[{"Name":"Exercises - Divergence Theorem","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Divergence Theorem","Duration":"11m 28s","ChapterTopicVideoID":8763,"CourseChapterTopicPlaylistID":4966,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8763.jpeg","UploadDate":"2020-02-26T12:29:25.7030000","DurationForVideoObject":"PT11M28S","Description":null,"MetaTitle":"The Divergence Theorem: Video + Workbook | Proprep","MetaDescription":"Divergence Theorem - Exercises - Divergence Theorem. 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/divergence-theorem/exercises-_-divergence-theorem/vid8831","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.855","Text":"Now we come to the divergence theorem."},{"Start":"00:03.855 ","End":"00:06.345","Text":"It\u0027s part of surface integrals."},{"Start":"00:06.345 ","End":"00:08.250","Text":"Together with Stokes\u0027 theorem,"},{"Start":"00:08.250 ","End":"00:10.380","Text":"it\u0027s one of the big theorems"},{"Start":"00:10.380 ","End":"00:14.640","Text":"and we\u0027ll conclude the chapter on surface integrals with this."},{"Start":"00:14.640 ","End":"00:18.600","Text":"Let me first of all present it and then I\u0027ll explain it."},{"Start":"00:18.600 ","End":"00:21.615","Text":"This is what it looks like,"},{"Start":"00:21.615 ","End":"00:24.605","Text":"and now I\u0027ll do the explaining."},{"Start":"00:24.605 ","End":"00:30.980","Text":"What it does is it equates a triple integral"},{"Start":"00:30.980 ","End":"00:36.530","Text":"over a certain volume to a surface integral."},{"Start":"00:36.530 ","End":"00:39.530","Text":"The symbol here is optional."},{"Start":"00:39.530 ","End":"00:41.779","Text":"Just like for a closed curve,"},{"Start":"00:41.779 ","End":"00:46.400","Text":"we wrote a little circle when we have a closed surface,"},{"Start":"00:46.400 ","End":"00:51.535","Text":"then sometimes you put this oval shape over it."},{"Start":"00:51.535 ","End":"00:54.940","Text":"Ignore that if you want."},{"Start":"00:54.940 ","End":"00:58.840","Text":"I have to tell you what S and V are."},{"Start":"00:59.780 ","End":"01:02.270","Text":"Here\u0027s a picture."},{"Start":"01:02.270 ","End":"01:04.805","Text":"V is a volume,"},{"Start":"01:04.805 ","End":"01:08.045","Text":"a solid in 3D,"},{"Start":"01:08.045 ","End":"01:14.745","Text":"and S is the boundary of this volume."},{"Start":"01:14.745 ","End":"01:17.430","Text":"That\u0027s what this symbol here is."},{"Start":"01:17.430 ","End":"01:22.689","Text":"This funny d is also used to express a boundary,"},{"Start":"01:22.689 ","End":"01:26.599","Text":"the surface is the boundary of the volume."},{"Start":"01:26.599 ","End":"01:33.380","Text":"The n that\u0027s in here is the outer facing normal"},{"Start":"01:33.380 ","End":"01:36.410","Text":"and we usually put an arrow over vectors,"},{"Start":"01:36.410 ","End":"01:40.440","Text":"so let\u0027s put arrows."},{"Start":"01:41.600 ","End":"01:48.590","Text":"This is a surface integral of a scalar because the dot product will give us a scalar."},{"Start":"01:48.590 ","End":"01:53.810","Text":"This is a triple integral."},{"Start":"01:53.810 ","End":"01:55.850","Text":"If you haven\u0027t studied triple integrals,"},{"Start":"01:55.850 ","End":"02:00.230","Text":"well, it\u0027s just the same as double integrals where typically,"},{"Start":"02:00.230 ","End":"02:03.700","Text":"just like we had the dA was dxdy,"},{"Start":"02:03.700 ","End":"02:10.865","Text":"dV is dxdydz in any order."},{"Start":"02:10.865 ","End":"02:12.620","Text":"It\u0027s very similar."},{"Start":"02:12.620 ","End":"02:15.290","Text":"If you haven\u0027t studied triple integrals,"},{"Start":"02:15.290 ","End":"02:19.440","Text":"just think of it as an extension of the double integral."},{"Start":"02:20.270 ","End":"02:26.230","Text":"What\u0027s important is that this surface should be piecewise smooth,"},{"Start":"02:26.230 ","End":"02:28.940","Text":"and there\u0027s various technical conditions,"},{"Start":"02:28.940 ","End":"02:33.080","Text":"continuous derivatives, and all that."},{"Start":"02:33.080 ","End":"02:36.815","Text":"But piecewise, in other words, it can be,"},{"Start":"02:36.815 ","End":"02:38.765","Text":"as in the picture,"},{"Start":"02:38.765 ","End":"02:40.595","Text":"made up of separate pieces,"},{"Start":"02:40.595 ","End":"02:42.860","Text":"but each one of them is got to be smooth."},{"Start":"02:42.860 ","End":"02:46.730","Text":"Just to give you an example of what is not a good surface."},{"Start":"02:46.730 ","End":"02:49.870","Text":"I brought in another picture,"},{"Start":"02:49.870 ","End":"02:56.435","Text":"the 3 pictures, 3 shapes on the left surfaces and these are good and these are bad."},{"Start":"02:56.435 ","End":"03:00.350","Text":"These are good because this is the boundary of a solid sphere."