"},{"Start":"02:15.650 ","End":"02:18.695","Text":"but if I cap it with D,"},{"Start":"02:18.695 ","End":"02:21.859","Text":"then together it will be a closed surface."},{"Start":"02:21.859 ","End":"02:25.820","Text":"The reason I\u0027m concerned with this is I want to use the divergence theorem,"},{"Start":"02:25.820 ","End":"02:31.459","Text":"and I want to take the whole surface of a 3D body,"},{"Start":"02:31.459 ","End":"02:34.920","Text":"to be like the solid paraboloid."},{"Start":"02:36.050 ","End":"02:43.445","Text":"What I\u0027m going to do is use the divergence theorem and say that the double integral"},{"Start":"02:43.445 ","End":"02:52.200","Text":"over S plus D could use the symbol plus,"},{"Start":"02:52.200 ","End":"02:56.310","Text":"sometimes use the union symbol,"},{"Start":"02:56.310 ","End":"02:59.400","Text":"perhaps better to use this is not a U,"},{"Start":"02:59.400 ","End":"03:02.250","Text":"this is the union."},{"Start":"03:02.250 ","End":"03:05.010","Text":"But they don\u0027t have any overlap,"},{"Start":"03:05.010 ","End":"03:08.190","Text":"except the circle itself."},{"Start":"03:08.190 ","End":"03:12.980","Text":"That\u0027s why the double integral will be the sum of the integral"},{"Start":"03:12.980 ","End":"03:18.350","Text":"over S plus the integral over D. Well, we\u0027ll get to that."},{"Start":"03:18.350 ","End":"03:23.240","Text":"But first of all, this is going to of"},{"Start":"03:23.240 ","End":"03:33.930","Text":"F.n dS will be by the divergence theorem, the triple integral."},{"Start":"03:34.000 ","End":"03:39.395","Text":"Let\u0027s give the 3D body,"},{"Start":"03:39.395 ","End":"03:49.600","Text":"meaning the volume trapped inside between S and D. I\u0027ll call that B for body."},{"Start":"03:49.700 ","End":"04:00.190","Text":"This solid object B of the divergence of F dV."},{"Start":"04:00.190 ","End":"04:03.815","Text":"That\u0027s what the divergence theorem will give us."},{"Start":"04:03.815 ","End":"04:08.270","Text":"The reason that this will help us is that,"},{"Start":"04:08.270 ","End":"04:10.370","Text":"lets show you with the side,"},{"Start":"04:10.370 ","End":"04:19.600","Text":"this surface integral will equal the surface integral"},{"Start":"04:19.600 ","End":"04:24.130","Text":"over S of whatever it is plus the surface integral"},{"Start":"04:24.130 ","End":"04:29.050","Text":"over the disk D. Because when we have this union,"},{"Start":"04:29.050 ","End":"04:32.880","Text":"since they don\u0027t really overlap except on the circle itself,"},{"Start":"04:32.880 ","End":"04:36.585","Text":"but that has no consequence,"},{"Start":"04:36.585 ","End":"04:45.450","Text":"this will equal this triple integral over B of whatever it is."},{"Start":"04:45.450 ","End":"04:50.545","Text":"Then what I can do is do a subtraction and compute this by saying"},{"Start":"04:50.545 ","End":"04:59.255","Text":"that this will be this minus this at the end."},{"Start":"04:59.255 ","End":"05:04.810","Text":"I\u0027ll be able to say, I just write down that the integral that"},{"Start":"05:04.810 ","End":"05:10.070","Text":"we want over S will be the triple integral over B."},{"Start":"05:10.070 ","End":"05:19.470","Text":"It\u0027s 1 computation minus the surface integral over the cap at the bottom,"},{"Start":"05:19.470 ","End":"05:24.380","Text":"D. Now this will be fairly easy to compute."},{"Start":"05:24.380 ","End":"05:29.200","Text":"It\u0027s better to do 2 integrals and subtract them rather than try and do"},{"Start":"05:29.200 ","End":"05:33.880","Text":"it directly over the open paraboloid,"},{"Start":"05:33.880 ","End":"05:36.950","Text":"this would be quite difficult."},{"Start":"05:37.880 ","End":"05:40.070","Text":"Let\u0027s get started."},{"Start":"05:40.070 ","End":"05:43.740","Text":"I\u0027ll do the triple integral first."},{"Start":"05:43.910 ","End":"05:47.170","Text":"The first step in computing the triple integral,"},{"Start":"05:47.170 ","End":"05:52.595","Text":"is to first see what is the divergence of the vector field."},{"Start":"05:52.595 ","End":"05:56.090","Text":"Let\u0027s call each of the components by a name."},{"Start":"05:56.090 ","End":"06:01.875","Text":"Let\u0027s call the first component P,"},{"Start":"06:01.875 ","End":"06:06.855","Text":"the second one, I\u0027ll call this one here Q."},{"Start":"06:06.855 ","End":"06:13.335","Text":"That\u0027s just what comes before the i What comes before the j"},{"Start":"06:13.335 ","End":"06:20.120","Text":"and R is what goes with the k. In general,"},{"Start":"06:20.120 ","End":"06:27.260","Text":"the divergence of such an F is just the first component, in our case,"},{"Start":"06:27.260 ","End":"06:32.990","Text":"P with respect to x plus second component with respect to y,"},{"Start":"06:32.990 ","End":"06:36.925","Text":"plus the third component with respect to z."},{"Start":"06:36.925 ","End":"06:39.030","Text":"Let\u0027s see what it comes out."},{"Start":"06:39.030 ","End":"06:43.880","Text":"In our case I need to do 3 partial derivatives."},{"Start":"06:43.880 ","End":"06:52.280","Text":"First of all, let\u0027s start with P. With respect to x since y is a constant,"},{"Start":"06:52.280 ","End":"06:57.440","Text":"this would be like just differentiating x squared and getting 2x and the rest stays."},{"Start":"06:57.440 ","End":"07:04.650","Text":"This is 2xy over 1 plus y squared."},{"Start":"07:04.650 ","End":"07:07.845","Text":"This with respect to x is nothing."},{"Start":"07:07.845 ","End":"07:15.920","Text":"Now I need Q with respect to y,"},{"Start":"07:15.920 ","End":"07:20.075","Text":"and I need the derivative of arctangent."},{"Start":"07:20.075 ","End":"07:28.560","Text":"The derivative of arctangent y is 1/1 plus y squared."},{"Start":"07:28.560 ","End":"07:31.905","Text":"The 2x we choose a constant just sticks,"},{"Start":"07:31.905 ","End":"07:36.090","Text":"so it\u0027s 2x over 1 plus y squared."},{"Start":"07:36.090 ","End":"07:38.035","Text":"Now the last bit,"},{"Start":"07:38.035 ","End":"07:43.835","Text":"the partial derivative of R with respect to z."},{"Start":"07:43.835 ","End":"07:48.170","Text":"The denominator is a constant as far as z goes."},{"Start":"07:48.170 ","End":"07:51.545","Text":"I can just leave it there."},{"Start":"07:51.545 ","End":"07:56.969","Text":"Now I just have to differentiate the numerator with respect to z."},{"Start":"07:58.370 ","End":"08:02.805","Text":"The only place that z appears is here."},{"Start":"08:02.805 ","End":"08:05.299","Text":"The rest of it is like a constant."},{"Start":"08:05.299 ","End":"08:08.600","Text":"It\u0027s a constant times z plus another constant."},{"Start":"08:08.600 ","End":"08:14.960","Text":"All I\u0027m left with is 2x times 1 plus y,"},{"Start":"08:14.960 ","End":"08:18.680","Text":"the coefficient of z, and this is the constant goes to 0."},{"Start":"08:18.680 ","End":"08:21.185","Text":"This is the divergence."},{"Start":"08:21.185 ","End":"08:27.780","Text":"Notice that all 3 denominators here are the same."},{"Start":"08:29.240 ","End":"08:32.275","Text":"Let\u0027s simplify."},{"Start":"08:32.275 ","End":"08:37.130","Text":"We have the denominator 1 plus y squared."},{"Start":"08:37.130 ","End":"08:45.620","Text":"Then the numerator is 2xy plus 2x."},{"Start":"08:45.620 ","End":"08:48.450","Text":"Let\u0027s multiply this out."},{"Start":"08:48.550 ","End":"08:51.725","Text":"I just noticed this here is a minus,"},{"Start":"08:51.725 ","End":"08:55.930","Text":"so yeah, fixed it just in time."},{"Start":"08:56.260 ","End":"09:02.945","Text":"I get minus 2x, minus 2xy."},{"Start":"09:02.945 ","End":"09:09.760","Text":"Look, this cancels with this and this cancels with this."},{"Start":"09:09.760 ","End":"09:12.780","Text":"This just comes out to be 0."},{"Start":"09:12.780 ","End":"09:14.385","Text":"Isn\u0027t that great?"},{"Start":"09:14.385 ","End":"09:18.080","Text":"I don\u0027t really need to compute the integral."},{"Start":"09:18.080 ","End":"09:24.265","Text":"This integral, in our case just comes out to be 0."},{"Start":"09:24.265 ","End":"09:27.030","Text":"I\u0027ll just write that,"},{"Start":"09:27.030 ","End":"09:33.310","Text":"in our scheme, this part is 0."},{"Start":"09:33.830 ","End":"09:37.370","Text":"Next we\u0027re going to compute this part here."},{"Start":"09:37.370 ","End":"09:39.980","Text":"When I have it, 0 minus it,"},{"Start":"09:39.980 ","End":"09:42.020","Text":"will give me the answer I want,"},{"Start":"09:42.020 ","End":"09:44.760","Text":"this is the bit that I want."},{"Start":"09:46.700 ","End":"09:49.815","Text":"Get some space here."},{"Start":"09:49.815 ","End":"09:52.520","Text":"I can just copy from here."},{"Start":"09:52.520 ","End":"09:55.055","Text":"I want the double integral,"},{"Start":"09:55.055 ","End":"09:59.300","Text":"but just over D of,"},{"Start":"09:59.300 ","End":"10:10.880","Text":"it should be a dot there to dot-product F.n dS."},{"Start":"10:10.880 ","End":"10:18.110","Text":"I\u0027d like to illustrate this normal vector on this part called D. It\u0027s outward."},{"Start":"10:18.110 ","End":"10:24.650","Text":"If I take a point on the disk here,"},{"Start":"10:24.650 ","End":"10:29.639","Text":"the normal, in this case it actually goes straight down."},{"Start":"10:30.280 ","End":"10:38.690","Text":"But all I need is for it to have a downward component because of the formula"},{"Start":"10:38.690 ","End":"10:47.690","Text":"that gives me one of 2 forms according to whether the normal is upward or downward."},{"Start":"10:47.690 ","End":"10:49.520","Text":"I\u0027m noting that it\u0027s downward."},{"Start":"10:49.520 ","End":"10:53.345","Text":"Even if it went down with an angle as long as it\u0027s downward."},{"Start":"10:53.345 ","End":"10:59.120","Text":"The formula says that this equals the double"},{"Start":"10:59.120 ","End":"11:09.870","Text":"integral of F. the vector."},{"Start":"11:20.100 ","End":"11:25.570","Text":"It is gx i plus"},{"Start":"11:25.570 ","End":"11:34.575","Text":"gy j minus k dA."},{"Start":"11:34.575 ","End":"11:41.485","Text":"Now I need to explain a few things here on several things I want to say."},{"Start":"11:41.485 ","End":"11:47.365","Text":"First of all usually you see it not so much in this form as with"},{"Start":"11:47.365 ","End":"11:52.270","Text":"the brackets form gx,"},{"Start":"11:52.270 ","End":"11:59.785","Text":"gy, negative 1."},{"Start":"11:59.785 ","End":"12:04.795","Text":"I just used the ijk to be consistent with the ijk here."},{"Start":"12:04.795 ","End":"12:08.305","Text":"The second thing I haven\u0027t even told you what g is."},{"Start":"12:08.305 ","End":"12:10.330","Text":"Let\u0027s get some space."},{"Start":"12:10.330 ","End":"12:18.445","Text":"G is the function that describes this bit of the surface,"},{"Start":"12:18.445 ","End":"12:24.805","Text":"the D part, the base as z as a function of x and y."},{"Start":"12:24.805 ","End":"12:29.110","Text":"In other words, this is described by z,"},{"Start":"12:29.110 ","End":"12:32.320","Text":"which is g of xy,"},{"Start":"12:32.320 ","End":"12:36.340","Text":"which in our particular case is just 0."},{"Start":"12:36.340 ","End":"12:40.255","Text":"The whole plane is z equals 0."},{"Start":"12:40.255 ","End":"12:49.420","Text":"That gives me g. The integral is the projection of D onto the x_y plane,"},{"Start":"12:49.420 ","End":"12:51.340","Text":"which is D itself."},{"Start":"12:51.340 ","End":"12:54.340","Text":"Technically you could say this is not the same thing,"},{"Start":"12:54.340 ","End":"12:55.615","Text":"this is the disk."},{"Start":"12:55.615 ","End":"13:01.060","Text":"But in 3-space, and this is the same disk only in 2-space."},{"Start":"13:01.060 ","End":"13:03.955","Text":"Maybe I\u0027ll pull this 1, I don\u0027t know,"},{"Start":"13:03.955 ","End":"13:07.960","Text":"D naught, just to be more precise."},{"Start":"13:07.960 ","End":"13:09.880","Text":"It\u0027s the same as D,"},{"Start":"13:09.880 ","End":"13:13.585","Text":"but just as considered only in 2-dimensional space."},{"Start":"13:13.585 ","End":"13:17.860","Text":"This D without the naught is in 3-dimensional space."},{"Start":"13:17.860 ","End":"13:20.390","Text":"It\u0027s a technicality."},{"Start":"13:22.890 ","End":"13:26.785","Text":"That\u0027s basically it."},{"Start":"13:26.785 ","End":"13:33.505","Text":"Only now I have to do the computation of what is this dot-product."},{"Start":"13:33.505 ","End":"13:41.270","Text":"Then we\u0027ll have a regular double integral over the disc with radius 2."},{"Start":"13:41.270 ","End":"13:46.710","Text":"Of course, before I do the dot-product I have to compute the partial derivatives."},{"Start":"13:46.710 ","End":"13:49.335","Text":"If g of xy is 0,"},{"Start":"13:49.335 ","End":"13:54.755","Text":"then gx is also equal to 0,"},{"Start":"13:54.755 ","End":"13:59.245","Text":"and gy is also equal to 0."},{"Start":"13:59.245 ","End":"14:02.094","Text":"Just 1 more little comment."},{"Start":"14:02.094 ","End":"14:08.215","Text":"The formula I produced comes from a theorem where there\u0027s actually 2 formulas."},{"Start":"14:08.215 ","End":"14:10.270","Text":"When it\u0027s a downward normal,"},{"Start":"14:10.270 ","End":"14:11.980","Text":"then this is the formula."},{"Start":"14:11.980 ","End":"14:13.690","Text":"If it was an upward normal,"},{"Start":"14:13.690 ","End":"14:16.420","Text":"then there\u0027d be a minus here and a minus here,"},{"Start":"14:16.420 ","End":"14:17.920","Text":"and a plus here."},{"Start":"14:17.920 ","End":"14:20.305","Text":"I\u0027m just putting it in contexts."},{"Start":"14:20.305 ","End":"14:22.420","Text":"But since r faces down,"},{"Start":"14:22.420 ","End":"14:28.670","Text":"then the plus-plus minus is what we want."},{"Start":"14:29.430 ","End":"14:33.145","Text":"I\u0027m going to continue with this over here."},{"Start":"14:33.145 ","End":"14:39.440","Text":"What we have is the double integral over the disk."},{"Start":"14:40.110 ","End":"14:43.015","Text":"Now I\u0027ve lost that, I scrolled up,"},{"Start":"14:43.015 ","End":"14:49.104","Text":"but I remember that it was something i plus something"},{"Start":"14:49.104 ","End":"14:56.199","Text":"j plus something k. We\u0027ll afterwards we\u0027ll scroll back up and see what it was."},{"Start":"14:56.199 ","End":"15:04.915","Text":"Dot, and this vector gx we said is 0, gy is 0."},{"Start":"15:04.915 ","End":"15:09.170","Text":"All I\u0027m left with is minus K_dA."},{"Start":"15:11.760 ","End":"15:17.575","Text":"Now, when I do the dot product with minus k,"},{"Start":"15:17.575 ","End":"15:28.370","Text":"all I\u0027m going to get is the double integral over D naught of just minus R_dA."},{"Start":"15:31.890 ","End":"15:35.950","Text":"I went and picked backup to see what R was,"},{"Start":"15:35.950 ","End":"15:38.660","Text":"it\u0027s not on the screen anymore."},{"Start":"15:38.970 ","End":"15:52.165","Text":"It was equal to 2xz, 1 plus y,"},{"Start":"15:52.165 ","End":"16:00.490","Text":"plus 1 plus y squared,"},{"Start":"16:00.490 ","End":"16:04.330","Text":"over 1 plus y squared,"},{"Start":"16:04.330 ","End":"16:08.240","Text":"and there\u0027s a minus in front of it."},{"Start":"16:09.750 ","End":"16:13.420","Text":"Notice though that in"},{"Start":"16:13.420 ","End":"16:23.335","Text":"R case z we have to substitute from the surface for g,"},{"Start":"16:23.335 ","End":"16:27.970","Text":"z here is equal to 0."},{"Start":"16:27.970 ","End":"16:34.450","Text":"Clearly, also the disc is in the x-y plane, so the z is 0."},{"Start":"16:34.450 ","End":"16:39.940","Text":"If this is 0, then"},{"Start":"16:39.940 ","End":"16:47.275","Text":"this whole first term on the numerator disappears."},{"Start":"16:47.275 ","End":"16:52.525","Text":"What I\u0027m left with is this over this and a minus here."},{"Start":"16:52.525 ","End":"16:57.610","Text":"It just comes out to be minus 1."},{"Start":"16:57.610 ","End":"17:00.865","Text":"The same numerator and denominator so that cancels."},{"Start":"17:00.865 ","End":"17:06.625","Text":"If I go back here and just to clear a bit more space,"},{"Start":"17:06.625 ","End":"17:13.675","Text":"then we have the double integral over the disk D naught."},{"Start":"17:13.675 ","End":"17:16.060","Text":"We can partly see it."},{"Start":"17:16.060 ","End":"17:18.880","Text":"Minus R is plus 1,"},{"Start":"17:18.880 ","End":"17:21.865","Text":"so it\u0027s just 1 dA."},{"Start":"17:21.865 ","End":"17:28.900","Text":"Now there\u0027s a theorem that the double integral over"},{"Start":"17:28.900 ","End":"17:36.610","Text":"a region in 2D of the function 1 is just the area of that region or domain."},{"Start":"17:36.610 ","End":"17:43.585","Text":"This is just the area of D naught."},{"Start":"17:43.585 ","End":"17:49.540","Text":"But D naught is a disk of radius 2,"},{"Start":"17:49.540 ","End":"17:55.405","Text":"and we know the formula for the area of a disk Pi r squared."},{"Start":"17:55.405 ","End":"18:05.080","Text":"This case it\u0027s Pi times the radius is 2 squared, which is 4Pi."},{"Start":"18:05.080 ","End":"18:08.725","Text":"Now we\u0027re ready to close."},{"Start":"18:08.725 ","End":"18:11.380","Text":"This is the place where we\u0027re at."},{"Start":"18:11.380 ","End":"18:13.795","Text":"This was the strategy."},{"Start":"18:13.795 ","End":"18:23.305","Text":"We found the triple integral over the solid body B."},{"Start":"18:23.305 ","End":"18:30.460","Text":"This is surface integral over the base D. We\u0027ve just computed it is"},{"Start":"18:30.460 ","End":"18:39.340","Text":"4Pi and 0 minus 4Pi is minus 4Pi."},{"Start":"18:39.340 ","End":"18:45.890","Text":"That\u0027s the bit that we wanted the surface integral over just S."},{"Start":"18:46.290 ","End":"18:50.755","Text":"The final answer is"},{"Start":"18:50.755 ","End":"19:00.140","Text":"this minus 4Pi. We\u0027re done."}],"ID":9660},{"Watched":false,"Name":"Exercise 9","Duration":"24m 4s","ChapterTopicVideoID":8794,"CourseChapterTopicPlaylistID":4974,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:09.180","Text":"In this exercise, we have a type 2 surface integral on a unit sphere."},{"Start":"00:09.180 ","End":"00:10.770","Text":"S is the unit sphere,"},{"Start":"00:10.770 ","End":"00:13.860","Text":"I\u0027ve tried to illustrate it here,"},{"Start":"00:13.860 ","End":"00:21.629","Text":"and f is a vector field given in the i, j, k form,"},{"Start":"00:21.629 ","End":"00:30.465","Text":"and n is the outward unit normal."},{"Start":"00:30.465 ","End":"00:32.804","Text":"At any given point,"},{"Start":"00:32.804 ","End":"00:40.890","Text":"I have a normal vector which is 1 in length and faces outwards."},{"Start":"00:40.890 ","End":"00:43.070","Text":"So here for example,"},{"Start":"00:43.070 ","End":"00:45.380","Text":"is the normal vector n,"},{"Start":"00:45.380 ","End":"00:47.990","Text":"and if I\u0027m here,"},{"Start":"00:47.990 ","End":"00:50.855","Text":"might go this way."},{"Start":"00:50.855 ","End":"00:55.475","Text":"In fact, I want to break the sphere up into 2 parts,"},{"Start":"00:55.475 ","End":"00:58.760","Text":"the upper hemisphere and the lower hemisphere."},{"Start":"00:58.760 ","End":"01:01.320","Text":"In fact, I want to call this,"},{"Start":"01:01.320 ","End":"01:07.360","Text":"say S_1 and the lower hemisphere S_2."},{"Start":"01:07.760 ","End":"01:10.550","Text":"1 of the reasons is,"},{"Start":"01:10.550 ","End":"01:17.320","Text":"the main reason is that then I can write z as a function of x and y,"},{"Start":"01:17.320 ","End":"01:21.005","Text":"and they\u0027ll have all kinds of theorems and formulas for that."},{"Start":"01:21.005 ","End":"01:28.140","Text":"So if I want an equation for S_1,"},{"Start":"01:28.140 ","End":"01:31.000","Text":"we can isolate it."},{"Start":"01:32.720 ","End":"01:40.174","Text":"First of all we would say that z squared is 1 minus x squared minus y squared,"},{"Start":"01:40.174 ","End":"01:42.710","Text":"and then we would take the square root."},{"Start":"01:42.710 ","End":"01:46.630","Text":"For S_1 we take the positive square root."},{"Start":"01:46.630 ","End":"01:52.430","Text":"For S_2 we would take the negative square root,"},{"Start":"01:52.430 ","End":"01:59.225","Text":"so that would be minus square root of 1 minus x squared minus y squared."},{"Start":"01:59.225 ","End":"02:04.060","Text":"Then the idea is to break the integral up over S,"},{"Start":"02:04.060 ","End":"02:11.450","Text":"to say that it\u0027s the integral over S_1 plus the integral over S_2,"},{"Start":"02:11.450 ","End":"02:14.530","Text":"and to do it in 2 bits."},{"Start":"02:14.530 ","End":"02:16.995","Text":"Let\u0027s begin with S_1,"},{"Start":"02:16.995 ","End":"02:22.310","Text":"and I can write this as z equals g of x,"},{"Start":"02:22.310 ","End":"02:24.350","Text":"y, where g of x,"},{"Start":"02:24.350 ","End":"02:26.340","Text":"y is this here."},{"Start":"02:26.340 ","End":"02:29.600","Text":"Reason I\u0027m doing this is that there\u0027s a theorem I can use."},{"Start":"02:29.600 ","End":"02:34.685","Text":"It\u0027s actually more convenient for me to use the angular bracket notation."},{"Start":"02:34.685 ","End":"02:39.170","Text":"Let me write f as x,"},{"Start":"02:39.170 ","End":"02:46.695","Text":"minus 2y, 3z, it\u0027ll be a bit more convenient."},{"Start":"02:46.695 ","End":"02:54.540","Text":"Then there\u0027s a theorem that says that the double integral,"},{"Start":"02:55.810 ","End":"02:59.870","Text":"in this case it would be S_1."},{"Start":"02:59.870 ","End":"03:07.775","Text":"But in general, the double integral over a surface of F.n"},{"Start":"03:07.775 ","End":"03:16.925","Text":"ds is equal to"},{"Start":"03:16.925 ","End":"03:21.020","Text":"the double integral over R,"},{"Start":"03:21.020 ","End":"03:24.710","Text":"and I\u0027ll take a break to show you what I mean by R."},{"Start":"03:24.710 ","End":"03:28.970","Text":"R is the projection of the surface onto the xy plane,"},{"Start":"03:28.970 ","End":"03:31.800","Text":"or if you like, the domain."},{"Start":"03:31.870 ","End":"03:41.030","Text":"Here, the domain would be where x squared plus y squared is less than or equal to 1."},{"Start":"03:41.030 ","End":"03:44.690","Text":"You could see this if this is the unit sphere where"},{"Start":"03:44.690 ","End":"03:52.395","Text":"this is x is 1,"},{"Start":"03:52.395 ","End":"03:54.915","Text":"this where y is 1, and where z is 1."},{"Start":"03:54.915 ","End":"04:03.170","Text":"This part in just the xy plane would be our region R,"},{"Start":"04:03.170 ","End":"04:05.635","Text":"maybe I\u0027ll put a separate picture."},{"Start":"04:05.635 ","End":"04:08.720","Text":"Here it is and this is the unit disk,"},{"Start":"04:08.720 ","End":"04:12.650","Text":"x squared plus y squared less than or equal to 1."},{"Start":"04:12.650 ","End":"04:17.840","Text":"It\u0027s equal to 1 on the circumference of the disk,"},{"Start":"04:17.840 ","End":"04:21.425","Text":"which is the circle. Back here."},{"Start":"04:21.425 ","End":"04:24.890","Text":"So in general, R is this region which is like"},{"Start":"04:24.890 ","End":"04:29.390","Text":"the domain of definition or the projection of S onto the xy plane."},{"Start":"04:29.390 ","End":"04:39.170","Text":"It\u0027s equal to f dot with the vector which is minus g,"},{"Start":"04:39.170 ","End":"04:42.935","Text":"partial derivative with respect to x,"},{"Start":"04:42.935 ","End":"04:47.215","Text":"minus partial derivative with respect to y,"},{"Start":"04:47.215 ","End":"04:51.670","Text":"1, and all this, dA."},{"Start":"04:52.640 ","End":"04:58.340","Text":"This is not quite precise as a condition that is provided that"},{"Start":"04:58.340 ","End":"05:03.815","Text":"the normal vector n has an upward component."},{"Start":"05:03.815 ","End":"05:06.605","Text":"It doesn\u0027t have to be exactly upward,"},{"Start":"05:06.605 ","End":"05:11.075","Text":"but the last component,"},{"Start":"05:11.075 ","End":"05:13.235","Text":"the k component, if you like,"},{"Start":"05:13.235 ","End":"05:14.810","Text":"has to be positive,"},{"Start":"05:14.810 ","End":"05:18.050","Text":"has to be partially upward."},{"Start":"05:18.050 ","End":"05:20.285","Text":"So that would work for S_1,"},{"Start":"05:20.285 ","End":"05:22.505","Text":"but it would not work for S_2."},{"Start":"05:22.505 ","End":"05:25.910","Text":"If the normal has a downward component,"},{"Start":"05:25.910 ","End":"05:31.560","Text":"then we have to replace all the signs here and this would be a plus,"},{"Start":"05:31.560 ","End":"05:32.865","Text":"this would be a plus,"},{"Start":"05:32.865 ","End":"05:35.640","Text":"this would be a minus 1,"},{"Start":"05:35.640 ","End":"05:38.070","Text":"in case we had it down with normal."},{"Start":"05:38.070 ","End":"05:42.660","Text":"But for S_1, we\u0027re okay as it is here."},{"Start":"05:42.660 ","End":"05:46.760","Text":"When we get to the lower hemisphere and we take this and"},{"Start":"05:46.760 ","End":"05:51.065","Text":"then we\u0027ll use the reverse signs."},{"Start":"05:51.065 ","End":"05:54.695","Text":"Now, here\u0027s the formula for g,"},{"Start":"05:54.695 ","End":"05:59.850","Text":"let\u0027s compute what are the partial derivatives."},{"Start":"05:59.850 ","End":"06:04.815","Text":"G with respect to x is equal to,"},{"Start":"06:04.815 ","End":"06:06.735","Text":"we have a square root,"},{"Start":"06:06.735 ","End":"06:13.970","Text":"so it\u0027s first of all twice the square root on the denominator of the same thing,"},{"Start":"06:13.970 ","End":"06:16.505","Text":"1 minus x squared minus y squared."},{"Start":"06:16.505 ","End":"06:19.025","Text":"On the numerator, the inner derivative,"},{"Start":"06:19.025 ","End":"06:22.450","Text":"which in this case is minus 2x,"},{"Start":"06:22.450 ","End":"06:25.245","Text":"and the 2 \u0027s cancel."},{"Start":"06:25.245 ","End":"06:27.270","Text":"Now for g_y."},{"Start":"06:27.270 ","End":"06:32.070","Text":"Well, same thing,"},{"Start":"06:32.070 ","End":"06:33.435","Text":"the 2 \u0027s are going to cancel,"},{"Start":"06:33.435 ","End":"06:34.980","Text":"I\u0027m not even going to bother writing it."},{"Start":"06:34.980 ","End":"06:38.029","Text":"On the denominator we\u0027re going to have the same thing."},{"Start":"06:38.029 ","End":"06:44.060","Text":"The only difference is that in the numerator we have a y instead of an x,"},{"Start":"06:44.060 ","End":"06:46.640","Text":"so that\u0027s this here."},{"Start":"06:46.640 ","End":"06:49.130","Text":"Now that we have these 2,"},{"Start":"06:49.130 ","End":"06:53.120","Text":"we can do the dot-product,"},{"Start":"06:53.120 ","End":"06:55.370","Text":"just to make it clear,"},{"Start":"06:55.370 ","End":"07:03.310","Text":"I want the dot product of this vector with this vector here."},{"Start":"07:03.310 ","End":"07:07.200","Text":"But I\u0027m going to use the g_x from here and here."},{"Start":"07:07.200 ","End":"07:09.375","Text":"So let\u0027s see what we get."},{"Start":"07:09.375 ","End":"07:15.260","Text":"So I\u0027m continuing from here equals and I come out here."},{"Start":"07:15.260 ","End":"07:23.805","Text":"What I get is the double integral over R, this unit disk."},{"Start":"07:23.805 ","End":"07:26.300","Text":"This of course will be the same when we get to S_2,"},{"Start":"07:26.300 ","End":"07:28.380","Text":"I\u0027m just mentioning it."},{"Start":"07:28.900 ","End":"07:33.380","Text":"Let\u0027s see now, we take the x component with the x component."},{"Start":"07:33.380 ","End":"07:35.700","Text":"I need minus x, g_x,"},{"Start":"07:35.920 ","End":"07:38.675","Text":"and g_x is this,"},{"Start":"07:38.675 ","End":"07:44.090","Text":"so we get the minus and the minus is a plus."},{"Start":"07:44.090 ","End":"07:49.910","Text":"We get x minus x times this will be x"},{"Start":"07:49.910 ","End":"07:58.920","Text":"squared over the square root of 1 minus x squared minus y squared."},{"Start":"07:58.920 ","End":"08:04.150","Text":"Then I need minus 2y with minus g_y."},{"Start":"08:07.350 ","End":"08:10.885","Text":"There\u0027s 3 minuses, the minus here,"},{"Start":"08:10.885 ","End":"08:12.310","Text":"there\u0027s a minus here,"},{"Start":"08:12.310 ","End":"08:14.320","Text":"and there\u0027s a minus in the gy."},{"Start":"08:14.320 ","End":"08:17.230","Text":"So minus, minus, minus will be minus,"},{"Start":"08:17.230 ","End":"08:20.125","Text":"and then it\u0027ll be 2y times y is"},{"Start":"08:20.125 ","End":"08:28.165","Text":"2y squared over the square root of 1 minus x squared minus y squared."},{"Start":"08:28.165 ","End":"08:34.400","Text":"Lastly, this with this is just 3z."},{"Start":"08:34.410 ","End":"08:36.910","Text":"That will be plus 3z,"},{"Start":"08:36.910 ","End":"08:42.715","Text":"and put this in a bracket,"},{"Start":"08:42.715 ","End":"08:45.490","Text":"and this is dA."},{"Start":"08:47.400 ","End":"08:51.130","Text":"I wanted to write 3z but I don\u0027t write the z,"},{"Start":"08:51.130 ","End":"08:54.340","Text":"I write it as a function of x and y,"},{"Start":"08:54.340 ","End":"08:55.570","Text":"and here it is."},{"Start":"08:55.570 ","End":"09:00.430","Text":"It\u0027s the square root of 1 minus x squared"},{"Start":"09:00.430 ","End":"09:06.770","Text":"minus y squared and this is dA,"},{"Start":"09:06.780 ","End":"09:09.700","Text":"it looks quite a mess."},{"Start":"09:09.700 ","End":"09:14.455","Text":"What I suggest is polar coordinates."},{"Start":"09:14.455 ","End":"09:20.065","Text":"First of all, the region R has a circular symmetry, it\u0027s a disc."},{"Start":"09:20.065 ","End":"09:23.890","Text":"Also there\u0027s lots of x squared plus y squared going around,"},{"Start":"09:23.890 ","End":"09:27.520","Text":"so all good indications of polar coordinates."},{"Start":"09:27.520 ","End":"09:29.140","Text":"But before we do the polar,"},{"Start":"09:29.140 ","End":"09:31.960","Text":"let\u0027s do some simplification."