},{"Start":"03:00.350 ","End":"03:04.895","Text":"This is the boundary of a solid donut or what we call a torus,"},{"Start":"03:04.895 ","End":"03:07.865","Text":"and this is the boundary of a cube."},{"Start":"03:07.865 ","End":"03:10.295","Text":"It\u0027s made up of 6 separate pieces,"},{"Start":"03:10.295 ","End":"03:12.140","Text":"and that\u0027s the piecewise smooth."},{"Start":"03:12.140 ","End":"03:14.945","Text":"Each plane is smooth and it\u0027s made up of separate pieces."},{"Start":"03:14.945 ","End":"03:17.930","Text":"These surfaces are not good because they have a line boundary,"},{"Start":"03:17.930 ","End":"03:22.980","Text":"they\u0027re not closed surfaces and they don\u0027t wrap a volume."},{"Start":"03:22.980 ","End":"03:25.875","Text":"This is what the theorem says,"},{"Start":"03:25.875 ","End":"03:29.150","Text":"and as most theorems with equations,"},{"Start":"03:29.150 ","End":"03:32.460","Text":"we can use them in either direction."},{"Start":"03:33.520 ","End":"03:36.020","Text":"Generally in this course,"},{"Start":"03:36.020 ","End":"03:40.770","Text":"I haven\u0027t been giving you intuition as to what it actually means."},{"Start":"03:40.880 ","End":"03:45.670","Text":"I\u0027ll show you what the Wikipedia says as the intuitive meaning of this."},{"Start":"03:45.670 ","End":"03:47.830","Text":"Here we are."},{"Start":"03:47.830 ","End":"03:51.400","Text":"I\u0027ll leave it up for a few seconds and you can refer to it later"},{"Start":"03:51.400 ","End":"03:55.105","Text":"and read it if you\u0027re interested in getting an intuition about it,"},{"Start":"03:55.105 ","End":"03:57.070","Text":"I don\u0027t always find it helpful."},{"Start":"03:57.070 ","End":"03:59.770","Text":"It takes awhile to absorb it,"},{"Start":"03:59.770 ","End":"04:00.820","Text":"but I\u0027ll leave it up there."},{"Start":"04:00.820 ","End":"04:02.870","Text":"That\u0027s enough."},{"Start":"04:03.600 ","End":"04:08.035","Text":"The main thing to do now would be an example."},{"Start":"04:08.035 ","End":"04:10.550","Text":"I\u0027ll clear some space."},{"Start":"04:10.550 ","End":"04:12.840","Text":"Before I get to the example,"},{"Start":"04:12.840 ","End":"04:15.850","Text":"I wanted to say there\u0027s a slight variations on this."},{"Start":"04:15.850 ","End":"04:19.660","Text":"For example, instead of writing del dot with F,"},{"Start":"04:19.660 ","End":"04:22.390","Text":"we can actually write div F."},{"Start":"04:22.390 ","End":"04:29.330","Text":"This integral can also be written as"},{"Start":"04:29.330 ","End":"04:35.675","Text":"a surface integral of a vector field instead of the dot n,"},{"Start":"04:35.675 ","End":"04:41.760","Text":"we can just by definition replace dS with a vector dS."},{"Start":"04:42.380 ","End":"04:45.800","Text":"In another place I found it written this way"},{"Start":"04:45.800 ","End":"04:48.950","Text":"and they\u0027ve also reverse the sides that"},{"Start":"04:48.950 ","End":"04:53.570","Text":"the divergence is on the right and the surface integrals on the left."},{"Start":"04:53.570 ","End":"04:57.085","Text":"I\u0027m just telling you, you can expect variations on this."},{"Start":"04:57.085 ","End":"05:03.920","Text":"Here they didn\u0027t even put the symbol for closed surface and so it doesn\u0027t matter."},{"Start":"05:03.920 ","End":"05:05.495","Text":"That\u0027s basically it."},{"Start":"05:05.495 ","End":"05:06.890","Text":"Those are the variations,"},{"Start":"05:06.890 ","End":"05:09.155","Text":"instead of the del to write div,"},{"Start":"05:09.155 ","End":"05:13.430","Text":"instead of F.n to write what it\u0027s equivalent to."},{"Start":"05:13.430 ","End":"05:15.890","Text":"This in fact is shorter, it\u0027s more compact."},{"Start":"05:15.890 ","End":"05:19.235","Text":"F.dS is defined to be just this."},{"Start":"05:19.235 ","End":"05:21.565","Text":"Let\u0027s get to the example."},{"Start":"05:21.565 ","End":"05:30.545","Text":"I need to tell you what are V and S and the vector field F and so on."},{"Start":"05:30.545 ","End":"05:39.440","Text":"Let\u0027s take, S will be the unit sphere"},{"Start":"05:39.440 ","End":"05:43.445","Text":"and S is the boundary of the volume V,"},{"Start":"05:43.445 ","End":"05:46.175","Text":"which is the unit ball,"},{"Start":"05:46.175 ","End":"05:50.310","Text":"which is the sphere, including its interior."},{"Start":"05:51.260 ","End":"05:56.120","Text":"This would be, for example, in Cartesian coordinates,"},{"Start":"05:56.120 ","End":"06:00.890","Text":"x squared plus y squared plus z squared equals 1"},{"Start":"06:00.890 ","End":"06:09.455","Text":"and the unit ball is x squared plus y squared plus z squared is less than or equal to 1."},{"Start":"06:09.455 ","End":"06:11.800","Text":"Maybe I\u0027ll bring in a picture."},{"Start":"06:11.800 ","End":"06:14.250","Text":"Here\u0027s one if it helps."