},{"Start":"09:31.960 ","End":"09:35.725","Text":"This is quite a mess really."},{"Start":"09:35.725 ","End":"09:38.290","Text":"Let me do the simplification at the side."},{"Start":"09:38.290 ","End":"09:41.485","Text":"I want to put this over a common denominator."},{"Start":"09:41.485 ","End":"09:49.420","Text":"The common denominator would be the square root of 1 minus x squared minus y squared."},{"Start":"09:49.420 ","End":"09:52.225","Text":"From here I\u0027d get x squared,"},{"Start":"09:52.225 ","End":"09:56.155","Text":"from here minus 2y squared,"},{"Start":"09:56.155 ","End":"09:58.135","Text":"and from the last,"},{"Start":"09:58.135 ","End":"10:01.329","Text":"I just multiply top and bottom."},{"Start":"10:01.329 ","End":"10:02.380","Text":"Well there is no bottom,"},{"Start":"10:02.380 ","End":"10:04.555","Text":"but I could think of it as over 1."},{"Start":"10:04.555 ","End":"10:10.510","Text":"I could multiply by the square root and get 3 times 1 minus x squared"},{"Start":"10:10.510 ","End":"10:17.155","Text":"minus y squared over the square root of 1 minus x squared minus y squared."},{"Start":"10:17.155 ","End":"10:20.840","Text":"Let\u0027s see what this comes out too."},{"Start":"10:22.550 ","End":"10:25.050","Text":"First of all, just numbers."},{"Start":"10:25.050 ","End":"10:28.500","Text":"I have 3 and now x squared."},{"Start":"10:28.500 ","End":"10:29.520","Text":"How many do I have?"},{"Start":"10:29.520 ","End":"10:31.535","Text":"I have 1x squared,"},{"Start":"10:31.535 ","End":"10:34.675","Text":"and I have minus 3x squared."},{"Start":"10:34.675 ","End":"10:42.175","Text":"That\u0027s minus 2x squared and as for y squared,"},{"Start":"10:42.175 ","End":"10:50.430","Text":"I have minus 2y squared minus 3y squared."},{"Start":"10:50.430 ","End":"10:56.865","Text":"That\u0027s minus 5y squared over same thing,"},{"Start":"10:56.865 ","End":"11:01.990","Text":"square root 1 minus x squared minus y squared."},{"Start":"11:01.990 ","End":"11:08.215","Text":"Actually, I\u0027d like to take this 1 step further in anticipation of a polar substitution,"},{"Start":"11:08.215 ","End":"11:12.610","Text":"because I know that x squared plus y squared is something in polar,"},{"Start":"11:12.610 ","End":"11:14.680","Text":"x squared plus y squared is r squared."},{"Start":"11:14.680 ","End":"11:17.650","Text":"I\u0027m going to write this as 3."},{"Start":"11:17.650 ","End":"11:24.205","Text":"Now look, I could write this minus twice x squared plus y squared."},{"Start":"11:24.205 ","End":"11:26.320","Text":"Let\u0027s see what\u0027s leftover."},{"Start":"11:26.320 ","End":"11:29.365","Text":"I have 3 minus 2x squared minus 2y squared."},{"Start":"11:29.365 ","End":"11:37.090","Text":"I have to put another minus 3y squared and then I\u0027ll be all right and it\u0027s still"},{"Start":"11:37.090 ","End":"11:46.450","Text":"over the same square root of 1 minus x squared minus y squared."},{"Start":"11:46.450 ","End":"11:52.330","Text":"Let me remind you now of the polar substitution equations."},{"Start":"11:52.330 ","End":"11:57.820","Text":"We let x equals r cosine Theta,"},{"Start":"11:57.820 ","End":"12:02.005","Text":"y equals r sine Theta,"},{"Start":"12:02.005 ","End":"12:09.745","Text":"and we substitute for dA rdrd Theta,"},{"Start":"12:09.745 ","End":"12:13.000","Text":"and then there\u0027s that extra equation that\u0027s very useful most"},{"Start":"12:13.000 ","End":"12:17.320","Text":"times is x squared plus y squared equals r squared."},{"Start":"12:17.320 ","End":"12:20.260","Text":"Certainly going to be useful in our case."},{"Start":"12:20.260 ","End":"12:24.160","Text":"We also have to describe the region in polar,"},{"Start":"12:24.160 ","End":"12:27.670","Text":"but the unit circle we\u0027ve done so many times."},{"Start":"12:27.670 ","End":"12:30.400","Text":"Remember we have r and Theta."},{"Start":"12:30.400 ","End":"12:34.870","Text":"Theta goes the whole way."},{"Start":"12:34.870 ","End":"12:46.210","Text":"So that theta goes from 0 all the way around to 2 Pi and r goes from 0 to 1,"},{"Start":"12:46.210 ","End":"12:49.045","Text":"and so when I convert to polar,"},{"Start":"12:49.045 ","End":"12:59.275","Text":"I get Theta from 0 to 2 Pi r from 0-1."},{"Start":"12:59.275 ","End":"13:01.855","Text":"Well, what\u0027s written here?"},{"Start":"13:01.855 ","End":"13:03.760","Text":"That goes up there,"},{"Start":"13:03.760 ","End":"13:11.260","Text":"and so we have 3 minus 2 r squared,"},{"Start":"13:11.260 ","End":"13:17.725","Text":"that\u0027s from the x squared plus y squared is r squared minus 3."},{"Start":"13:17.725 ","End":"13:24.340","Text":"Now y squared from here is r squared sine squared Theta."},{"Start":"13:24.340 ","End":"13:29.335","Text":"R squared sine squared Theta."},{"Start":"13:29.335 ","End":"13:37.225","Text":"The denominator is 1 minus x squared minus y squared square root,"},{"Start":"13:37.225 ","End":"13:41.440","Text":"which is the square root of 1 minus r squared."},{"Start":"13:41.440 ","End":"13:44.545","Text":"Again, I\u0027m using the x squared plus y squared equals r squared,"},{"Start":"13:44.545 ","End":"13:53.125","Text":"and then I need the dA to be rdrd Theta."},{"Start":"13:53.125 ","End":"13:56.770","Text":"I\u0027d like to do a slight rewrite if I can take"},{"Start":"13:56.770 ","End":"14:02.830","Text":"the minus out the brackets and make this a plus."},{"Start":"14:02.830 ","End":"14:07.630","Text":"Now what I want to do is split this up into 2 separate integrals"},{"Start":"14:07.630 ","End":"14:14.155","Text":"and splitting it up to something minus something based on this minus."},{"Start":"14:14.155 ","End":"14:17.725","Text":"This comes out to be the first bit,"},{"Start":"14:17.725 ","End":"14:22.255","Text":"I can take the 3 outside the brackets,"},{"Start":"14:22.255 ","End":"14:24.145","Text":"there\u0027s an r here."},{"Start":"14:24.145 ","End":"14:31.735","Text":"I get the double integral 0 to 2 Pi for Theta,"},{"Start":"14:31.735 ","End":"14:38.935","Text":"0-1 for r of r over"},{"Start":"14:38.935 ","End":"14:47.725","Text":"the square root of 1 minus r squared drd Theta."},{"Start":"14:47.725 ","End":"14:53.335","Text":"That\u0027s the first bit then minus from this minus here."},{"Start":"14:53.335 ","End":"14:56.770","Text":"Now notice that I have here r squared then here r"},{"Start":"14:56.770 ","End":"14:59.995","Text":"squared shouldn\u0027t really taken this outside the brackets."},{"Start":"14:59.995 ","End":"15:04.615","Text":"But they combine with this r and I\u0027ll get an r cubed."},{"Start":"15:04.615 ","End":"15:16.465","Text":"What I get is the double integral, 0 to 2 Pi 0-1,"},{"Start":"15:16.465 ","End":"15:25.435","Text":"and then I can first of all take the 2 plus 3 sine squared"},{"Start":"15:25.435 ","End":"15:34.720","Text":"Theta and then the stuff"},{"Start":"15:34.720 ","End":"15:36.400","Text":"with the r like I said,"},{"Start":"15:36.400 ","End":"15:46.610","Text":"it\u0027s r cubed over 1 minus r squared square root,"},{"Start":"15:47.130 ","End":"15:53.170","Text":"and then drd Theta."},{"Start":"15:53.170 ","End":"15:59.365","Text":"For this integral actually I could take the part with just Theta."},{"Start":"15:59.365 ","End":"16:04.720","Text":"I can imagine I\u0027ve put this here in front of the integral sign."},{"Start":"16:04.720 ","End":"16:10.540","Text":"All I have to do for the first integral is the stuff with the r."},{"Start":"16:10.540 ","End":"16:17.150","Text":"What I\u0027m going to get is 2 side exercises."},{"Start":"16:17.190 ","End":"16:22.735","Text":"In both cases, both bits I do the dr integral first."},{"Start":"16:22.735 ","End":"16:32.470","Text":"I need to know what is the integral from 0-1 of r over the square root of 1 minus r"},{"Start":"16:32.470 ","End":"16:36.295","Text":"squared dr. That\u0027s 1 exercise that will help me"},{"Start":"16:36.295 ","End":"16:41.320","Text":"here and the other bit will be the integral from"},{"Start":"16:41.320 ","End":"16:50.710","Text":"0-1 of r cubed over square root of 1 minus r squared dr. Now,"},{"Start":"16:50.710 ","End":"16:52.960","Text":"I don\u0027t want to go into these and all the details."},{"Start":"16:52.960 ","End":"16:56.800","Text":"I\u0027m going to just quote some results."},{"Start":"16:56.800 ","End":"16:58.570","Text":"I\u0027ll start with the 2nd 1,"},{"Start":"16:58.570 ","End":"16:59.800","Text":"I\u0027ll just tell you the idea."},{"Start":"16:59.800 ","End":"17:04.960","Text":"The idea is to substitute t equals square root of 1 minus r squared."},{"Start":"17:04.960 ","End":"17:07.255","Text":"I\u0027m not going to do all the computations."},{"Start":"17:07.255 ","End":"17:11.710","Text":"The indefinite integral comes out to be"},{"Start":"17:11.710 ","End":"17:18.190","Text":"1/3 square root of 1 minus r squared cubed,"},{"Start":"17:18.190 ","End":"17:21.655","Text":"and that\u0027s not all,"},{"Start":"17:21.655 ","End":"17:26.875","Text":"minus the square root of 1 minus r squared."},{"Start":"17:26.875 ","End":"17:29.200","Text":"That\u0027s the indefinite integral plus c,"},{"Start":"17:29.200 ","End":"17:30.625","Text":"which we don\u0027t need."},{"Start":"17:30.625 ","End":"17:36.550","Text":"This we have to take between 0 and 1."},{"Start":"17:36.550 ","End":"17:42.010","Text":"The same substitution works in the first integral and in this case we"},{"Start":"17:42.010 ","End":"17:47.020","Text":"get minus the square root of 1 minus r squared,"},{"Start":"17:47.020 ","End":"17:52.090","Text":"which we also have to take between 0 and 1."},{"Start":"17:52.090 ","End":"17:54.100","Text":"Let\u0027s see what happens here."},{"Start":"17:54.100 ","End":"17:57.010","Text":"When r is 1, 1 minus r squared is 0."},{"Start":"17:57.010 ","End":"18:05.725","Text":"So this whole thing comes out to be 0 and if we let r equals 0,"},{"Start":"18:05.725 ","End":"18:07.870","Text":"then we get the square root of 1."},{"Start":"18:07.870 ","End":"18:10.450","Text":"So this is 1/3 minus 1,"},{"Start":"18:10.450 ","End":"18:13.390","Text":"which is minus 2/3,"},{"Start":"18:13.390 ","End":"18:16.045","Text":"so this is 2/3."},{"Start":"18:16.045 ","End":"18:19.330","Text":"It\u0027s getting a bit cramped,"},{"Start":"18:19.330 ","End":"18:21.385","Text":"I\u0027ll just separate these."},{"Start":"18:21.385 ","End":"18:26.605","Text":"Now, this 1, if I let r equals 1,"},{"Start":"18:26.605 ","End":"18:29.590","Text":"then this is just 0."},{"Start":"18:29.590 ","End":"18:36.010","Text":"If I let r equals 0,"},{"Start":"18:36.010 ","End":"18:40.595","Text":"then I get minus 1,"},{"Start":"18:40.595 ","End":"18:45.655","Text":"so I\u0027ve got 0 minus minus 1 is 1."},{"Start":"18:45.655 ","End":"18:48.655","Text":"Now I\u0027ve got both pieces,"},{"Start":"18:48.655 ","End":"18:54.790","Text":"the main pieces that I need and now I\u0027m going to go back here and I\u0027ll just"},{"Start":"18:54.790 ","End":"19:02.440","Text":"scroll down a bit and so here I\u0027m going to continue."},{"Start":"19:02.440 ","End":"19:10.270","Text":"Over here, I\u0027ve got 3 times the integral from"},{"Start":"19:10.270 ","End":"19:18.130","Text":"0-2 Pi d Theta"},{"Start":"19:18.130 ","End":"19:23.155","Text":"and we got this integral already from the first bit is 1,"},{"Start":"19:23.155 ","End":"19:31.480","Text":"so it\u0027s just 1 d Theta and the second bit minus,"},{"Start":"19:31.480 ","End":"19:36.400","Text":"I\u0027ve got the integral from 0-2 Pi."},{"Start":"19:36.400 ","End":"19:40.105","Text":"I have to take this bit,"},{"Start":"19:40.105 ","End":"19:45.505","Text":"which is 2 plus 3 sine squared"},{"Start":"19:45.505 ","End":"19:53.245","Text":"Theta and this integral came out to be 2/3,"},{"Start":"19:53.245 ","End":"20:02.540","Text":"which I can put in front and d Theta also."},{"Start":"20:02.670 ","End":"20:12.865","Text":"Now, I\u0027d like to combine these into a single integral from 0-2 Pi d Theta."},{"Start":"20:12.865 ","End":"20:16.045","Text":"Let\u0027s see, I have 3 here,"},{"Start":"20:16.045 ","End":"20:19.075","Text":"minus 2/3 times 2,"},{"Start":"20:19.075 ","End":"20:23.800","Text":"3 minus 4/3 comes out to be"},{"Start":"20:23.800 ","End":"20:30.400","Text":"5/3 and 2/3 times 3 is 2."},{"Start":"20:30.400 ","End":"20:38.590","Text":"So it\u0027s minus 2 sine squared Theta d Theta."},{"Start":"20:38.590 ","End":"20:45.730","Text":"Now I\u0027m going to quote a trigonometrical identity that sine squared Theta is"},{"Start":"20:45.730 ","End":"20:53.245","Text":"1/2 of 1 minus cosine 2 Theta and put this in here,"},{"Start":"20:53.245 ","End":"20:56.210","Text":"for sine squared Theta."},{"Start":"20:56.700 ","End":"21:00.535","Text":"The 2 will cancel with the 1/2 here."},{"Start":"21:00.535 ","End":"21:08.890","Text":"So it\u0027s 5/3 minus 1 plus cosine 2 Theta."},{"Start":"21:08.890 ","End":"21:12.805","Text":"In other words, we get the 0-2 Pi,"},{"Start":"21:12.805 ","End":"21:16.390","Text":"5/3 minus 1 is 2/3 and like we said,"},{"Start":"21:16.390 ","End":"21:26.395","Text":"we get the plus from the minus minus cosine 2 Theta d Theta and this is equal to,"},{"Start":"21:26.395 ","End":"21:32.215","Text":"on we go, 2/3 gives me 2/3 Theta,"},{"Start":"21:32.215 ","End":"21:38.590","Text":"cosine 2 Theta is not quite sine 2 Theta and the integral we also have to"},{"Start":"21:38.590 ","End":"21:46.555","Text":"divide by the 2 and this to take from 0-2 Pi."},{"Start":"21:46.555 ","End":"21:53.770","Text":"Now, when I plug in 2 Pi,"},{"Start":"21:53.770 ","End":"21:57.550","Text":"the sine is going to give me 0 in either case,"},{"Start":"21:57.550 ","End":"22:01.915","Text":"because sine of 0 is 0 and sine of 4 Pi is also 0."},{"Start":"22:01.915 ","End":"22:04.780","Text":"So you just have to relate to the first term."},{"Start":"22:04.780 ","End":"22:09.590","Text":"It\u0027s 2/3 of 2 Pi minus 0."},{"Start":"22:13.080 ","End":"22:16.915","Text":"2/3 times 2 Pi,"},{"Start":"22:16.915 ","End":"22:26.050","Text":"which is 4 Pi over 3."},{"Start":"22:26.050 ","End":"22:28.540","Text":"Now, let me highlight this,"},{"Start":"22:28.540 ","End":"22:33.385","Text":"but we\u0027re not done because this was just the upper hemisphere."},{"Start":"22:33.385 ","End":"22:37.465","Text":"Let\u0027s scroll back and see what would happen."},{"Start":"22:37.465 ","End":"22:42.100","Text":"Do we do need to do all this work for the lower hemisphere?"},{"Start":"22:42.100 ","End":"22:48.460","Text":"I claim not because what happens is"},{"Start":"22:48.460 ","End":"22:56.860","Text":"that g this time is not the square root but minus the square root."},{"Start":"22:56.860 ","End":"23:02.545","Text":"So I have to reverse g to make it minus."},{"Start":"23:02.545 ","End":"23:07.300","Text":"But I also have this reversal where this whole thing,"},{"Start":"23:07.300 ","End":"23:10.345","Text":"it\u0027s like putting a minus in front here."