},{"Start":"06:14.250 ","End":"06:20.535","Text":"The S would be the boundary and V,"},{"Start":"06:20.535 ","End":"06:27.810","Text":"I\u0027ll just explain here, is the whole interior of the sphere."},{"Start":"06:27.810 ","End":"06:32.640","Text":"That\u0027s the volume and the surface is the sphere."},{"Start":"06:32.640 ","End":"06:35.580","Text":"Indeed S is the boundary of V."},{"Start":"06:35.580 ","End":"06:39.280","Text":"Now I need to give you what F is."},{"Start":"06:40.190 ","End":"06:42.560","Text":"This is a vector field,"},{"Start":"06:42.560 ","End":"06:46.760","Text":"so I need to tell you what it is for each x, y, z,"},{"Start":"06:46.760 ","End":"06:53.820","Text":"and it is 2x comma y squared comma z squared."},{"Start":"06:53.820 ","End":"06:56.670","Text":"I\u0027ll give it an angular brackets notation."},{"Start":"06:56.670 ","End":"07:03.365","Text":"The question as phrased is to find the surface integral"},{"Start":"07:03.365 ","End":"07:10.170","Text":"over the unit sphere of vector field F,"},{"Start":"07:10.170 ","End":"07:14.655","Text":"and let me write it in the form of dS."},{"Start":"07:14.655 ","End":"07:18.965","Text":"You know what, I\u0027ll use this new notation to indicate that"},{"Start":"07:18.965 ","End":"07:24.125","Text":"the sphere is indeed a closed surface and it\u0027s the boundary of a certain volume."},{"Start":"07:24.125 ","End":"07:26.500","Text":"Now, the obvious thing to do,"},{"Start":"07:26.500 ","End":"07:29.130","Text":"we\u0027re in the chapter on divergence theorem,"},{"Start":"07:29.130 ","End":"07:33.480","Text":"is to evaluate it by using the divergence theorem."},{"Start":"07:33.580 ","End":"07:41.300","Text":"We will say that this is going to equal the triple integral along the volume V,"},{"Start":"07:41.300 ","End":"07:45.700","Text":"which is the unit ball of,"},{"Start":"07:45.700 ","End":"07:49.580","Text":"and I\u0027ll write it in the form with the words div,"},{"Start":"07:49.580 ","End":"07:53.780","Text":"divergence of the vector field."},{"Start":"07:53.780 ","End":"07:58.415","Text":"This will be dV."},{"Start":"07:58.415 ","End":"08:04.430","Text":"In general, the divergence of F is going to be,"},{"Start":"08:04.430 ","End":"08:07.670","Text":"we take the first component and differentiate with respect to x,"},{"Start":"08:07.670 ","End":"08:10.670","Text":"this with respect to y, this with respect to z."},{"Start":"08:10.670 ","End":"08:15.400","Text":"What we get is 2x with respect to x is 2."},{"Start":"08:15.400 ","End":"08:20.060","Text":"The derivative of y squared with respect to y is 2y"},{"Start":"08:20.060 ","End":"08:24.370","Text":"and the derivative of z squared with respect to z is 2z."},{"Start":"08:24.370 ","End":"08:28.875","Text":"That\u0027s the divergence."},{"Start":"08:28.875 ","End":"08:33.710","Text":"What we need now is the triple integral of this"},{"Start":"08:33.710 ","End":"08:35.660","Text":"and we can break it up and we can say,"},{"Start":"08:35.660 ","End":"08:38.705","Text":"it\u0027s this will be equal to,"},{"Start":"08:38.705 ","End":"08:41.515","Text":"and I\u0027m continuing down here."},{"Start":"08:41.515 ","End":"08:49.275","Text":"The twice the triple integral of 1dV."},{"Start":"08:49.275 ","End":"08:50.685","Text":"I\u0027ll write the 1 in,"},{"Start":"08:50.685 ","End":"08:55.845","Text":"1dV plus twice the triple integral"},{"Start":"08:55.845 ","End":"09:06.010","Text":"of ydV plus twice the triple integral of zdV."},{"Start":"09:06.650 ","End":"09:12.930","Text":"All this is over the ball V, the solid ball."},{"Start":"09:13.040 ","End":"09:16.910","Text":"It actually turns out that this is not going to be so hard to compute,"},{"Start":"09:16.910 ","End":"09:22.205","Text":"but if you tried to compute the surface integral, it gets messy."},{"Start":"09:22.205 ","End":"09:24.440","Text":"Now they\u0027re going to be some tricks we\u0027re going to use here."},{"Start":"09:24.440 ","End":"09:28.690","Text":"These are useful tricks which use symmetry."},{"Start":"09:28.690 ","End":"09:30.510","Text":"Let\u0027s take this 1,"},{"Start":"09:30.510 ","End":"09:33.005","Text":"maybe it\u0027s easier to demonstrate it."},{"Start":"09:33.005 ","End":"09:35.905","Text":"Z is positive above the xy plane."},{"Start":"09:35.905 ","End":"09:39.285","Text":"Maybe we can take V as 2 separate volumes,"},{"Start":"09:39.285 ","End":"09:46.320","Text":"V_1 and V_2 above and below, 2 solid hemispheres."},{"Start":"09:46.930 ","End":"09:52.145","Text":"What happens is that z is totally symmetrical but opposite."},{"Start":"09:52.145 ","End":"09:53.930","Text":"For each z here,"},{"Start":"09:53.930 ","End":"10:00.900","Text":"there\u0027s a minus z down here and so the 2 integrals,"},{"Start":"10:00.900 ","End":"10:03.410","Text":"the 2 hemispheres would cancel each other out."},{"Start":"10:03.410 ","End":"10:07.955","Text":"This will turn out this bit is equal to 0 by symmetry"},{"Start":"10:07.955 ","End":"10:11.360","Text":"because we have the equal and opposite z and minus z."},{"Start":"10:11.360 ","End":"10:13.115","Text":"Similarly for y,"},{"Start":"10:13.115 ","End":"10:21.260","Text":"if we just break it up as V_1 and V_2 to the 1 side of the xz plane on the other side,"},{"Start":"10:21.260 ","End":"10:24.560","Text":"also we have like it\u0027s an odd function."},{"Start":"10:24.560 ","End":"10:31.595","Text":"It\u0027s also equal and opposite on each side of the xz plane,"},{"Start":"10:31.595 ","End":"10:33.880","Text":"2 hemispheres, they cancel each other out,"},{"Start":"10:33.880 ","End":"10:35.995","Text":"y and minus y."},{"Start":"10:35.995 ","End":"10:39.520","Text":"That\u0027s also going to come out to be 0."},{"Start":"10:39.520 ","End":"10:45.180","Text":"The last trick is that just like when we had an integral dA of 1,"},{"Start":"10:45.180 ","End":"10:48.255","Text":"it\u0027s just the area of the shape."},{"Start":"10:48.255 ","End":"10:55.410","Text":"This bit here is the volume of the ball V,"},{"Start":"10:55.410 ","End":"11:00.550","Text":"so all we need now is twice the volume of the ball, which is twice."},{"Start":"11:00.550 ","End":"11:04.990","Text":"Now the volume of a ball you might remember, is in general,"},{"Start":"11:04.990 ","End":"11:11.220","Text":"we have 4/3 Pi r cubed with radius r."},{"Start":"11:11.220 ","End":"11:12.705","Text":"But here r is 1,"},{"Start":"11:12.705 ","End":"11:17.790","Text":"so it\u0027s twice 4/3 Pi times 1 cubed,"},{"Start":"11:17.790 ","End":"11:24.824","Text":"and so the final answer is 8 Pi over 3."},{"Start":"11:24.824 ","End":"11:28.870","Text":"Let\u0027s just highlight that and we are done."}],"ID":8831},{"Watched":false,"Name":"Exercise 1 – Verified one direction","Duration":"6m 46s","ChapterTopicVideoID":8765,"CourseChapterTopicPlaylistID":4966,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.090","Text":"In this exercise, we want to verify the divergence theorem,"},{"Start":"00:06.090 ","End":"00:14.340","Text":"which equates the triple integral along over a 3D region R"},{"Start":"00:14.340 ","End":"00:20.580","Text":"with a surface integral of the boundary of that region,"},{"Start":"00:20.580 ","End":"00:25.500","Text":"so that here S is the surface which is the boundary of the region R."},{"Start":"00:25.500 ","End":"00:30.810","Text":"In this case we\u0027re taking a cube and determined by these planes,"},{"Start":"00:30.810 ","End":"00:36.640","Text":"which means it\u0027s basically the unit cube and here is the sketch of it."},{"Start":"00:36.640 ","End":"00:39.525","Text":"Everything coincide 1."},{"Start":"00:39.525 ","End":"00:44.195","Text":"The whole region including the inside is R,"},{"Start":"00:44.195 ","End":"00:47.975","Text":"but S is just the surface of it."},{"Start":"00:47.975 ","End":"00:51.709","Text":"For the triple integral over this region"},{"Start":"00:51.709 ","End":"00:56.240","Text":"we\u0027ll need to compute the divergence of the vector field F first,"},{"Start":"00:56.240 ","End":"00:58.520","Text":"and this is where F is given."},{"Start":"00:58.520 ","End":"01:01.640","Text":"Let\u0027s label the component functions."},{"Start":"01:01.640 ","End":"01:03.390","Text":"This one it\u0027ll be P."},{"Start":"01:03.390 ","End":"01:07.440","Text":"This one we\u0027ll call Q,"},{"Start":"01:07.440 ","End":"01:10.280","Text":"and This one we\u0027ll call R."},{"Start":"01:10.280 ","End":"01:17.645","Text":"In general, the divergence of F is P with respect to x,"},{"Start":"01:17.645 ","End":"01:23.155","Text":"partial derivative plus Q with respect to y plus R with respect to z."},{"Start":"01:23.155 ","End":"01:26.150","Text":"In our case let\u0027s see what it comes out to."},{"Start":"01:26.150 ","End":"01:29.490","Text":"P with respect to x is 2,"},{"Start":"01:29.490 ","End":"01:34.470","Text":"Q with respect to y is x squared,"},{"Start":"01:34.470 ","End":"01:41.560","Text":"and R with respect to z will be minus 2xz."},{"Start":"01:43.690 ","End":"01:47.840","Text":"Now this cube is a very nice shape for converting"},{"Start":"01:47.840 ","End":"01:53.854","Text":"the integral over the whole cube 2 as an iterated integral,"},{"Start":"01:53.854 ","End":"02:00.495","Text":"we could take, let\u0027s say, x from 0-1."},{"Start":"02:00.495 ","End":"02:06.050","Text":"For each x, we have also y goes from 0-1"},{"Start":"02:06.050 ","End":"02:12.500","Text":"and z from 0-1 of this divergence,"},{"Start":"02:12.500 ","End":"02:23.620","Text":"which is 2 plus x squared minus 2xz, dzdydx."},{"Start":"02:23.620 ","End":"02:25.890","Text":"We\u0027ll do the innermost one first,"},{"Start":"02:25.890 ","End":"02:28.810","Text":"that\u0027s the dz integral."