},{"Start":"23:10.345 ","End":"23:14.710","Text":"So the 2 minuses cancel each other out and"},{"Start":"23:14.710 ","End":"23:18.550","Text":"I\u0027m going to get exactly the same answer in the end."},{"Start":"23:18.550 ","End":"23:21.070","Text":"So if I go back here,"},{"Start":"23:21.070 ","End":"23:27.610","Text":"that was for S_1 and it\u0027s going to be the"},{"Start":"23:27.610 ","End":"23:33.520","Text":"same for S_2 because"},{"Start":"23:33.520 ","End":"23:39.235","Text":"of the minus minus and so the final answer,"},{"Start":"23:39.235 ","End":"23:42.790","Text":"the double integral over S,"},{"Start":"23:42.790 ","End":"23:50.290","Text":"is going to be 4 Pi over 3 plus 4 Pi over 3,"},{"Start":"23:50.290 ","End":"23:59.395","Text":"which is 8 Pi over 3 and so the final answer is this,"},{"Start":"23:59.395 ","End":"24:04.790","Text":"not this and we\u0027re finally done."}],"ID":9661},{"Watched":false,"Name":"Exercise 10 part a","Duration":"21m 44s","ChapterTopicVideoID":8795,"CourseChapterTopicPlaylistID":4974,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.180 ","End":"00:07.015","Text":"In this exercise, we have to compute the surface integral F.ndS,"},{"Start":"00:07.015 ","End":"00:11.005","Text":"where we\u0027re given the vector field F as follows."},{"Start":"00:11.005 ","End":"00:16.605","Text":"We\u0027re told that S is the surface of the pyramid given by 4 planes."},{"Start":"00:16.605 ","End":"00:22.455","Text":"Well, 3 of the planes are simply the coordinate planes,"},{"Start":"00:22.455 ","End":"00:26.755","Text":"when x is 0, it\u0027s the yz-plane, and so on."},{"Start":"00:26.755 ","End":"00:29.155","Text":"This one is the more interesting one,"},{"Start":"00:29.155 ","End":"00:32.325","Text":"that\u0027s this skew plane,"},{"Start":"00:32.325 ","End":"00:37.840","Text":"and it cuts all 3 of the coordinate axis at points A,"},{"Start":"00:37.840 ","End":"00:41.245","Text":"B, and C. This is the origin."},{"Start":"00:41.245 ","End":"00:45.480","Text":"I just want to compute the coordinates of A, B,"},{"Start":"00:45.480 ","End":"00:52.815","Text":"and C. When I let x and y equal 0,"},{"Start":"00:52.815 ","End":"00:55.110","Text":"then I just get z equals 6."},{"Start":"00:55.110 ","End":"01:01.365","Text":"I\u0027m just going to write the z component because x and y is 6."},{"Start":"01:01.365 ","End":"01:04.245","Text":"If I want B,"},{"Start":"01:04.245 ","End":"01:07.740","Text":"I will let x and z be 0,"},{"Start":"01:07.740 ","End":"01:11.850","Text":"I get 2y equals 6, y equals 3."},{"Start":"01:11.850 ","End":"01:13.995","Text":"If I want the point C,"},{"Start":"01:13.995 ","End":"01:16.740","Text":"I let y and z equal 0,"},{"Start":"01:16.740 ","End":"01:21.449","Text":"2x equal 6, x equals 3."},{"Start":"01:21.449 ","End":"01:25.275","Text":"I have these points, this is of course is the point 0, 0, 0,"},{"Start":"01:25.275 ","End":"01:32.600","Text":"and the idea is to take the double integral of S over S,"},{"Start":"01:32.600 ","End":"01:40.210","Text":"which is the whole pyramid into 4 separate integrals,"},{"Start":"01:42.260 ","End":"01:45.195","Text":"and I\u0027ll tell you which order I\u0027ll do them in."},{"Start":"01:45.195 ","End":"01:49.550","Text":"Let me first of all do the difficult one which is the skew one,"},{"Start":"01:49.550 ","End":"01:53.510","Text":"which will be the ABC triangle."},{"Start":"01:53.510 ","End":"01:57.965","Text":"Plus, we\u0027ll take the double integral over,"},{"Start":"01:57.965 ","End":"01:59.525","Text":"I want this one,"},{"Start":"01:59.525 ","End":"02:04.660","Text":"the one in the xy-plane that will be OBC."},{"Start":"02:04.660 ","End":"02:07.980","Text":"Then the other 2,"},{"Start":"02:07.980 ","End":"02:15.430","Text":"I\u0027ll take the double integral of the one in the yz-plane,"},{"Start":"02:15.430 ","End":"02:20.690","Text":"that would be OAB."},{"Start":"02:20.690 ","End":"02:25.435","Text":"Finally, we\u0027ll take the double integral of this one here,"},{"Start":"02:25.435 ","End":"02:28.220","Text":"which will be OAC."},{"Start":"02:28.220 ","End":"02:32.340","Text":"That\u0027s the general idea to do 4 separate calculations."},{"Start":"02:32.340 ","End":"02:35.140","Text":"As I said, I\u0027m going to start with the difficult one,"},{"Start":"02:35.140 ","End":"02:37.415","Text":"which is the ABC."},{"Start":"02:37.415 ","End":"02:41.410","Text":"So ABC, and for this,"},{"Start":"02:41.410 ","End":"02:45.960","Text":"I would like to rewrite this plane in"},{"Start":"02:45.960 ","End":"02:51.860","Text":"the terms of z as a function of x and y. I\u0027ll write from here,"},{"Start":"02:51.860 ","End":"02:56.989","Text":"if I extract z which is a function of x and y,"},{"Start":"02:56.989 ","End":"03:04.515","Text":"but specifically, it\u0027s equal to 6 minus 2x minus 2y."},{"Start":"03:04.515 ","End":"03:07.250","Text":"We could have done it with any of the other coordinates too,"},{"Start":"03:07.250 ","End":"03:10.400","Text":"I prefer to have z as a function of x and y."},{"Start":"03:10.400 ","End":"03:14.420","Text":"Notice that the projection of this surface ABC,"},{"Start":"03:14.420 ","End":"03:21.875","Text":"this triangle here onto the xy-plane is this here which I\u0027ll shade,"},{"Start":"03:21.875 ","End":"03:28.025","Text":"and I\u0027ll label this as R. I want to bring in"},{"Start":"03:28.025 ","End":"03:35.890","Text":"a formula which works when we have z extracted as a function of x and y."},{"Start":"03:35.890 ","End":"03:40.820","Text":"This theorem or formula says that we can"},{"Start":"03:40.820 ","End":"03:47.625","Text":"calculate the double integral over sum S,"},{"Start":"03:47.625 ","End":"03:50.505","Text":"not the same as is the pyramid,"},{"Start":"03:50.505 ","End":"03:54.465","Text":"this S will be one of these planes."},{"Start":"03:54.465 ","End":"03:56.609","Text":"This is in general"},{"Start":"03:56.609 ","End":"04:05.320","Text":"of F.ndS"},{"Start":"04:05.320 ","End":"04:07.190","Text":"is what I meant to write,"},{"Start":"04:07.190 ","End":"04:12.140","Text":"is equal to the double integral over the region R,"},{"Start":"04:12.140 ","End":"04:15.470","Text":"which is the projection of that surface onto"},{"Start":"04:15.470 ","End":"04:23.700","Text":"the xy-plane of F dot,"},{"Start":"04:23.700 ","End":"04:28.580","Text":"and here I write one of 2 things."},{"Start":"04:28.580 ","End":"04:31.180","Text":"Well, let me write one of them."},{"Start":"04:31.180 ","End":"04:34.395","Text":"I\u0027ll use the angular bracket form,"},{"Start":"04:34.395 ","End":"04:36.155","Text":"it\u0027ll be easier for me."},{"Start":"04:36.155 ","End":"04:41.370","Text":"What we have is minus g with respect to x,"},{"Start":"04:41.370 ","End":"04:47.250","Text":"where g is the function that defines z over the region R,"},{"Start":"04:47.250 ","End":"04:53.790","Text":"partial derivative minus g with respect to y,"},{"Start":"04:53.790 ","End":"04:58.000","Text":"and then 1, and this is dA."},{"Start":"04:59.540 ","End":"05:02.160","Text":"I\u0027m using some angular brackets here,"},{"Start":"05:02.160 ","End":"05:11.270","Text":"let me rewrite F as an angular bracket form is 2xy plus z in the first component,"},{"Start":"05:11.270 ","End":"05:14.045","Text":"y squared in the second component,"},{"Start":"05:14.045 ","End":"05:22.575","Text":"and this is a minus x plus 3y in the third component,"},{"Start":"05:22.575 ","End":"05:27.885","Text":"so it\u0027ll be easier rather than i, j, k. Here,"},{"Start":"05:27.885 ","End":"05:30.825","Text":"when we substitute x, y, and z,"},{"Start":"05:30.825 ","End":"05:35.330","Text":"instead of z, we have to put in g of xy, we\u0027ll see this."},{"Start":"05:35.330 ","End":"05:38.680","Text":"Now, I was saying that there\u0027s 2 cases."},{"Start":"05:38.680 ","End":"05:41.885","Text":"It all depends on whether the normal vector"},{"Start":"05:41.885 ","End":"05:46.100","Text":"has an upward component or a downward component."},{"Start":"05:46.100 ","End":"05:47.990","Text":"In the case of this plane,"},{"Start":"05:47.990 ","End":"05:51.185","Text":"if I took a point on this plane ABC,"},{"Start":"05:51.185 ","End":"05:57.320","Text":"the normal, it doesn\u0027t go upwards in the sense of in the z direction,"},{"Start":"05:57.320 ","End":"05:59.270","Text":"but it has an upward component."},{"Start":"05:59.270 ","End":"06:01.190","Text":"As we go outward,"},{"Start":"06:01.190 ","End":"06:03.725","Text":"we also go somewhat higher,"},{"Start":"06:03.725 ","End":"06:07.285","Text":"so this has an upward normal component."},{"Start":"06:07.285 ","End":"06:11.960","Text":"Whereas when we get to the triangle OBC,"},{"Start":"06:11.960 ","End":"06:14.120","Text":"which is this and I take a point,"},{"Start":"06:14.120 ","End":"06:18.004","Text":"it\u0027s normal vector is actually strictly downwards,"},{"Start":"06:18.004 ","End":"06:19.894","Text":"but it has a downward component."},{"Start":"06:19.894 ","End":"06:28.575","Text":"This is the formula that works when the normal tilts upwards,"},{"Start":"06:28.575 ","End":"06:34.720","Text":"and there\u0027s another formula for when the normal tilts downwards."},{"Start":"06:34.720 ","End":"06:39.050","Text":"I copy pasted this and now let me just make a small change for"},{"Start":"06:39.050 ","End":"06:45.350","Text":"the downward case is that I just take the minus of this by reversing the signs,"},{"Start":"06:45.350 ","End":"06:47.375","Text":"make this minus into a plus,"},{"Start":"06:47.375 ","End":"06:48.665","Text":"this into a plus,"},{"Start":"06:48.665 ","End":"06:50.720","Text":"and this to a minus."},{"Start":"06:50.720 ","End":"06:53.394","Text":"Actually, I\u0027ll erase the pluses."},{"Start":"06:53.394 ","End":"07:02.730","Text":"For ABC, since this normal has a positive K component,"},{"Start":"07:02.730 ","End":"07:05.520","Text":"we\u0027ll be using this formula here."},{"Start":"07:05.520 ","End":"07:07.695","Text":"Later when we get to OBC,"},{"Start":"07:07.695 ","End":"07:10.690","Text":"we\u0027ll be using this formula here."},{"Start":"07:11.150 ","End":"07:13.530","Text":"Let\u0027s see then."},{"Start":"07:13.530 ","End":"07:16.300","Text":"We need g_x and g_y,"},{"Start":"07:17.150 ","End":"07:20.085","Text":"and those are easy."},{"Start":"07:20.085 ","End":"07:25.425","Text":"G with respect to x is just a constant minus 2,"},{"Start":"07:25.425 ","End":"07:31.810","Text":"and g with respect to y is also the constant minus 2."},{"Start":"07:34.070 ","End":"07:39.200","Text":"What I\u0027m saying is that the S in our case is"},{"Start":"07:39.200 ","End":"07:47.935","Text":"just the ABC triangle of F.ndS,"},{"Start":"07:47.935 ","End":"07:50.920","Text":"is equal by the formula and I\u0027m using the top one,"},{"Start":"07:50.920 ","End":"07:54.440","Text":"is equal to the double integral over the projection,"},{"Start":"07:54.440 ","End":"08:02.265","Text":"which is the region R of F got,"},{"Start":"08:02.265 ","End":"08:05.700","Text":"and I have this vector already,"},{"Start":"08:05.700 ","End":"08:09.600","Text":"g_x is minus 2."},{"Start":"08:09.600 ","End":"08:12.444","Text":"Sorry, I\u0027m using the top formula."},{"Start":"08:12.444 ","End":"08:16.055","Text":"It\u0027s going to be plus 2 because it\u0027s minus g_x,"},{"Start":"08:16.055 ","End":"08:19.835","Text":"and then minus g_y will also be 2,"},{"Start":"08:19.835 ","End":"08:22.770","Text":"and here we just have a 1dA."},{"Start":"08:24.980 ","End":"08:29.840","Text":"Next, I want to compute the dot product of this with F,"},{"Start":"08:29.840 ","End":"08:31.190","Text":"which I didn\u0027t copy,"},{"Start":"08:31.190 ","End":"08:33.050","Text":"but it\u0027s over here."},{"Start":"08:33.050 ","End":"08:36.090","Text":"The dot product of these 2,"},{"Start":"08:36.650 ","End":"08:41.250","Text":"so I get the double integral."},{"Start":"08:41.250 ","End":"08:43.910","Text":"I\u0027m not writing R here because I\u0027m going to"},{"Start":"08:43.910 ","End":"08:46.880","Text":"replace this by an iterated integral in a moment."},{"Start":"08:46.880 ","End":"08:48.725","Text":"Let\u0027s just do the dot-product."},{"Start":"08:48.725 ","End":"08:56.050","Text":"I\u0027ve got 2 times this, 2xy plus z."},{"Start":"08:56.050 ","End":"08:58.295","Text":"But wherever I see z,"},{"Start":"08:58.295 ","End":"09:03.750","Text":"I replace z by 6 minus,"},{"Start":"09:04.820 ","End":"09:08.160","Text":"welll, wouldn\u0027t have to highlight this also,"},{"Start":"09:08.160 ","End":"09:10.660","Text":"that\u0027s what z is."},{"Start":"09:11.450 ","End":"09:16.890","Text":"Here I have 6 minus 2x minus 2y,"},{"Start":"09:16.890 ","End":"09:19.485","Text":"that\u0027s the first component, this with this."},{"Start":"09:19.485 ","End":"09:24.135","Text":"Now, 2 with y squared is just 2y squared."},{"Start":"09:24.135 ","End":"09:33.490","Text":"Then 1 with this gives me minus x plus 3y."},{"Start":"09:35.120 ","End":"09:43.430","Text":"All this is dA. You know what?"},{"Start":"09:43.430 ","End":"09:45.725","Text":"I\u0027ll write it first of all like this,"},{"Start":"09:45.725 ","End":"09:49.040","Text":"and now next step, I\u0027ll simplify this,"},{"Start":"09:49.040 ","End":"09:51.800","Text":"but I also want to change this integral over"},{"Start":"09:51.800 ","End":"09:56.460","Text":"the triangle to an iterated integral, something dx dy."},{"Start":"09:56.680 ","End":"10:01.535","Text":"I\u0027m thinking I\u0027ll bring in an extra sketch for R over here."},{"Start":"10:01.535 ","End":"10:06.810","Text":"Here is the sketch of the R in the xy-plane,"},{"Start":"10:06.810 ","End":"10:08.774","Text":"I\u0027ll just label it R,"},{"Start":"10:08.774 ","End":"10:11.660","Text":"and we already computed that this was 3,"},{"Start":"10:11.660 ","End":"10:13.055","Text":"and this was 3."},{"Start":"10:13.055 ","End":"10:17.020","Text":"What I need is the equation of this line here."},{"Start":"10:17.020 ","End":"10:22.335","Text":"Well, this is just where this plane cuts the xy-plane,"},{"Start":"10:22.335 ","End":"10:25.110","Text":"so if I said z equals 0 here,"},{"Start":"10:25.110 ","End":"10:29.040","Text":"and divide by 2, I get x plus y equals 3."},{"Start":"10:29.040 ","End":"10:33.495","Text":"This line is x plus y equals 3."},{"Start":"10:33.495 ","End":"10:36.620","Text":"In fact, since I\u0027m going to do it as an iterated integral,"},{"Start":"10:36.620 ","End":"10:39.725","Text":"I prefer to have 1 variable in terms of the others,"},{"Start":"10:39.725 ","End":"10:48.645","Text":"let\u0027s make it y in terms of x. I\u0027ll write it as y equals 3 minus x."},{"Start":"10:48.645 ","End":"10:54.630","Text":"Now we can take this region as a type 1 region,"},{"Start":"10:54.630 ","End":"11:00.650","Text":"meaning take vertical slices and different color,"},{"Start":"11:00.650 ","End":"11:04.445","Text":"and then when we cut through the region,"},{"Start":"11:04.445 ","End":"11:06.530","Text":"we cut here and here."