},{"Start":"02:29.350 ","End":"02:33.030","Text":"I prefer to do this separately at the side ,"},{"Start":"02:33.030 ","End":"02:35.450","Text":"so let\u0027s see, with respect to z,"},{"Start":"02:35.450 ","End":"02:46.010","Text":"we get 2z plus x squared z minus x is a constant,"},{"Start":"02:46.010 ","End":"02:51.140","Text":"the integral of 2z is z squared so we have xz squared,"},{"Start":"02:51.140 ","End":"02:56.790","Text":"and we have to evaluate this from 0-1,"},{"Start":"02:56.790 ","End":"02:58.965","Text":"that\u0027s z of course."},{"Start":"02:58.965 ","End":"03:02.580","Text":"When z is 0, they\u0027re all zeros that doesn\u0027t give anything,"},{"Start":"03:02.580 ","End":"03:13.470","Text":"when z is 1, we get 2 plus x squared minus x"},{"Start":"03:13.470 ","End":"03:20.725","Text":"and so I\u0027ll go back here and we now get the integral from 0-1,"},{"Start":"03:20.725 ","End":"03:26.370","Text":"and that\u0027s x and y also from 0-1."},{"Start":"03:26.370 ","End":"03:40.550","Text":"We have this, which is x squared minus x plus 2 dydx."},{"Start":"03:40.550 ","End":"03:45.010","Text":"Notice that there is no y in this."},{"Start":"03:45.010 ","End":"03:50.395","Text":"I could actually reverse."},{"Start":"03:50.395 ","End":"03:57.040","Text":"I could pull this part in front of the integral."},{"Start":"03:57.040 ","End":"04:01.180","Text":"Let\u0027s imagine that this is here and that all they left here is 1"},{"Start":"04:01.180 ","End":"04:11.090","Text":"so the inner integral in this case just becomes the integral of 1dy from 0-1"},{"Start":"04:11.090 ","End":"04:16.680","Text":"and that is equal to d to c is just equal to 1."},{"Start":"04:18.290 ","End":"04:29.740","Text":"What I\u0027m left with now is the integral x goes from 0-1,"},{"Start":"04:29.740 ","End":"04:35.210","Text":"x squared minus x plus 2."},{"Start":"04:35.210 ","End":"04:38.670","Text":"All this came out to be 1."},{"Start":"04:39.170 ","End":"04:41.520","Text":"We\u0027ll put times 1,"},{"Start":"04:41.520 ","End":"04:45.240","Text":"that\u0027s the 1 here and then dx,"},{"Start":"04:45.240 ","End":"04:47.275","Text":"throw the 1 away."},{"Start":"04:47.275 ","End":"04:52.700","Text":"Now we have just a straightforward integral in 1 variable."},{"Start":"04:53.530 ","End":"04:56.450","Text":"No need to do it at the side."},{"Start":"04:56.450 ","End":"05:08.270","Text":"This is just x cubed over 3 minus x squared over 2 plus 2x from 0-1"},{"Start":"05:08.270 ","End":"05:11.685","Text":"and so at 0 we get nothing."},{"Start":"05:11.685 ","End":"05:19.410","Text":"At 1 we get 1/3 minus a 1/2 plus 2"},{"Start":"05:19.410 ","End":"05:24.895","Text":"and that comes out to be let see,"},{"Start":"05:24.895 ","End":"05:27.785","Text":"1/2 minus a 1/3 is a 1/6,"},{"Start":"05:27.785 ","End":"05:30.695","Text":"this is minus 1/6 plus 2,"},{"Start":"05:30.695 ","End":"05:38.935","Text":"1 and 5/6 or 11 over 6, whichever you prefer."},{"Start":"05:38.935 ","End":"05:43.400","Text":"I\u0027ll take it as a mixed number."},{"Start":"05:43.400 ","End":"05:49.170","Text":"That just completes half the exercise."},{"Start":"05:49.210 ","End":"05:53.119","Text":"It just does this half."},{"Start":"05:53.119 ","End":"05:56.975","Text":"Now, we have to do the other half."},{"Start":"05:56.975 ","End":"06:00.230","Text":"The surface integral."},{"Start":"06:00.230 ","End":"06:06.650","Text":"Now, this exact same exercise was given in"},{"Start":"06:06.650 ","End":"06:13.820","Text":"the chapter on surface integrals and if you check there,"},{"Start":"06:13.820 ","End":"06:18.410","Text":"you\u0027ll see that we got the same answer, 1 and 5/6."},{"Start":"06:18.410 ","End":"06:23.340","Text":"But just in case you don\u0027t have it or for whatever reason,"},{"Start":"06:23.340 ","End":"06:28.330","Text":"I\u0027ve copied the exercise to the following clip."},{"Start":"06:28.330 ","End":"06:30.410","Text":"Just go to the next clip"},{"Start":"06:30.410 ","End":"06:33.830","Text":"and there I\u0027ll do the surface integral"},{"Start":"06:33.830 ","End":"06:36.785","Text":"and the answer does come out to be the same."},{"Start":"06:36.785 ","End":"06:38.870","Text":"It also 1 and 5/6,"},{"Start":"06:38.870 ","End":"06:42.480","Text":"which verifies the divergence theorem."},{"Start":"06:42.640 ","End":"06:45.960","Text":"That\u0027s it for this half."}],"ID":8832},{"Watched":false,"Name":"Exercise 1 – Verified second direction","Duration":"22m ","ChapterTopicVideoID":8764,"CourseChapterTopicPlaylistID":4966,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.590","Text":"In this exercise, we have to compute a surface integral."},{"Start":"00:04.590 ","End":"00:07.439","Text":"F is a vector field in 3D."