},{"Start":"11:06.530 ","End":"11:09.200","Text":"This is y equals 3 minus x,"},{"Start":"11:09.200 ","End":"11:14.070","Text":"this is just the x-axis which is y equals 0,"},{"Start":"11:14.070 ","End":"11:16.965","Text":"and we notice that x goes from 0-3,"},{"Start":"11:16.965 ","End":"11:22.970","Text":"y goes from 0-3 minus x. I\u0027m going to rewrite the"},{"Start":"11:22.970 ","End":"11:33.025","Text":"integral as the integral where x goes from 0-3,"},{"Start":"11:33.025 ","End":"11:34.950","Text":"and then inside that,"},{"Start":"11:34.950 ","End":"11:42.015","Text":"y goes from 0-3 minus x"},{"Start":"11:42.015 ","End":"11:47.590","Text":"of all this dy dx."},{"Start":"11:49.230 ","End":"11:53.600","Text":"Now all I have to do is simplify this."},{"Start":"11:53.700 ","End":"11:57.145","Text":"Well, let\u0027s see what this comes out to."},{"Start":"11:57.145 ","End":"12:02.420","Text":"Let\u0027s see 2 times 2xy, that is 4xy."},{"Start":"12:02.910 ","End":"12:07.675","Text":"2 times 6 is 12."},{"Start":"12:07.675 ","End":"12:11.200","Text":"Now, from here we get minus 4x,"},{"Start":"12:11.200 ","End":"12:14.605","Text":"but we also have a minus x here,"},{"Start":"12:14.605 ","End":"12:18.055","Text":"so it\u0027ll be minus 5x."},{"Start":"12:18.055 ","End":"12:22.015","Text":"Let\u0027s see, for y we get minus 4y,"},{"Start":"12:22.015 ","End":"12:26.575","Text":"but we also have minus 3y,"},{"Start":"12:26.575 ","End":"12:36.820","Text":"that will be minus 7y and finally, plus 2y squared."},{"Start":"12:36.820 ","End":"12:46.435","Text":"This is the integral we have to compute to get the 1 of 4 just the ABC part."},{"Start":"12:46.435 ","End":"12:49.750","Text":"I\u0027m going to need some more space."},{"Start":"12:49.750 ","End":"12:53.995","Text":"This is equal to the integral"},{"Start":"12:53.995 ","End":"13:03.595","Text":"from x equals 0 to 3 of."},{"Start":"13:03.595 ","End":"13:07.490","Text":"Now let\u0027s see, with respect to y,"},{"Start":"13:08.280 ","End":"13:13.120","Text":"x is a constant so we get"},{"Start":"13:13.120 ","End":"13:19.990","Text":"4xy squared over 2 which is 2xy squared,"},{"Start":"13:19.990 ","End":"13:25.280","Text":"then plus 12y minus 5xy."},{"Start":"13:25.680 ","End":"13:31.930","Text":"Then for this we get minus 7 over"},{"Start":"13:31.930 ","End":"13:40.460","Text":"2y squared and here plus 2/3y cubed."},{"Start":"13:43.260 ","End":"13:46.670","Text":"That already is the integral."},{"Start":"13:48.780 ","End":"13:57.805","Text":"This bit has to be taken from 0"},{"Start":"13:57.805 ","End":"14:07.670","Text":"to 3 minus x for y and then you still have the integral dx."},{"Start":"14:08.700 ","End":"14:12.430","Text":"Should have done this as a side exercise."},{"Start":"14:12.430 ","End":"14:15.280","Text":"When y equals 0,"},{"Start":"14:15.280 ","End":"14:16.870","Text":"everything here is 0,"},{"Start":"14:16.870 ","End":"14:28.000","Text":"but we still have to plug in 3 minus x so we get the integral from 0 to 3 of 2x,"},{"Start":"14:28.000 ","End":"14:31.255","Text":"3 minus x squared,"},{"Start":"14:31.255 ","End":"14:33.745","Text":"need another brackets here,"},{"Start":"14:33.745 ","End":"14:40.495","Text":"plus 12 times 3 minus x minus 5x,"},{"Start":"14:40.495 ","End":"14:46.225","Text":"3 minus x minus 7 over 2,"},{"Start":"14:46.225 ","End":"14:55.045","Text":"3 minus x squared and then plus 2/3,"},{"Start":"14:55.045 ","End":"15:01.510","Text":"3 minus x cubed, dx."},{"Start":"15:01.510 ","End":"15:04.360","Text":"Yes, this is getting pretty messy."},{"Start":"15:04.360 ","End":"15:07.900","Text":"Let\u0027s see. Let\u0027s continue here."},{"Start":"15:07.900 ","End":"15:12.520","Text":"We get the integral from 0 to"},{"Start":"15:12.520 ","End":"15:20.050","Text":"3 and let me see if I can multiply this out."},{"Start":"15:20.050 ","End":"15:26.755","Text":"I\u0027d like to do these 2 separately at the side."},{"Start":"15:26.755 ","End":"15:30.190","Text":"The rest of them I can cope with."},{"Start":"15:30.190 ","End":"15:39.940","Text":"This 1 is going to be 2x times using the special binomial expansion,"},{"Start":"15:39.940 ","End":"15:48.910","Text":"this is going to be 3 squared is 9 minus twice 3 times x is 6x plus x squared"},{"Start":"15:48.910 ","End":"15:58.405","Text":"and this is equal to just 2x times 9 is"},{"Start":"15:58.405 ","End":"16:09.230","Text":"18x minus 12x squared plus 2x cubed."},{"Start":"16:09.240 ","End":"16:12.055","Text":"That\u0027s this 1."},{"Start":"16:12.055 ","End":"16:18.475","Text":"This 1 will give me minus"},{"Start":"16:18.475 ","End":"16:28.820","Text":"15x plus 5x squared."},{"Start":"16:29.730 ","End":"16:37.035","Text":"If I combine these 2 together and put them here, what do I get?"},{"Start":"16:37.035 ","End":"16:44.385","Text":"18x minus 15x is 3x minus 12x squared plus 5x"},{"Start":"16:44.385 ","End":"16:52.590","Text":"squared is minus 7x squared and then plus 2x cubed."},{"Start":"16:52.590 ","End":"16:54.535","Text":"Then the rest of them,"},{"Start":"16:54.535 ","End":"16:56.710","Text":"I\u0027ll leave as is,"},{"Start":"16:56.710 ","End":"17:02.590","Text":"I have 12, 3 minus x minus,"},{"Start":"17:02.590 ","End":"17:05.125","Text":"just copying the rest, 7 over 2,"},{"Start":"17:05.125 ","End":"17:09.710","Text":"3 minus x squared plus 2/3,"},{"Start":"17:09.710 ","End":"17:14.350","Text":"3 minus x cubed dx."},{"Start":"17:14.350 ","End":"17:17.395","Text":"Now ready to do the actual integral."},{"Start":"17:17.395 ","End":"17:20.740","Text":"For this part we just as usual,"},{"Start":"17:20.740 ","End":"17:27.760","Text":"3 over 2x squared minus 7 over 3x cubed here,"},{"Start":"17:27.760 ","End":"17:33.820","Text":"2 over 4, I can write as 1/2 x^4."},{"Start":"17:33.820 ","End":"17:35.350","Text":"Now in these remaining bits,"},{"Start":"17:35.350 ","End":"17:40.450","Text":"I have 3 minus x instead of x. I can treat it as if it was x,"},{"Start":"17:40.450 ","End":"17:43.390","Text":"but because it\u0027s an inner derivative of minus 1,"},{"Start":"17:43.390 ","End":"17:45.025","Text":"I\u0027ll have to divide by that."},{"Start":"17:45.025 ","End":"17:54.010","Text":"What I\u0027ll get is I\u0027ll take the 3 minus x squared over 2,"},{"Start":"17:54.010 ","End":"17:59.050","Text":"which will give me just 6 but instead of a plus I\u0027ll write a minus because of that"},{"Start":"17:59.050 ","End":"18:04.270","Text":"inner derivative and similarly here I\u0027m going to get the minus is going to become a plus."},{"Start":"18:04.270 ","End":"18:06.070","Text":"That takes care of the minus."},{"Start":"18:06.070 ","End":"18:07.840","Text":"I\u0027ll raise the power by 1,"},{"Start":"18:07.840 ","End":"18:12.535","Text":"that\u0027s 3 divide by 3 so it\u0027s 7 over 2 times 3,"},{"Start":"18:12.535 ","End":"18:17.860","Text":"7 over 6, 3 minus x cubed."},{"Start":"18:17.860 ","End":"18:21.790","Text":"Similar idea here, going to become a minus,"},{"Start":"18:21.790 ","End":"18:24.710","Text":"there\u0027s going to be a 4 here."},{"Start":"18:24.810 ","End":"18:30.640","Text":"I\u0027m going to divide by the 4 so I\u0027ll end up with 2 over 4"},{"Start":"18:30.640 ","End":"18:36.505","Text":"times 3 is just going to be 2 over 12 is 1/6."},{"Start":"18:36.505 ","End":"18:39.460","Text":"That\u0027s the integral and"},{"Start":"18:39.460 ","End":"18:49.135","Text":"now have to evaluate this between 0 and 3."},{"Start":"18:49.135 ","End":"18:52.960","Text":"Let\u0027s start with the 3."},{"Start":"18:52.960 ","End":"18:55.600","Text":"If I plug in 3,"},{"Start":"18:55.600 ","End":"19:01.615","Text":"I\u0027ll get 3 over 2 times 3 squared"},{"Start":"19:01.615 ","End":"19:10.165","Text":"is 27 over 2 is 13 and 1/2."},{"Start":"19:10.165 ","End":"19:13.390","Text":"Next, 3 cubed over 3 is like 3 squared,"},{"Start":"19:13.390 ","End":"19:17.570","Text":"which is 9 times 7 is 63."},{"Start":"19:17.760 ","End":"19:26.095","Text":"3^4 is 81, divided by 2 is 40 and 1/2."},{"Start":"19:26.095 ","End":"19:30.280","Text":"All the rest of them come out 0 when x is 3."},{"Start":"19:30.280 ","End":"19:35.590","Text":"This is the 3 part and now I have to subtract the 0 part."},{"Start":"19:35.590 ","End":"19:39.220","Text":"For the 0, this 3 come out 0."},{"Start":"19:39.220 ","End":"19:43.795","Text":"Now here, when x is 0,"},{"Start":"19:43.795 ","End":"19:47.260","Text":"I get 3 squared is 9"},{"Start":"19:47.260 ","End":"19:57.500","Text":"times minus 6 is minus 54."},{"Start":"19:57.570 ","End":"20:05.710","Text":"Here I got 7 over 6 times 3 cubed comes out."},{"Start":"20:05.710 ","End":"20:14.170","Text":"Here, I make it 31 and 1/2 and here I\u0027ll take"},{"Start":"20:14.170 ","End":"20:25.495","Text":"81 over 6 which is like 27 over 2,"},{"Start":"20:25.495 ","End":"20:29.575","Text":"is 13 and 1/2."},{"Start":"20:29.575 ","End":"20:32.185","Text":"Well, let\u0027s see now,"},{"Start":"20:32.185 ","End":"20:40.180","Text":"13 and 1/2 plus 40 and 1/2, I make that 54,"},{"Start":"20:40.180 ","End":"20:49.975","Text":"54 minus 63, that would be minus 9 minus,"},{"Start":"20:49.975 ","End":"20:53.180","Text":"now let\u0027s see what I get here."},{"Start":"20:53.670 ","End":"21:02.830","Text":"The minuses are 67 and 1/2,"},{"Start":"21:02.830 ","End":"21:13.480","Text":"31 and 1/2 minus 67 and 1/2 is minus 36 so what do I get altogether?"},{"Start":"21:13.480 ","End":"21:16.510","Text":"Plus 36 minus 9,"},{"Start":"21:16.510 ","End":"21:22.540","Text":"I make that 27 but that\u0027s not the answer."},{"Start":"21:22.540 ","End":"21:26.920","Text":"That\u0027s just the part for the triangle ABC,"},{"Start":"21:26.920 ","End":"21:29.740","Text":"1 of the 4 faces although it\u0027s the most difficult 1."},{"Start":"21:29.740 ","End":"21:32.630","Text":"Now let\u0027s scroll back up."},{"Start":"21:37.770 ","End":"21:44.600","Text":"I\u0027ll record this as 27 and now we\u0027ll tackle the other 3."}],"ID":9662},{"Watched":false,"Name":"Exercise 10 part b","Duration":"19m 54s","ChapterTopicVideoID":8796,"CourseChapterTopicPlaylistID":4974,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.930","Text":"Now we\u0027ll go and do the OBC,"},{"Start":"00:03.930 ","End":"00:07.425","Text":"and that would be this one."},{"Start":"00:07.425 ","End":"00:09.060","Text":"Now it\u0027s very similar."},{"Start":"00:09.060 ","End":"00:11.685","Text":"Let me erase what I don\u0027t need."},{"Start":"00:11.685 ","End":"00:16.680","Text":"The equation of this plane is z equals 0,"},{"Start":"00:16.680 ","End":"00:18.270","Text":"so our g of x,"},{"Start":"00:18.270 ","End":"00:20.825","Text":"y is equal to 0,"},{"Start":"00:20.825 ","End":"00:24.875","Text":"and then the partial derivatives will also be 0."},{"Start":"00:24.875 ","End":"00:31.970","Text":"The other difference is that here the outward normal has a downward component,"},{"Start":"00:31.970 ","End":"00:38.190","Text":"so instead of this formula we\u0027ll be using this formula."},{"Start":"00:38.330 ","End":"00:43.925","Text":"Then what we get is F. g_x,"},{"Start":"00:43.925 ","End":"00:53.165","Text":"g_y minus 1 is 0, 0, minus 1."},{"Start":"00:53.165 ","End":"01:01.660","Text":"Here\u0027s my F. When I do the dot product,"},{"Start":"01:02.060 ","End":"01:05.340","Text":"we will get the double integral,"},{"Start":"01:05.340 ","End":"01:12.269","Text":"and it\u0027s going to be the same R. This projection is the thing itself."},{"Start":"01:12.269 ","End":"01:13.830","Text":"One another small thing,"},{"Start":"01:13.830 ","End":"01:15.000","Text":"this is not an A now,"},{"Start":"01:15.000 ","End":"01:16.740","Text":"this is an O."},{"Start":"01:16.740 ","End":"01:20.700","Text":"It\u0027s OBC, and it\u0027s OBC."},{"Start":"01:20.700 ","End":"01:26.740","Text":"That\u0027s our piece of surface S. What did I say?"},{"Start":"01:26.740 ","End":"01:31.020","Text":"We just take the minus 1 of the last component of F,"},{"Start":"01:31.020 ","End":"01:33.520","Text":"so it\u0027s going to obliterate this minus,"},{"Start":"01:33.520 ","End":"01:40.045","Text":"and we\u0027ll just get x plus 3y."},{"Start":"01:40.045 ","End":"01:50.570","Text":"The minus 1 times this is this and the others is 0. dA,"},{"Start":"01:50.570 ","End":"01:52.470","Text":"and as I said it\u0027s the same R,"},{"Start":"01:52.470 ","End":"01:58.930","Text":"so we can break it up the same way into a double integral."},{"Start":"01:58.930 ","End":"02:08.440","Text":"This will be if I do it iterated will be the integral x goes from 0-3."},{"Start":"02:08.440 ","End":"02:10.840","Text":"Then for each such x,"},{"Start":"02:10.840 ","End":"02:15.565","Text":"y goes from 0-3 minus x."},{"Start":"02:15.565 ","End":"02:17.665","Text":"The difference is that it\u0027s this,"},{"Start":"02:17.665 ","End":"02:27.670","Text":"it\u0027s x plus 3y dy, dx."},{"Start":"02:27.670 ","End":"02:31.055","Text":"We\u0027ll start from the inside."},{"Start":"02:31.055 ","End":"02:35.040","Text":"I\u0027ll get some more space here."},{"Start":"02:35.040 ","End":"02:37.930","Text":"Let\u0027s continue here."},{"Start":"02:37.930 ","End":"02:41.970","Text":"We\u0027ve got the double integral,"},{"Start":"02:43.000 ","End":"02:46.455","Text":"x goes from 0 to 3."},{"Start":"02:46.455 ","End":"02:49.545","Text":"Now I\u0027m going to evaluate this one,"},{"Start":"02:49.545 ","End":"02:53.140","Text":"I mean the dy_1."},{"Start":"02:55.160 ","End":"03:06.375","Text":"What I have is xy plus 3/2 y squared."},{"Start":"03:06.375 ","End":"03:13.755","Text":"This taken, that is for y,"},{"Start":"03:13.755 ","End":"03:17.310","Text":"from 0-3 minus x,"},{"Start":"03:17.310 ","End":"03:20.910","Text":"dx and this is equal"},{"Start":"03:20.910 ","End":"03:29.240","Text":"to the integral from 0 to 3."},{"Start":"03:29.240 ","End":"03:32.680","Text":"If I put y equals 0, I don\u0027t get anything,"},{"Start":"03:32.680 ","End":"03:42.385","Text":"so it\u0027s just the 3 minus x. I get x times 3 minus x plus"},{"Start":"03:42.385 ","End":"03:45.430","Text":"3 over 2 times 3"},{"Start":"03:45.430 ","End":"03:54.410","Text":"minus x squared, dx."},{"Start":"03:54.410 ","End":"03:57.690","Text":"Perhaps I\u0027ll do this at the side."},{"Start":"03:59.180 ","End":"04:01.935","Text":"Well, maybe not."},{"Start":"04:01.935 ","End":"04:04.380","Text":"I think we\u0027ll just go ahead and do it here,"},{"Start":"04:04.380 ","End":"04:09.780","Text":"get some space, get integral from 0 to 3."},{"Start":"04:09.780 ","End":"04:20.260","Text":"Here I have 3x minus x squared,"},{"Start":"04:24.640 ","End":"04:27.815","Text":"I\u0027ll just copy this as it is,"},{"Start":"04:27.815 ","End":"04:37.380","Text":"plus 3/2 times 3"},{"Start":"04:37.380 ","End":"04:44.675","Text":"minus x squared, this dx."},{"Start":"04:44.675 ","End":"04:47.375","Text":"Now do the actual integral."},{"Start":"04:47.375 ","End":"04:58.095","Text":"Here I\u0027ve got 3/2 x squared minus 1/3x cubed."},{"Start":"04:58.095 ","End":"05:03.740","Text":"Here I\u0027ll look at it as a function of 3 minus x."},{"Start":"05:05.000 ","End":"05:10.305","Text":"What I do is I take the 3 minus x and instead of 2,"},{"Start":"05:10.305 ","End":"05:12.360","Text":"make it to the power of 3,"},{"Start":"05:12.360 ","End":"05:15.940","Text":"and then I have to divide by 3."},{"Start":"05:16.430 ","End":"05:19.290","Text":"Now instead of putting over 3,"},{"Start":"05:19.290 ","End":"05:21.495","Text":"I can just cancel with this 3."},{"Start":"05:21.495 ","End":"05:23.955","Text":"This just becomes 1/2."},{"Start":"05:23.955 ","End":"05:26.540","Text":"Then there\u0027s also the matter of the inner derivative."},{"Start":"05:26.540 ","End":"05:31.580","Text":"It\u0027s not x, it\u0027s 3 minus x. I need to also divide by minus 1,"},{"Start":"05:31.580 ","End":"05:34.325","Text":"so I put a minus here."},{"Start":"05:34.325 ","End":"05:39.660","Text":"Now that\u0027s the integral already."},{"Start":"05:39.660 ","End":"05:48.300","Text":"I have to just evaluate this from 0-3 for x."},{"Start":"05:48.300 ","End":"05:50.370","Text":"What we get is,"},{"Start":"05:50.370 ","End":"05:52.935","Text":"if we put in 3,"},{"Start":"05:52.935 ","End":"05:59.190","Text":"we get 3/2 times 3 squared is 27/2."},{"Start":"05:59.190 ","End":"06:02.010","Text":"I\u0027ll write it as 13 and 1/2."},{"Start":"06:02.010 ","End":"06:04.424","Text":"Here, if x is 3,"},{"Start":"06:04.424 ","End":"06:10.200","Text":"x cubed over 3 is like just 3 cubed over 3 is 3 squared,"},{"Start":"06:10.200 ","End":"06:13.665","Text":"that\u0027s 9, and that\u0027s a minus."},{"Start":"06:13.665 ","End":"06:18.735","Text":"Here when x is 3, it\u0027s just 0."},{"Start":"06:18.735 ","End":"06:20.745","Text":"This is the 3 part,"},{"Start":"06:20.745 ","End":"06:23.870","Text":"and now I have to subtract the 0 part."},{"Start":"06:23.870 ","End":"06:26.344","Text":"The 0 part, this is 0,"},{"Start":"06:26.344 ","End":"06:31.715","Text":"this is 0, 3 minus 0 is 3,"},{"Start":"06:31.715 ","End":"06:35.960","Text":"so it\u0027s minus 1/2 3 cubed,"},{"Start":"06:35.960 ","End":"06:45.145","Text":"minus 27/2 minus 13 and 1/2."},{"Start":"06:45.145 ","End":"06:46.680","Text":"Altogether, what do I get?"},{"Start":"06:46.680 ","End":"06:50.880","Text":"It\u0027s 13.5 plus 13.5 is 27,"},{"Start":"06:50.880 ","End":"06:55.845","Text":"27 minus 9 is 18,"},{"Start":"06:55.845 ","End":"07:03.750","Text":"and this would be the answer for the OBC triangle."},{"Start":"07:03.750 ","End":"07:06.285","Text":"Now I\u0027ll just highlight that."},{"Start":"07:06.285 ","End":"07:10.110","Text":"That\u0027s the second of 4."},{"Start":"07:10.110 ","End":"07:16.550","Text":"I\u0027ll just go back up and record that result that we have."},{"Start":"07:16.550 ","End":"07:23.040","Text":"Over here, we have 18 was the answer."},{"Start":"07:23.040 ","End":"07:25.850","Text":"Now let\u0027s move on to the next one,"},{"Start":"07:25.850 ","End":"07:31.830","Text":"which is OAB, which is this one here."},{"Start":"07:32.200 ","End":"07:35.755","Text":"Let me erase what I don\u0027t need."},{"Start":"07:35.755 ","End":"07:39.320","Text":"Here we are. I erased some, I replaced some."},{"Start":"07:39.320 ","End":"07:41.750","Text":"This time we\u0027re on this face here,"},{"Start":"07:41.750 ","End":"07:46.620","Text":"which is the x equals 0 face."},{"Start":"07:46.880 ","End":"07:52.310","Text":"Its projection is just itself because it isn\u0027t the z-y plane."},{"Start":"07:52.310 ","End":"07:56.494","Text":"But this time we\u0027re going to have x as a function of y and z."},{"Start":"07:56.494 ","End":"07:59.810","Text":"This time we have x equals some function,"},{"Start":"07:59.810 ","End":"08:03.620","Text":"I\u0027ll also call it g of y and z,"},{"Start":"08:03.620 ","End":"08:07.120","Text":"and that function is just 0."},{"Start":"08:07.120 ","End":"08:13.759","Text":"Here\u0027s the picture in the z-y plane of this projection."},{"Start":"08:13.759 ","End":"08:17.930","Text":"This is the region R. I also have to replace these formulas."},{"Start":"08:17.930 ","End":"08:21.830","Text":"These were based on a function z equals g of x and"},{"Start":"08:21.830 ","End":"08:29.555","Text":"y. I\u0027ve just replaced them for the case where x is a function of y and z,"},{"Start":"08:29.555 ","End":"08:35.555","Text":"which we also call g. Slight differences."},{"Start":"08:35.555 ","End":"08:38.435","Text":"If you look at the previous, it\u0027s just analogous."},{"Start":"08:38.435 ","End":"08:40.670","Text":"Which of these two do we need?"},{"Start":"08:40.670 ","End":"08:44.750","Text":"Well, the first formula applies for when the"},{"Start":"08:44.750 ","End":"08:49.940","Text":"normal has a component in the negative x direction,"},{"Start":"08:49.940 ","End":"08:56.129","Text":"so that the first component of the 3 in the vector is negative."},{"Start":"08:56.129 ","End":"08:58.670","Text":"This one\u0027s for the positive,"},{"Start":"08:58.670 ","End":"09:02.225","Text":"and this one\u0027s for the negative x component."},{"Start":"09:02.225 ","End":"09:05.315","Text":"Now in our case, if I pick a point here,"},{"Start":"09:05.315 ","End":"09:08.825","Text":"the normal is going to be outwards."},{"Start":"09:08.825 ","End":"09:13.135","Text":"It\u0027s going to be going against the direction of x,"},{"Start":"09:13.135 ","End":"09:16.240","Text":"so it\u0027s a negative x,"},{"Start":"09:16.240 ","End":"09:18.290","Text":"and so we\u0027re going to use this formula."},{"Start":"09:18.290 ","End":"09:20.450","Text":"The arrow was there up before, but that\u0027s right."},{"Start":"09:20.450 ","End":"09:23.225","Text":"This is the formula we\u0027re going to be using."},{"Start":"09:23.225 ","End":"09:26.550","Text":"We need gy and gz."},{"Start":"09:26.930 ","End":"09:30.870","Text":"Obviously since this is the 0 function,"},{"Start":"09:30.870 ","End":"09:35.565","Text":"the derivatives are also gy and gz,"},{"Start":"09:35.565 ","End":"09:40.160","Text":"are both 0, so that this vector that I\u0027m going to be"},{"Start":"09:40.160 ","End":"09:46.150","Text":"using in our case is going to be minus 1, 0, 0."},{"Start":"09:50.590 ","End":"09:59.840","Text":"The integral over OAB of F. n dS for"},{"Start":"09:59.840 ","End":"10:08.690","Text":"our case where x is a function of y and z and the normal is in the negative x direction,"},{"Start":"10:08.690 ","End":"10:16.220","Text":"this is equal to the double integral over R. The R"},{"Start":"10:16.220 ","End":"10:24.780","Text":"is actually the same as the side of the pyramid but it\u0027s R as in this picture."},{"Start":"10:25.870 ","End":"10:30.890","Text":"The dot product, since it\u0027s only a minus 1 and these two are 0,"},{"Start":"10:30.890 ","End":"10:39.830","Text":"I just have to take minus 1 times the x component, which is this,"},{"Start":"10:39.830 ","End":"10:43.580","Text":"which is minus"},{"Start":"10:45.670 ","End":"10:57.735","Text":"1 times 2xy plus z."},{"Start":"10:57.735 ","End":"11:00.490","Text":"But that\u0027s not quite right,"},{"Start":"11:00.490 ","End":"11:02.980","Text":"because I don\u0027t want x,"},{"Start":"11:02.980 ","End":"11:10.790","Text":"I want what x is equal to and x is equal to 0."},{"Start":"11:12.360 ","End":"11:16.130","Text":"I\u0027ll leave it like this for the moment."},{"Start":"11:17.940 ","End":"11:24.715","Text":"I replace x because it\u0027s just going to be a function of y and z by 0."},{"Start":"11:24.715 ","End":"11:26.545","Text":"I get z with the minus,"},{"Start":"11:26.545 ","End":"11:32.240","Text":"it\u0027s the double integral over R of minus zdA."},{"Start":"11:33.780 ","End":"11:36.400","Text":"To do this as an iterated integral,"},{"Start":"11:36.400 ","End":"11:39.475","Text":"I\u0027ll be needing the equation of this line here."},{"Start":"11:39.475 ","End":"11:44.785","Text":"I can get this from here by letting x equal 0."},{"Start":"11:44.785 ","End":"11:49.795","Text":"For that x equals 0, I get that z equals 6 minus 2y."},{"Start":"11:49.795 ","End":"11:58.120","Text":"This line here is given by z equals 6 minus 2y,"},{"Start":"11:58.120 ","End":"12:03.295","Text":"and as before, this is not y equals 0."},{"Start":"12:03.295 ","End":"12:07.850","Text":"This is z equals 0 in the z, y plane."},{"Start":"12:07.860 ","End":"12:14.185","Text":"Our vertical slices, go from 0-6 minus 2y,"},{"Start":"12:14.185 ","End":"12:17.965","Text":"and y goes from 0-3."},{"Start":"12:17.965 ","End":"12:23.754","Text":"I can rewrite this as an iterated double integral."},{"Start":"12:23.754 ","End":"12:29.740","Text":"The outer loop on y going from 0-3."},{"Start":"12:29.740 ","End":"12:39.670","Text":"For each y, z goes from 0-6 minus 2y."},{"Start":"12:39.670 ","End":"12:49.280","Text":"I still have the minus z, and it\u0027s dz, dy."},{"Start":"12:50.520 ","End":"12:54.710","Text":"It getting cramped. Let\u0027s move over."},{"Start":"12:56.040 ","End":"12:58.900","Text":"This was equal to this,"},{"Start":"12:58.900 ","End":"13:03.865","Text":"and now let\u0027s move down here and let\u0027s see what it\u0027s equal to."},{"Start":"13:03.865 ","End":"13:06.950","Text":"We\u0027ll do the inner one first."},{"Start":"13:10.040 ","End":"13:18.129","Text":"This bit. I could take the minus outside the integral,"},{"Start":"13:18.129 ","End":"13:22.540","Text":"and get y goes from 0-3."},{"Start":"13:22.540 ","End":"13:29.215","Text":"The integral of z, dz is 1 1/2 z squared,"},{"Start":"13:29.215 ","End":"13:37.390","Text":"which I have to evaluate from 0-6 minus 2y."},{"Start":"13:37.390 ","End":"13:41.270","Text":"Then that\u0027s going to be dy."},{"Start":"13:41.370 ","End":"13:50.125","Text":"This will equal minus 1/2."},{"Start":"13:50.125 ","End":"13:52.810","Text":"I can also take the 1/2 out,"},{"Start":"13:52.810 ","End":"14:03.865","Text":"of just z squared from the integral from 0-3."},{"Start":"14:03.865 ","End":"14:07.750","Text":"Z squared from here to here,"},{"Start":"14:07.750 ","End":"14:15.220","Text":"is going to be just 6 minus 2y squared, minus 0 squared."},{"Start":"14:15.220 ","End":"14:17.380","Text":"The zeros squared doesn\u0027t matter,"},{"Start":"14:17.380 ","End":"14:21.860","Text":"this is basically what we\u0027re left with, dy."},{"Start":"14:23.700 ","End":"14:28.600","Text":"Let\u0027s see, I can do this integral right away."},{"Start":"14:28.600 ","End":"14:32.830","Text":"It could do a substitution 6 minus 2y, but we don\u0027t need to."},{"Start":"14:32.830 ","End":"14:34.915","Text":"It\u0027s a linear function of y."},{"Start":"14:34.915 ","End":"14:38.785","Text":"We start out pretending that this whole thing was like y,"},{"Start":"14:38.785 ","End":"14:44.690","Text":"and we would get this thing cubed over 3."},{"Start":"14:46.170 ","End":"14:49.450","Text":"But then we would say, it wasn\u0027t,"},{"Start":"14:49.450 ","End":"14:53.230","Text":"it was 6 minus 2y while the [inaudible] derivative is minus 2,"},{"Start":"14:53.230 ","End":"14:56.815","Text":"so I have to divide that by minus 2."},{"Start":"14:56.815 ","End":"15:00.260","Text":"Let\u0027s write minus 1/2."},{"Start":"15:00.600 ","End":"15:04.090","Text":"Then I also have a minus 1/2 here,"},{"Start":"15:04.090 ","End":"15:06.715","Text":"so it\u0027s minus 1/2, minus 1/2,"},{"Start":"15:06.715 ","End":"15:11.425","Text":"times 1/3 times 6 minus 2y cubed,"},{"Start":"15:11.425 ","End":"15:16.070","Text":"and all this from 0-3."},{"Start":"15:17.010 ","End":"15:23.400","Text":"This is now easy to evaluate."},{"Start":"15:23.400 ","End":"15:26.310","Text":"When we put in 3,"},{"Start":"15:26.310 ","End":"15:28.665","Text":"6 minus twice 3 is 0,"},{"Start":"15:28.665 ","End":"15:31.455","Text":"so this all comes out to be 0."},{"Start":"15:31.455 ","End":"15:35.860","Text":"When I put in 0, let\u0027s see what I get."},{"Start":"15:35.860 ","End":"15:41.170","Text":"If I put in 0, well, I\u0027ll just write it."},{"Start":"15:41.170 ","End":"15:43.795","Text":"It\u0027s 1/2 times 1/2."},{"Start":"15:43.795 ","End":"15:46.539","Text":"I can forget, the minus will cancel with the minus,"},{"Start":"15:46.539 ","End":"15:52.190","Text":"times 1/3 times 6 cubed."},{"Start":"15:53.310 ","End":"15:59.560","Text":"Any event it\u0027s going to come out negative. Now let\u0027s see."},{"Start":"15:59.560 ","End":"16:04.405","Text":"One of the 6s will cancel with a 2 and a 3,"},{"Start":"16:04.405 ","End":"16:09.535","Text":"so it\u0027s 6 squared over 2, 36 over 2."},{"Start":"16:09.535 ","End":"16:14.120","Text":"I make it minus 18,"},{"Start":"16:14.220 ","End":"16:17.425","Text":"and I\u0027ll highlight it."},{"Start":"16:17.425 ","End":"16:21.655","Text":"This was the answer for OAB."},{"Start":"16:21.655 ","End":"16:32.680","Text":"I\u0027m going to go back up and mark that as this was AOB, that\u0027s minus 18."},{"Start":"16:32.680 ","End":"16:35.290","Text":"We only have one more to go,"},{"Start":"16:35.290 ","End":"16:38.215","Text":"which is the OAC."},{"Start":"16:38.215 ","End":"16:42.040","Text":"I\u0027m going to erase what I don\u0027t need."},{"Start":"16:42.040 ","End":"16:46.300","Text":"Now some replacements, I\u0027ll change the B to a"},{"Start":"16:46.300 ","End":"16:53.169","Text":"C. I\u0027ll also change this because now y is 0,"},{"Start":"16:53.169 ","End":"16:56.485","Text":"and y is a function of x and z."},{"Start":"16:56.485 ","End":"17:00.550","Text":"Here y, here x."},{"Start":"17:00.550 ","End":"17:05.980","Text":"Of course, we\u0027ll need the partial derivative g with respect to x is 0,"},{"Start":"17:05.980 ","End":"17:10.930","Text":"and g with respect to z equals 0. What else?"},{"Start":"17:10.930 ","End":"17:15.580","Text":"I want to highlight what we\u0027re talking about."},{"Start":"17:15.580 ","End":"17:19.270","Text":"Here this is the side we\u0027re talking about,"},{"Start":"17:19.270 ","End":"17:21.370","Text":"the face of the pyramid."},{"Start":"17:21.370 ","End":"17:23.350","Text":"This picture will still work,"},{"Start":"17:23.350 ","End":"17:30.880","Text":"but this one has to be z and this one has to be x. I have to change."},{"Start":"17:30.880 ","End":"17:35.630","Text":"Now this formula, revise this one."},{"Start":"17:35.910 ","End":"17:39.610","Text":"I\u0027ve made the modifications."},{"Start":"17:39.610 ","End":"17:42.250","Text":"We have the g_x and the g_z."},{"Start":"17:42.250 ","End":"17:43.945","Text":"Wherever there\u0027s a g_y,"},{"Start":"17:43.945 ","End":"17:45.715","Text":"it\u0027s 1 or minus 1."},{"Start":"17:45.715 ","End":"17:54.280","Text":"We use the top formula when the normal has a positive y middle component."},{"Start":"17:54.280 ","End":"17:59.499","Text":"We use this one when it has a negative y component or j component."},{"Start":"17:59.499 ","End":"18:05.935","Text":"In our case, if we take a point on this plane,"},{"Start":"18:05.935 ","End":"18:12.295","Text":"the normal goes exactly in the opposite direction as the y-direction."},{"Start":"18:12.295 ","End":"18:17.425","Text":"This is correct, this is the formula we\u0027re going to be using."