},{"Start":"00:07.439 ","End":"00:08.610","Text":"It\u0027s given as follows,"},{"Start":"00:08.610 ","End":"00:10.995","Text":"something i, something j, something k,"},{"Start":"00:10.995 ","End":"00:13.620","Text":"and S is the closed surface;"},{"Start":"00:13.620 ","End":"00:18.780","Text":"it\u0027s the surface of the cube is a picture of it."},{"Start":"00:18.780 ","End":"00:25.545","Text":"One way of defining it is just by giving the equations of each of the 6 planes,"},{"Start":"00:25.545 ","End":"00:27.690","Text":"which are x equals 0, x equals 1,"},{"Start":"00:27.690 ","End":"00:32.570","Text":"and so on and n is the positive orientation,"},{"Start":"00:32.570 ","End":"00:36.935","Text":"which is the outward unit normal."},{"Start":"00:36.935 ","End":"00:40.775","Text":"Let\u0027s look at this and what we want to do,"},{"Start":"00:40.775 ","End":"00:43.490","Text":"is do it in pairs."},{"Start":"00:43.490 ","End":"00:45.260","Text":"I brought in a picture,"},{"Start":"00:45.260 ","End":"00:47.615","Text":"it\u0027s easier to explain with the picture."},{"Start":"00:47.615 ","End":"00:50.030","Text":"Talking about these 2 that I\u0027ve shaded,"},{"Start":"00:50.030 ","End":"00:51.665","Text":"the top 1 and on the bottom 1,"},{"Start":"00:51.665 ","End":"00:53.615","Text":"I\u0027ll do this pair first."},{"Start":"00:53.615 ","End":"00:55.750","Text":"Let me just label them,"},{"Start":"00:55.750 ","End":"01:01.110","Text":"maybe the lower 1 is S_1 and the upper 1 is S_2,"},{"Start":"01:01.110 ","End":"01:05.175","Text":"that says z equals 0, z equals 1."},{"Start":"01:05.175 ","End":"01:08.989","Text":"Notice that the outward normal for the top surface,"},{"Start":"01:08.989 ","End":"01:13.695","Text":"it\u0027s the vector k, upward face upward vector,"},{"Start":"01:13.695 ","End":"01:17.280","Text":"but for the bottom surface, it\u0027s minus k."},{"Start":"01:17.280 ","End":"01:20.535","Text":"Let\u0027s start with one of them,"},{"Start":"01:20.535 ","End":"01:22.875","Text":"I\u0027ll take S_2 for starters."},{"Start":"01:22.875 ","End":"01:30.800","Text":"S_2, I can describe it\u0027s equation as it\u0027s z equals 1,"},{"Start":"01:30.800 ","End":"01:35.109","Text":"but I want to write it as z equals g of x, y in general,"},{"Start":"01:35.109 ","End":"01:39.320","Text":"z is a function of x and y and just happens to equal 1 because"},{"Start":"01:39.320 ","End":"01:41.420","Text":"we know how to deal with such surfaces"},{"Start":"01:41.420 ","End":"01:46.220","Text":"when we have 1 variable as a function of the other 2."},{"Start":"01:46.220 ","End":"01:54.870","Text":"Notice that it\u0027s very easy to compute F.n in this case."},{"Start":"01:54.870 ","End":"02:01.560","Text":"For S_2, F.n is simply this particular F dot with k."},{"Start":"02:01.560 ","End":"02:05.105","Text":"I\u0027m not going to copy the whole F out again,"},{"Start":"02:05.105 ","End":"02:09.380","Text":"because when you take a dot product with a unit vector k,"},{"Start":"02:09.380 ","End":"02:11.990","Text":"it\u0027s just the last component."},{"Start":"02:11.990 ","End":"02:17.135","Text":"This will be minus xz squared."},{"Start":"02:17.135 ","End":"02:30.810","Text":"What we want is the double integral over S of minus xz squared dS."},{"Start":"02:30.810 ","End":"02:34.000","Text":"Now, the way we interpret dS,"},{"Start":"02:35.170 ","End":"02:39.265","Text":"maybe a better just write a general formula at the side."},{"Start":"02:39.265 ","End":"02:41.269","Text":"Whenever I have the double integral"},{"Start":"02:41.269 ","End":"02:46.070","Text":"over surface S of some function of x, y, and z,"},{"Start":"02:46.070 ","End":"02:52.270","Text":"which will vary in each of the 6 parts."},{"Start":"02:52.270 ","End":"02:53.600","Text":"But here at least,"},{"Start":"02:53.600 ","End":"02:56.570","Text":"we have f of x, y, and z is minus x squared."},{"Start":"02:56.570 ","End":"03:03.440","Text":"But in general, this dS is the double integral over D,"},{"Start":"03:03.440 ","End":"03:14.100","Text":"where D is the projection onto the xy plane in cases like this of F of x, y"},{"Start":"03:14.100 ","End":"03:16.710","Text":"and instead of z, g of x, y"},{"Start":"03:16.710 ","End":"03:21.770","Text":"and instead of dS, we put this peculiar expression,"},{"Start":"03:21.770 ","End":"03:25.460","Text":"the square root of partial derivative of g"},{"Start":"03:25.460 ","End":"03:29.959","Text":"with respect to x squared plus gy squared plus 1,"},{"Start":"03:29.959 ","End":"03:34.129","Text":"and then dA, and that\u0027s a regular double integral."},{"Start":"03:34.129 ","End":"03:39.290","Text":"What is this D, what is the projection of this onto the xy plane?"},{"Start":"03:39.290 ","End":"03:42.110","Text":"Here\u0027s a sketch to show what it looks like."},{"Start":"03:42.110 ","End":"03:50.240","Text":"It actually coincidentally turns out to be just S_1 considered as 2D."},{"Start":"03:50.240 ","End":"03:53.040","Text":"It\u0027s a unit square."