},{"Start":"18:17.425 ","End":"18:26.150","Text":"In fact, this one comes out to be 0 minus 1, 0."},{"Start":"18:26.370 ","End":"18:36.340","Text":"Here we need the double integral over this R of F"},{"Start":"18:36.340 ","End":"18:40.810","Text":"dot with 0 minus 1,"},{"Start":"18:40.810 ","End":"18:47.740","Text":"0 dA."},{"Start":"18:47.740 ","End":"18:55.000","Text":"Now if we look at F dot product [inaudible] with this."},{"Start":"18:55.000 ","End":"18:57.640","Text":"We\u0027ll get minus y squared."},{"Start":"18:57.640 ","End":"18:58.960","Text":"The minus 1 on the y squared,"},{"Start":"18:58.960 ","End":"19:01.340","Text":"all the rest are 0."},{"Start":"19:01.500 ","End":"19:09.115","Text":"This is equal to the double integral over R of minus y squared dA."},{"Start":"19:09.115 ","End":"19:12.820","Text":"But y equals 0."},{"Start":"19:12.820 ","End":"19:16.225","Text":"Since y equals 0,"},{"Start":"19:16.225 ","End":"19:18.985","Text":"this is just equal to 0."},{"Start":"19:18.985 ","End":"19:21.595","Text":"Note that because y equals 0,"},{"Start":"19:21.595 ","End":"19:24.595","Text":"that\u0027s the plane we\u0027re on here."},{"Start":"19:24.595 ","End":"19:27.490","Text":"One of them came out easy,"},{"Start":"19:27.490 ","End":"19:35.730","Text":"and I can just say that for this one, was for OAC."},{"Start":"19:35.730 ","End":"19:37.380","Text":"We\u0027ve done the 4th one out of 4."},{"Start":"19:37.380 ","End":"19:41.700","Text":"This one is a 0, and so the grand total,"},{"Start":"19:41.700 ","End":"19:43.140","Text":"adding all these up."},{"Start":"19:43.140 ","End":"19:46.290","Text":"This is 0, these 2 cancel each other out."},{"Start":"19:46.290 ","End":"19:49.605","Text":"We just get 27,"},{"Start":"19:49.605 ","End":"19:53.950","Text":"and that is our final answer."}],"ID":9663},{"Watched":false,"Name":"Exercise 11","Duration":"10m 56s","ChapterTopicVideoID":8797,"CourseChapterTopicPlaylistID":4974,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.700","Text":"In this exercise, we want to compute this type to surface integral,"},{"Start":"00:05.700 ","End":"00:10.260","Text":"where the vector field F is given as follows,"},{"Start":"00:10.260 ","End":"00:14.370","Text":"it\u0027s actually a constant vector field."},{"Start":"00:14.370 ","End":"00:22.500","Text":"S is part of the paraboloid that\u0027s above the x, y plane."},{"Start":"00:22.500 ","End":"00:25.515","Text":"In other words, z is bigger or equal to 0."},{"Start":"00:25.515 ","End":"00:28.630","Text":"Sketch will really help here."},{"Start":"00:28.670 ","End":"00:31.545","Text":"Here\u0027s the picture."},{"Start":"00:31.545 ","End":"00:35.100","Text":"What we\u0027re missing is the,"},{"Start":"00:35.100 ","End":"00:39.910","Text":"I need a sketch of the projection onto the x, y plane."},{"Start":"00:39.910 ","End":"00:44.375","Text":"What I can do is to say because z is bigger or equal to 0,"},{"Start":"00:44.375 ","End":"00:49.490","Text":"that 4 minus x squared minus y squared bigger or equal to 0,"},{"Start":"00:49.490 ","End":"00:55.475","Text":"and that will give me that x squared plus y squared is less than or equal to 4,"},{"Start":"00:55.475 ","End":"00:58.140","Text":"which is 2 squared."},{"Start":"00:58.910 ","End":"01:02.180","Text":"This will give this region,"},{"Start":"01:02.180 ","End":"01:05.120","Text":"I\u0027ll call it R. On here,"},{"Start":"01:05.120 ","End":"01:12.340","Text":"it would be like if I trace this out here and then there\u0027s an invisible hidden part here,"},{"Start":"01:12.340 ","End":"01:14.465","Text":"anyway, you get the idea,"},{"Start":"01:14.465 ","End":"01:16.820","Text":"project this down, I get this."},{"Start":"01:16.820 ","End":"01:19.130","Text":"On here I have this as the function."},{"Start":"01:19.130 ","End":"01:21.290","Text":"vector 1, I give this a letter,"},{"Start":"01:21.290 ","End":"01:25.190","Text":"I will call this g of x and y,"},{"Start":"01:25.190 ","End":"01:27.870","Text":"so that z is g of xy."},{"Start":"01:27.870 ","End":"01:32.890","Text":"We have formulas for the case where z is a function of x and y,"},{"Start":"01:32.890 ","End":"01:35.570","Text":"actually there\u0027s 2 formulas."},{"Start":"01:35.570 ","End":"01:37.160","Text":"It depends on whether the normal,"},{"Start":"01:37.160 ","End":"01:41.010","Text":"I\u0027ll take a point on this paraboloid,"},{"Start":"01:41.170 ","End":"01:44.090","Text":"has an upward component."},{"Start":"01:44.090 ","End":"01:45.875","Text":"I\u0027m not saying it\u0027s vertical,"},{"Start":"01:45.875 ","End":"01:49.670","Text":"but it has a positive z component."},{"Start":"01:49.670 ","End":"01:51.485","Text":"When this is the case,"},{"Start":"01:51.485 ","End":"01:55.880","Text":"then the formula that applies is from the theorem."},{"Start":"01:55.880 ","End":"01:58.280","Text":"I\u0027ll just label this, S is the surface,"},{"Start":"01:58.280 ","End":"02:05.345","Text":"is that the integral over S of"},{"Start":"02:05.345 ","End":"02:15.754","Text":"F.n ds is equal to the regular,"},{"Start":"02:15.754 ","End":"02:17.090","Text":"not the surface integral,"},{"Start":"02:17.090 ","End":"02:24.375","Text":"the regular double integral over the region R of f.,"},{"Start":"02:24.375 ","End":"02:27.980","Text":"and I\u0027m going to use angular brackets notation."},{"Start":"02:27.980 ","End":"02:35.040","Text":"For example, f in angular brackets notation would be the vector 5,2,3,"},{"Start":"02:35.040 ","End":"02:38.410","Text":"some people use round brackets."},{"Start":"02:38.530 ","End":"02:49.375","Text":"Here we have the formula is minus g_x minus g_y 1."},{"Start":"02:49.375 ","End":"02:51.320","Text":"If it was downward facing,"},{"Start":"02:51.320 ","End":"02:53.540","Text":"if it had a downward component on the normal,"},{"Start":"02:53.540 ","End":"02:57.290","Text":"we would just reverse all the signs here, plus, plus, minus."},{"Start":"02:57.290 ","End":"02:58.490","Text":"But we don\u0027t need that,"},{"Start":"02:58.490 ","End":"02:59.840","Text":"we just need this."},{"Start":"02:59.840 ","End":"03:02.330","Text":"Let\u0027s see what we get in our case."},{"Start":"03:02.330 ","End":"03:08.285","Text":"We get the double integral over S of F.n"},{"Start":"03:08.285 ","End":"03:16.530","Text":"ds equals the double integral over R of,"},{"Start":"03:16.530 ","End":"03:22.780","Text":"F is the vector 5,2,3.,"},{"Start":"03:27.410 ","End":"03:34.855","Text":"now let\u0027s see, minus g with respect to x is minus 2x."},{"Start":"03:34.855 ","End":"03:37.965","Text":"But I need it minus,"},{"Start":"03:37.965 ","End":"03:41.270","Text":"so I\u0027m going to actually erase this minus,"},{"Start":"03:41.270 ","End":"03:45.050","Text":"I\u0027ll just write it as plus to show that it\u0027s minus, minus."},{"Start":"03:45.050 ","End":"03:48.110","Text":"Then I need minus g_y,"},{"Start":"03:48.110 ","End":"03:57.720","Text":"which is plus 2y, and then 1."},{"Start":"03:58.990 ","End":"04:10.550","Text":"All that is, da."},{"Start":"04:10.550 ","End":"04:13.260","Text":"This is equal to,"},{"Start":"04:13.390 ","End":"04:18.530","Text":"let\u0027s see, just do the dot product."},{"Start":"04:18.530 ","End":"04:25.289","Text":"We have 5 times 2x is 10x,"},{"Start":"04:25.289 ","End":"04:31.320","Text":"and then 2 times 2y is 4y,"},{"Start":"04:31.320 ","End":"04:35.960","Text":"and then 3 times 1 is 3."},{"Start":"04:35.960 ","End":"04:41.645","Text":"We want this integral, da of A."},{"Start":"04:41.645 ","End":"04:47.470","Text":"Now, I want to convert this to polar coordinates,"},{"Start":"04:47.470 ","End":"04:51.640","Text":"and I\u0027ll remind you of the equations for polar conversion,"},{"Start":"04:51.640 ","End":"04:53.110","Text":"I\u0027ll write them over here."},{"Start":"04:53.110 ","End":"04:56.885","Text":"We have that x equals r cosine Theta."},{"Start":"04:56.885 ","End":"04:59.195","Text":"We go from xy to r and Theta."},{"Start":"04:59.195 ","End":"05:03.460","Text":"We put y equals r sine Theta,"},{"Start":"05:03.460 ","End":"05:09.775","Text":"instead of da, we put r dr dTheta."},{"Start":"05:09.775 ","End":"05:17.820","Text":"There\u0027s another useful formula that x squared plus y squared equals r squared,"},{"Start":"05:17.820 ","End":"05:20.530","Text":"I don\u0027t know if we\u0027ll use that one here."},{"Start":"05:20.540 ","End":"05:26.110","Text":"Anyway, so back here and we\u0027ll get,"},{"Start":"05:26.110 ","End":"05:29.450","Text":"and it will change color for polar."},{"Start":"05:29.450 ","End":"05:32.030","Text":"I also want to change the region."},{"Start":"05:32.030 ","End":"05:36.725","Text":"We\u0027ve seen this region before when we have a disc,"},{"Start":"05:36.725 ","End":"05:38.660","Text":"I\u0027ll just quickly remind you,"},{"Start":"05:38.660 ","End":"05:41.030","Text":"Theta goes all the way around,"},{"Start":"05:41.030 ","End":"05:45.110","Text":"it goes from 0 all the way around to 2 Pi."},{"Start":"05:45.110 ","End":"05:46.850","Text":"From here, it goes all the way around,"},{"Start":"05:46.850 ","End":"05:49.204","Text":"comes back here, and r,"},{"Start":"05:49.204 ","End":"05:54.665","Text":"the radius goes from being 0 here to being 2 here."},{"Start":"05:54.665 ","End":"06:01.110","Text":"So we\u0027ve got the integral from 0-2 Pi for Theta,"},{"Start":"06:01.390 ","End":"06:06.265","Text":"for r we go from 0-2,"},{"Start":"06:06.265 ","End":"06:10.445","Text":"and then we have this which we substitute as 10."},{"Start":"06:10.445 ","End":"06:13.925","Text":"Now x is r cosine Theta."},{"Start":"06:13.925 ","End":"06:22.770","Text":"Here we have 4r sine Theta plus 3rdrdTheta,"},{"Start":"06:25.280 ","End":"06:27.735","Text":"notice the r here,"},{"Start":"06:27.735 ","End":"06:29.220","Text":"one tends to forget it,"},{"Start":"06:29.220 ","End":"06:31.900","Text":"then see what we get."},{"Start":"06:32.600 ","End":"06:35.610","Text":"Let\u0027s multiply this by r,"},{"Start":"06:35.610 ","End":"06:39.240","Text":"we go from 0-2 Pi with Theta,"},{"Start":"06:39.240 ","End":"06:44.820","Text":"and then we have multiplying everything by r 10r squared"},{"Start":"06:44.820 ","End":"06:51.915","Text":"cosine Theta plus 4 squared sine Theta,"},{"Start":"06:51.915 ","End":"06:55.470","Text":"and then plus 3r."},{"Start":"06:55.470 ","End":"07:01.980","Text":"All this, dr dTheta still not the integral."},{"Start":"07:01.980 ","End":"07:04.359","Text":"Now we\u0027ll do the integral."},{"Start":"07:04.550 ","End":"07:09.645","Text":"Sorry, I forgot to write r goes from 0-2,"},{"Start":"07:09.645 ","End":"07:13.900","Text":"and then this equals the integral from 0-2 Pi."},{"Start":"07:13.900 ","End":"07:15.800","Text":"The integral of this,"},{"Start":"07:15.800 ","End":"07:17.570","Text":"I raise the power by 1,"},{"Start":"07:17.570 ","End":"07:19.505","Text":"cosine Theta is a constant of course,"},{"Start":"07:19.505 ","End":"07:29.264","Text":"so I get 10 over 3r cubed cosine Theta,"},{"Start":"07:29.264 ","End":"07:36.029","Text":"and here 4 over 3r cubed sine Theta,"},{"Start":"07:36.029 ","End":"07:40.570","Text":"and here 3 over 2 squared."},{"Start":"07:40.570 ","End":"07:42.580","Text":"That\u0027s really the integral, dr."},{"Start":"07:42.580 ","End":"07:47.569","Text":"But I do have to substitute that it goes from 0-2,"},{"Start":"07:48.590 ","End":"07:53.560","Text":"and then after that I still have to do an integral d Theta will get rid of r this way,"},{"Start":"07:53.560 ","End":"07:56.695","Text":"this is r from 0-2."},{"Start":"07:56.695 ","End":"08:01.120","Text":"Now, I plug in r equals 0, everything\u0027s going to be 0,"},{"Start":"08:01.120 ","End":"08:04.135","Text":"so I just need to plug in r equals 2,"},{"Start":"08:04.135 ","End":"08:08.095","Text":"anyway, integral from 0-2 Pi."},{"Start":"08:08.095 ","End":"08:11.500","Text":"When r is 2, 2 cubed is 8,"},{"Start":"08:11.500 ","End":"08:17.014","Text":"so it\u0027s 80 over 3 cosine Theta,"},{"Start":"08:17.014 ","End":"08:21.510","Text":"and then also here, r cubed is 8,"},{"Start":"08:21.510 ","End":"08:31.785","Text":"8 times 4 is 32 over 3 sine Theta."},{"Start":"08:31.785 ","End":"08:38.130","Text":"Here, when r is 2, 2 squared is 4,"},{"Start":"08:38.130 ","End":"08:41.175","Text":"4 times 3 over 2 is 6,"},{"Start":"08:41.175 ","End":"08:44.355","Text":"so I\u0027ve got just 6,"},{"Start":"08:44.355 ","End":"08:48.075","Text":"and this is d Theta."},{"Start":"08:48.075 ","End":"08:52.930","Text":"Continuing. Now we can do the integral."},{"Start":"08:52.930 ","End":"08:56.560","Text":"The integral of cosine is minus sine."},{"Start":"08:56.560 ","End":"09:00.799","Text":"I have minus 80 over 3."},{"Start":"09:00.820 ","End":"09:10.710","Text":"Sorry, the integral of cosine is sine Theta without the minus,"},{"Start":"09:11.000 ","End":"09:14.705","Text":"the integral of sine is minus cosine."},{"Start":"09:14.705 ","End":"09:17.800","Text":"So minus 32 over 3 cosine,"},{"Start":"09:17.800 ","End":"09:21.690","Text":"the term, the integral of 6 is just 6 Theta."},{"Start":"09:21.690 ","End":"09:26.700","Text":"All this has to be taken from 0-2 Pi."},{"Start":"09:26.700 ","End":"09:29.055","Text":"Let\u0027s see what we get."},{"Start":"09:29.055 ","End":"09:32.549","Text":"If we put Theta equals 2 Pi,"},{"Start":"09:32.549 ","End":"09:40.030","Text":"sine of 2 Pi is 0 and cosine of 2 Pi is the same as cosine 0,"},{"Start":"09:41.410 ","End":"09:44.345","Text":"cosine 0 is 1."},{"Start":"09:44.345 ","End":"09:49.759","Text":"So this is minus 32 over 3,"},{"Start":"09:49.759 ","End":"09:53.890","Text":"and 6 Theta is just 12 Pi."},{"Start":"09:53.890 ","End":"09:56.255","Text":"That\u0027s the 2 Pi part."},{"Start":"09:56.255 ","End":"09:59.644","Text":"Now the 0 part, which I subtract,"},{"Start":"09:59.644 ","End":"10:02.074","Text":"sine of 0 is 0,"},{"Start":"10:02.074 ","End":"10:05.135","Text":"cosine of 0 is 1,"},{"Start":"10:05.135 ","End":"10:16.920","Text":"it\u0027s minus 32 over 3,"},{"Start":"10:16.960 ","End":"10:20.000","Text":"and here it should be a minus of course,"},{"Start":"10:20.000 ","End":"10:23.720","Text":"because cosine Theta, cosine 2 Pi is also one,"},{"Start":"10:23.720 ","End":"10:25.670","Text":"but the minus was there."},{"Start":"10:25.670 ","End":"10:29.280","Text":"Cosine 2 Pi cosine 0 is the same thing."},{"Start":"10:31.780 ","End":"10:35.120","Text":"Also 0 and Theta 0,"},{"Start":"10:35.120 ","End":"10:37.745","Text":"so this minus this,"},{"Start":"10:37.745 ","End":"10:40.490","Text":"what does this give me?"},{"Start":"10:40.490 ","End":"10:45.350","Text":"Minus 32 over 3 cancels with plus 32 over 3, if you like."},{"Start":"10:45.350 ","End":"10:48.515","Text":"It\u0027s this thing minus the same thing anyway, this cancels,"},{"Start":"10:48.515 ","End":"10:52.630","Text":"and all we\u0027re left with is 12 Pi,"},{"Start":"10:52.630 ","End":"10:57.310","Text":"which I\u0027ll highlight, and we are done."