},{"Start":"03:53.900 ","End":"03:58.400","Text":"Now we can go ahead and this will be our R"},{"Start":"03:58.400 ","End":"04:05.795","Text":"and rewrite this again as the double integral over D,"},{"Start":"04:05.795 ","End":"04:07.580","Text":"this particular D."},{"Start":"04:07.580 ","End":"04:13.339","Text":"What we do is we replace z by g of x, y."},{"Start":"04:13.339 ","End":"04:15.110","Text":"That\u0027s what this means here."},{"Start":"04:15.110 ","End":"04:18.470","Text":"That is z, g of x, y, which happens to be 1."},{"Start":"04:18.470 ","End":"04:21.605","Text":"Basically, I\u0027m just saying let z equals 1 here."},{"Start":"04:21.605 ","End":"04:28.980","Text":"We\u0027ve got minus x and then we need this expression with the square root."},{"Start":"04:29.200 ","End":"04:36.045","Text":"Now, g with respect to everything, g is a constant,"},{"Start":"04:36.045 ","End":"04:39.080","Text":"so g with respect to x and y is 0."},{"Start":"04:39.080 ","End":"04:47.060","Text":"It is 0 squared plus 0 squared plus the one that\u0027s here and dA."},{"Start":"04:50.090 ","End":"04:56.250","Text":"I can write this as an iterative integral dx dy or dy dx."},{"Start":"04:56.300 ","End":"05:05.500","Text":"Let\u0027s take this the outer one as x and the inner one as y."},{"Start":"05:05.500 ","End":"05:08.055","Text":"They both go from 0-1."},{"Start":"05:08.055 ","End":"05:16.540","Text":"Then I have this expression which is just minus x and dy dx."},{"Start":"05:16.540 ","End":"05:19.845","Text":"This is straightforward to compute."},{"Start":"05:19.845 ","End":"05:23.309","Text":"This is equal to,"},{"Start":"05:23.309 ","End":"05:27.630","Text":"so we get the integral from 0-1 dx."},{"Start":"05:27.630 ","End":"05:30.915","Text":"I\u0027m talking about this bit here."},{"Start":"05:30.915 ","End":"05:34.490","Text":"This shaded bit as just minus xy,"},{"Start":"05:34.490 ","End":"05:37.130","Text":"which is the integral of minus x with respect to y,"},{"Start":"05:37.130 ","End":"05:41.505","Text":"evaluated from y equals 0-1."},{"Start":"05:41.505 ","End":"05:43.855","Text":"When I put y equals 1,"},{"Start":"05:43.855 ","End":"05:46.415","Text":"I get minus x."},{"Start":"05:46.415 ","End":"05:50.165","Text":"When I put y equals 0, I get nothing."},{"Start":"05:50.165 ","End":"05:53.820","Text":"This is just minus x."},{"Start":"05:53.820 ","End":"06:06.370","Text":"Now I have the integral from 0-1 of minus x dx."},{"Start":"06:07.670 ","End":"06:17.245","Text":"This is equal to minus x squared over 2 from 0-1,"},{"Start":"06:17.245 ","End":"06:22.455","Text":"0 gives nothing, 1 gives me minus 1/2."},{"Start":"06:22.455 ","End":"06:28.050","Text":"That\u0027s for S_2 and I\u0027ll highlight this"},{"Start":"06:28.050 ","End":"06:31.190","Text":"and we\u0027ll get 6 of these results altogether,"},{"Start":"06:31.190 ","End":"06:33.020","Text":"and then we\u0027ll add them all up."},{"Start":"06:33.020 ","End":"06:37.735","Text":"Next one, let\u0027s do S_1."},{"Start":"06:37.735 ","End":"06:42.670","Text":"Now, it\u0027s going to be very similar in many ways to the S_2."},{"Start":"06:42.670 ","End":"06:44.630","Text":"I just want to indicate the differences."},{"Start":"06:44.630 ","End":"06:49.200","Text":"What will be the difference if I do S_1 instead of S_2?"},{"Start":"06:49.200 ","End":"06:56.930","Text":"Well, 1 thing will change will be the function g of x, y is not 1 anymore, it\u0027s now 0."},{"Start":"06:56.930 ","End":"07:02.900","Text":"The other difference is that the normal is not k, it\u0027s minus k,"},{"Start":"07:02.900 ","End":"07:06.095","Text":"and the dot product will be,"},{"Start":"07:06.095 ","End":"07:10.360","Text":"I just emphasize it, plus xz squared."},{"Start":"07:10.360 ","End":"07:13.880","Text":"Instead of minus x, which we got,"},{"Start":"07:13.880 ","End":"07:19.460","Text":"we\u0027re going to get just 0 because z equals 0,"},{"Start":"07:19.460 ","End":"07:21.725","Text":"the g of xy is 0."},{"Start":"07:21.725 ","End":"07:26.510","Text":"Because we get a 0, the whole integral will just come out to be 0"},{"Start":"07:26.510 ","End":"07:33.860","Text":"and if this was the case for S_2,"},{"Start":"07:33.860 ","End":"07:41.925","Text":"then for S_1 we\u0027re going to get 0 and I\u0027ll highlight that."},{"Start":"07:41.925 ","End":"07:42.950","Text":"That\u0027s the second one."},{"Start":"07:42.950 ","End":"07:46.590","Text":"It was just so similar, didn\u0027t want to do it all again."},{"Start":"07:47.590 ","End":"07:52.035","Text":"We\u0027ve done S_1 and S_2."},{"Start":"07:52.035 ","End":"07:54.170","Text":"Now, let\u0027s go to this pair,"},{"Start":"07:54.170 ","End":"07:57.845","Text":"and I\u0027ll call this S_3 and S_4."},{"Start":"07:57.845 ","End":"08:02.475","Text":"I\u0027d like to save some of the work here, not start all over again."