}],"ID":9664},{"Watched":false,"Name":"Exercise 12","Duration":"13m ","ChapterTopicVideoID":8798,"CourseChapterTopicPlaylistID":4974,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.770","Text":"In this exercise, we have to compute this type 2 surface integral F.ndS."},{"Start":"00:07.770 ","End":"00:10.515","Text":"We\u0027re given the vector field F,"},{"Start":"00:10.515 ","End":"00:13.709","Text":"given the i, j and k components,"},{"Start":"00:13.709 ","End":"00:18.930","Text":"so though I might decide to use the angular brackets notation,"},{"Start":"00:18.930 ","End":"00:28.500","Text":"in which case it\u0027s 0 minus 2z and minus 3y minus 1."},{"Start":"00:28.500 ","End":"00:30.990","Text":"Use both notations, brackets or i, j,"},{"Start":"00:30.990 ","End":"00:39.285","Text":"k. S is the hemisphere that\u0027s above the xy-plane, 1/2 a sphere."},{"Start":"00:39.285 ","End":"00:45.379","Text":"One of the reasons we have a hemisphere or that it\u0027s useful to have a hemispheres,"},{"Start":"00:45.379 ","End":"00:50.165","Text":"is that we can then write z as a function of x and y."},{"Start":"00:50.165 ","End":"00:52.700","Text":"In fact, for radius 4,"},{"Start":"00:52.700 ","End":"00:56.850","Text":"we could write the equation that z equals"},{"Start":"00:56.850 ","End":"01:07.045","Text":"the square root of 4 squared minus x squared minus y squared."},{"Start":"01:07.045 ","End":"01:11.330","Text":"This comes from the equation of the sphere where x"},{"Start":"01:11.330 ","End":"01:15.185","Text":"squared plus y squared plus z squared equals r squared."},{"Start":"01:15.185 ","End":"01:17.595","Text":"But in our case,"},{"Start":"01:17.595 ","End":"01:22.925","Text":"z is bigger or equal to 0 because we\u0027re above the xy-plane."},{"Start":"01:22.925 ","End":"01:28.730","Text":"We take the positive square root after we transfer the x and y to the other side."},{"Start":"01:28.730 ","End":"01:31.520","Text":"This will be our g of x,"},{"Start":"01:31.520 ","End":"01:33.455","Text":"y, which we\u0027ll use,"},{"Start":"01:33.455 ","End":"01:36.950","Text":"and there are formulas that tell us how to"},{"Start":"01:36.950 ","End":"01:41.700","Text":"convert from the surface integral to a regular double integral."},{"Start":"01:41.830 ","End":"01:44.810","Text":"There are actually 2 formulas,"},{"Start":"01:44.810 ","End":"01:46.700","Text":"and they depend on whether"},{"Start":"01:46.700 ","End":"01:52.820","Text":"the normal vector has an upward component or a downward component."},{"Start":"01:52.820 ","End":"01:56.855","Text":"In this case, you can see that everywhere on the upper hemisphere, the normal,"},{"Start":"01:56.855 ","End":"01:58.250","Text":"I\u0027m not saying it\u0027s vertical,"},{"Start":"01:58.250 ","End":"02:03.230","Text":"but it has a vertical component that is in the direction of the positive z-axis."},{"Start":"02:03.230 ","End":"02:06.545","Text":"I\u0027ll write down the formula for this case."},{"Start":"02:06.545 ","End":"02:08.450","Text":"It turned out to be similar."},{"Start":"02:08.450 ","End":"02:16.099","Text":"I just minus when the normal is going downwards as it would say in the lower hemisphere."},{"Start":"02:16.099 ","End":"02:21.890","Text":"Anyway, the formula tells us that the double integral over a surface"},{"Start":"02:21.890 ","End":"02:31.865","Text":"of F.ndS is the double integral over R,"},{"Start":"02:31.865 ","End":"02:34.910","Text":"where R is the projection of this surface down onto"},{"Start":"02:34.910 ","End":"02:45.045","Text":"the xy-plane of the following of F dot,"},{"Start":"02:45.045 ","End":"02:52.520","Text":"and this will be minus g with respect to x. G is this function here,"},{"Start":"02:52.520 ","End":"03:00.150","Text":"partial derivative with respect to x minus g with respect to y, and 1 dA."},{"Start":"03:00.150 ","End":"03:03.709","Text":"Like I said, if the normal\u0027s facing downwards,"},{"Start":"03:03.709 ","End":"03:06.080","Text":"we\u0027d reverse all the signs here."},{"Start":"03:06.080 ","End":"03:10.320","Text":"But I\u0027m not going to write the other formula because you don\u0027t need it."},{"Start":"03:10.450 ","End":"03:14.880","Text":"Let\u0027s do this dot product."},{"Start":"03:15.010 ","End":"03:17.440","Text":"Now, go over here."},{"Start":"03:17.440 ","End":"03:20.180","Text":"What we need is the double integral over R,"},{"Start":"03:20.180 ","End":"03:26.760","Text":"where R is this in the xy-plane disk of radius 4."},{"Start":"03:26.780 ","End":"03:30.510","Text":"We need F. I\u0027ll do it in"},{"Start":"03:30.510 ","End":"03:38.760","Text":"the brackets notation 0"},{"Start":"03:38.760 ","End":"03:45.495","Text":"minus 2z minus 3y minus 1 dot."},{"Start":"03:45.495 ","End":"03:50.160","Text":"Now, what is gx?"},{"Start":"03:50.160 ","End":"03:56.975","Text":"Gx will be not so simple but not so difficult."},{"Start":"03:56.975 ","End":"03:59.015","Text":"Have a square root."},{"Start":"03:59.015 ","End":"04:07.140","Text":"1 over twice the square root of 16 minus x squared minus y squared,"},{"Start":"04:07.140 ","End":"04:08.825","Text":"and on the numerator,"},{"Start":"04:08.825 ","End":"04:11.450","Text":"the derivative, the inner derivative,"},{"Start":"04:11.450 ","End":"04:13.220","Text":"and it\u0027s with respect to x."},{"Start":"04:13.220 ","End":"04:16.195","Text":"It\u0027s minus 2x."},{"Start":"04:16.195 ","End":"04:19.455","Text":"But the two cancels."},{"Start":"04:19.455 ","End":"04:22.800","Text":"Then same thing just with y on the top,"},{"Start":"04:22.800 ","End":"04:25.100","Text":"it\u0027s clear that we\u0027ll get the same thing."},{"Start":"04:25.100 ","End":"04:30.455","Text":"Square root of 16 minus x squared minus y squared."},{"Start":"04:30.455 ","End":"04:35.130","Text":"Lastly, just 1 dA."},{"Start":"04:37.790 ","End":"04:44.510","Text":"This equals double integral over R. Now,"},{"Start":"04:44.510 ","End":"04:46.850","Text":"if we multiply out the first one,"},{"Start":"04:46.850 ","End":"04:49.850","Text":"0 times this is nothing,"},{"Start":"04:49.850 ","End":"04:54.530","Text":"then I have minus 2z times minus y over this."},{"Start":"04:54.530 ","End":"04:59.460","Text":"It\u0027s just 2zy over"},{"Start":"04:59.460 ","End":"05:05.940","Text":"the square root of 16 minus x squared minus y squared."},{"Start":"05:05.940 ","End":"05:17.105","Text":"Then the last component is just minus 3y minus 1,"},{"Start":"05:17.105 ","End":"05:21.170","Text":"and all this is dA."},{"Start":"05:21.170 ","End":"05:26.165","Text":"However, we don\u0027t want z here."},{"Start":"05:26.165 ","End":"05:30.390","Text":"We\u0027re replacing z by g of x,"},{"Start":"05:30.390 ","End":"05:32.205","Text":"y, which is this."},{"Start":"05:32.205 ","End":"05:40.350","Text":"Actually z equals the denominator here."},{"Start":"05:40.350 ","End":"05:46.415","Text":"What I\u0027m saying is that I can cancel because z is equal to this."},{"Start":"05:46.415 ","End":"05:49.140","Text":"This will cancel out,"},{"Start":"05:49.730 ","End":"05:56.960","Text":"and we\u0027ll be left with just the double integral over R,"},{"Start":"05:56.960 ","End":"06:05.100","Text":"2y minus 3y is minus y minus 1dA."},{"Start":"06:07.820 ","End":"06:12.250","Text":"What I wrote here is g with respect to x."},{"Start":"06:12.250 ","End":"06:16.535","Text":"I almost forgot that I need minus gx from the formula."},{"Start":"06:16.535 ","End":"06:22.130","Text":"The minus will knock out this minus,"},{"Start":"06:22.130 ","End":"06:25.935","Text":"and 2 will cancel with 2."},{"Start":"06:25.935 ","End":"06:29.890","Text":"Next. I\u0027ll get, well,"},{"Start":"06:29.890 ","End":"06:32.890","Text":"it\u0027ll be the same thing just with y instead of x,"},{"Start":"06:32.890 ","End":"06:35.850","Text":"because the inner derivative would be minus 2y,"},{"Start":"06:35.850 ","End":"06:36.935","Text":"the 2 would cancel again,"},{"Start":"06:36.935 ","End":"06:38.545","Text":"the minus with the minus."},{"Start":"06:38.545 ","End":"06:49.220","Text":"I just get y over square root of 16 minus x squared minus y squared."},{"Start":"06:49.820 ","End":"06:56.170","Text":"The last component is just 1 and that\u0027s dA."},{"Start":"06:58.640 ","End":"07:04.310","Text":"Actually what I have to do here is replace z."},{"Start":"07:05.150 ","End":"07:08.775","Text":"We only want x and y over this region,"},{"Start":"07:08.775 ","End":"07:11.860","Text":"or replace z by what it\u0027s equal to,"},{"Start":"07:11.860 ","End":"07:20.115","Text":"which is the square root of 16 minus x squared minus y squared."},{"Start":"07:20.115 ","End":"07:23.085","Text":"Let\u0027s do the multiplication."},{"Start":"07:23.085 ","End":"07:27.360","Text":"The dot-product, we get."},{"Start":"07:27.360 ","End":"07:31.525","Text":"The first component, 0 times anything is 0,"},{"Start":"07:31.525 ","End":"07:34.225","Text":"that\u0027s just 0 plus."},{"Start":"07:34.225 ","End":"07:36.100","Text":"Now let\u0027s see what else do I get?"},{"Start":"07:36.100 ","End":"07:43.030","Text":"I get minus 2 square root times y over square root."},{"Start":"07:43.030 ","End":"07:47.480","Text":"It\u0027s just minus 2y"},{"Start":"07:47.480 ","End":"07:51.835","Text":"because the square root in the numerator and the denominator will cancel each other."},{"Start":"07:51.835 ","End":"07:54.250","Text":"We\u0027ll get the minus 2 with the y."},{"Start":"07:54.250 ","End":"07:56.290","Text":"Then the last component,"},{"Start":"07:56.290 ","End":"08:00.920","Text":"I get minus 3y minus 1,"},{"Start":"08:02.180 ","End":"08:06.485","Text":"and all this dA,"},{"Start":"08:06.485 ","End":"08:09.250","Text":"which is just the integral."},{"Start":"08:09.250 ","End":"08:17.365","Text":"Let\u0027s just simplify this as we have minus 5y minus 1 dA."},{"Start":"08:17.365 ","End":"08:20.770","Text":"Now how am I going to do this integral?"},{"Start":"08:20.770 ","End":"08:24.220","Text":"I say it\u0027s best we do it in polar coordinates"},{"Start":"08:24.220 ","End":"08:29.060","Text":"because the region is a circle centered at the origin or a disk."},{"Start":"08:32.450 ","End":"08:39.489","Text":"We\u0027ll do it in polar and I\u0027ll remind you what the equations is of a polar."},{"Start":"08:39.489 ","End":"08:44.855","Text":"We have x equals r cosine Theta,"},{"Start":"08:44.855 ","End":"08:50.275","Text":"y equals r sine Theta,"},{"Start":"08:50.275 ","End":"08:57.765","Text":"and dA is equal to r, dr, d Theta."},{"Start":"08:57.765 ","End":"09:00.600","Text":"That\u0027s a fourth formula for x squared plus y squared,"},{"Start":"09:00.600 ","End":"09:02.885","Text":"but we won\u0027t need it here."},{"Start":"09:02.885 ","End":"09:11.500","Text":"The other thing we have to do is convert or describe the disk of radius 4 in polar terms."},{"Start":"09:11.500 ","End":"09:14.400","Text":"We\u0027ve done this so many times before."},{"Start":"09:14.400 ","End":"09:16.400","Text":"Let me just write it."},{"Start":"09:16.400 ","End":"09:18.565","Text":"We go the full circle,"},{"Start":"09:18.565 ","End":"09:22.985","Text":"Theta goes from 0-2 Pi or 360."},{"Start":"09:22.985 ","End":"09:25.925","Text":"R goes from the center to the circumference."},{"Start":"09:25.925 ","End":"09:28.770","Text":"It\u0027s 0-4."},{"Start":"09:28.770 ","End":"09:32.125","Text":"R goes from 0-4."},{"Start":"09:32.125 ","End":"09:35.780","Text":"Then I replace this,"},{"Start":"09:35.780 ","End":"09:45.120","Text":"I get minus 5y is r sine Theta and minus 1,"},{"Start":"09:45.120 ","End":"09:51.130","Text":"and dA, is r, dr, d Theta."},{"Start":"09:52.220 ","End":"10:00.120","Text":"Continuing, got the integral from 0-2 Pi."},{"Start":"10:00.120 ","End":"10:02.640","Text":"If I multiply out the r,"},{"Start":"10:02.640 ","End":"10:10.320","Text":"I\u0027ve got the integral from 0-4 minus"},{"Start":"10:10.320 ","End":"10:21.010","Text":"5r squared sine Theta minus r, dr, d Theta."},{"Start":"10:21.770 ","End":"10:26.370","Text":"Then I get the integral from 0-2 Pi."},{"Start":"10:26.370 ","End":"10:30.315","Text":"This integral with respect to r,"},{"Start":"10:30.315 ","End":"10:33.315","Text":"I raise the power by 1 and divide by it."},{"Start":"10:33.315 ","End":"10:40.110","Text":"I\u0027ve got minus 5/3 r cubed,"},{"Start":"10:40.110 ","End":"10:43.605","Text":"and sine Theta is a constant, just days."},{"Start":"10:43.605 ","End":"10:47.640","Text":"Here I have minus r squared over 2."},{"Start":"10:47.640 ","End":"10:56.350","Text":"I\u0027ve to take this from 0-4 and all this d Theta."},{"Start":"10:57.880 ","End":"11:01.210","Text":"If we plug in r equals 0,"},{"Start":"11:01.210 ","End":"11:02.775","Text":"we just get 0."},{"Start":"11:02.775 ","End":"11:05.040","Text":"That doesn\u0027t give us anything."},{"Start":"11:05.040 ","End":"11:08.190","Text":"If I plug in r equals 4."},{"Start":"11:08.190 ","End":"11:10.725","Text":"Let\u0027s see what we\u0027ll get."},{"Start":"11:10.725 ","End":"11:14.325","Text":"At 4, we get"},{"Start":"11:14.325 ","End":"11:22.110","Text":"4 cubed is 64."},{"Start":"11:22.110 ","End":"11:26.670","Text":"64 times 5 over 3."},{"Start":"11:26.670 ","End":"11:37.630","Text":"That\u0027s 320 over 3 sine Theta."},{"Start":"11:38.930 ","End":"11:41.510","Text":"Here r is 4,"},{"Start":"11:41.510 ","End":"11:43.730","Text":"I\u0027ve got minus 16 over 2,"},{"Start":"11:43.730 ","End":"11:46.820","Text":"which is minus 8."},{"Start":"11:46.820 ","End":"11:52.290","Text":"This integral d Theta."},{"Start":"11:52.540 ","End":"11:56.945","Text":"This equals continuing."},{"Start":"11:56.945 ","End":"12:02.910","Text":"The integral of sine is minus cosine."},{"Start":"12:02.910 ","End":"12:10.270","Text":"I\u0027ve got 320 over 3 cosine Theta."},{"Start":"12:11.300 ","End":"12:16.780","Text":"From here I\u0027ve got minus 8 Theta."},{"Start":"12:17.600 ","End":"12:23.755","Text":"This is from 0-2 Pi."},{"Start":"12:23.755 ","End":"12:29.300","Text":"Now, cosine of 0 and cosine of 2 Pi are the same."},{"Start":"12:29.300 ","End":"12:31.380","Text":"They both happened to equal to 1,"},{"Start":"12:31.380 ","End":"12:33.075","Text":"but it does matter that the same,"},{"Start":"12:33.075 ","End":"12:35.430","Text":"they will cancel each other out."},{"Start":"12:35.430 ","End":"12:38.805","Text":"Only the minus 8 Theta will matter,"},{"Start":"12:38.805 ","End":"12:44.400","Text":"and what I will get will be 8 Theta."},{"Start":"12:44.400 ","End":"12:49.650","Text":"When Theta is 2 Pi is minus 16 Pi,"},{"Start":"12:49.650 ","End":"12:51.900","Text":"and when Theta is 0, it\u0027s nothing."},{"Start":"12:51.900 ","End":"12:55.755","Text":"I\u0027m just left with minus 16 Pi."},{"Start":"12:55.755 ","End":"12:58.785","Text":"That is the final answer,"},{"Start":"12:58.785 ","End":"13:01.900","Text":"and we are done."}],"ID":9665}],"Thumbnail":null,"ID":4974}]

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