},{"Start":"08:02.475 ","End":"08:05.700","Text":"The obvious thing we have to do is,"},{"Start":"08:05.700 ","End":"08:08.890","Text":"first of all I\u0027ll replace this picture."},{"Start":"08:09.420 ","End":"08:11.995","Text":"I\u0027ll explain it in a moment."},{"Start":"08:11.995 ","End":"08:14.635","Text":"Let\u0027s just get rid of some of the clutter."},{"Start":"08:14.635 ","End":"08:21.599","Text":"I\u0027ll label this one, S_3, that\u0027s the x equals 0"},{"Start":"08:21.599 ","End":"08:26.490","Text":"and this one S_4, where x equals 1."},{"Start":"08:26.490 ","End":"08:29.010","Text":"The normal vector here,"},{"Start":"08:29.010 ","End":"08:31.300","Text":"the outward 1 is minus i,"},{"Start":"08:31.300 ","End":"08:34.330","Text":"it\u0027s in the direction of the minus x-axis."},{"Start":"08:34.330 ","End":"08:37.120","Text":"This is the positive direction for x"},{"Start":"08:37.120 ","End":"08:42.045","Text":"and it\u0027s going to be plus i for S_4."},{"Start":"08:42.045 ","End":"08:45.100","Text":"Let\u0027s start with S_4."},{"Start":"08:45.550 ","End":"08:50.000","Text":"But here, x is the odd one out,"},{"Start":"08:50.000 ","End":"08:52.310","Text":"x as a function of y and z."},{"Start":"08:52.310 ","End":"08:56.435","Text":"I\u0027ll reuse the letter g,"},{"Start":"08:56.435 ","End":"09:02.185","Text":"different g than the one before, g of y and z."},{"Start":"09:02.185 ","End":"09:05.970","Text":"This will be true for both S_3 and S_4,"},{"Start":"09:05.970 ","End":"09:08.105","Text":"but in the case of S_4,"},{"Start":"09:08.105 ","End":"09:10.230","Text":"it\u0027s going to equal 1."},{"Start":"09:10.230 ","End":"09:12.720","Text":"In the case of S_3, it will be 0."},{"Start":"09:12.720 ","End":"09:21.030","Text":"F.n is F. the normal here we said was i."},{"Start":"09:21.030 ","End":"09:23.585","Text":"When we dot product with i,"},{"Start":"09:23.585 ","End":"09:25.795","Text":"we just take the first component,"},{"Start":"09:25.795 ","End":"09:30.730","Text":"which will be 2x minus z."},{"Start":"09:31.120 ","End":"09:36.095","Text":"We\u0027re also going to have to modify this formula here."},{"Start":"09:36.095 ","End":"09:43.745","Text":"It\u0027s just completely analogous to the one where z was the function of x and y."},{"Start":"09:43.745 ","End":"09:46.010","Text":"There\u0027s also one other thing I have to replace,"},{"Start":"09:46.010 ","End":"09:48.500","Text":"this is no longer x and y."},{"Start":"09:48.500 ","End":"09:51.260","Text":"We are now a domain."},{"Start":"09:51.260 ","End":"09:57.740","Text":"The region is in the y, z plane, but it\u0027s actually the same picture."},{"Start":"09:57.740 ","End":"10:02.120","Text":"This is the projection of S_4 and it will work for S_3 also"},{"Start":"10:02.120 ","End":"10:05.660","Text":"onto the yz plane, it\u0027s just this unit square."},{"Start":"10:05.660 ","End":"10:18.870","Text":"What we need now is the double integral of 2x minus z over S_4,"},{"Start":"10:18.870 ","End":"10:26.310","Text":"dS will be the double integral over D."},{"Start":"10:26.310 ","End":"10:35.210","Text":"Now here, this just says replace x by what it is in terms of y and z"},{"Start":"10:35.210 ","End":"10:38.370","Text":"and this is what it is."},{"Start":"10:38.370 ","End":"10:43.710","Text":"It\u0027s 2 times x,"},{"Start":"10:43.710 ","End":"10:44.970","Text":"which is just 1."},{"Start":"10:44.970 ","End":"10:46.800","Text":"It\u0027s a constant function 2 times,"},{"Start":"10:46.800 ","End":"10:54.610","Text":"I\u0027ll write the 1 because that\u0027s what x is, minus z."},{"Start":"10:54.950 ","End":"10:58.150","Text":"Maybe I\u0027ll erase the 1."},{"Start":"10:58.340 ","End":"11:03.965","Text":"Just like before this square root comes out to be just 1,"},{"Start":"11:03.965 ","End":"11:09.155","Text":"because g is a constant, gy, gz a 0, square root of 1 is 1."},{"Start":"11:09.155 ","End":"11:14.640","Text":"We can basically just forget about the square root part."},{"Start":"11:15.460 ","End":"11:19.740","Text":"We just need to add dA."},{"Start":"11:22.080 ","End":"11:24.820","Text":"I want to write it as an iterated integral."},{"Start":"11:24.820 ","End":"11:28.220","Text":"Doesn\u0027t really matter, dy dz or dz dy."},{"Start":"11:28.890 ","End":"11:36.370","Text":"I\u0027ll go first of all from y equals 0-1,"},{"Start":"11:36.370 ","End":"11:41.140","Text":"and then the integral of z from 0-1,"},{"Start":"11:41.140 ","End":"11:42.715","Text":"that\u0027s the inner one."},{"Start":"11:42.715 ","End":"11:47.110","Text":"Then 2 minus z,"},{"Start":"11:47.110 ","End":"11:52.030","Text":"and in this case it has to be dz dy."},{"Start":"11:52.030 ","End":"11:57.400","Text":"Have to be, but we usually put the inner one with the inner one."},{"Start":"11:57.400 ","End":"12:02.020","Text":"Let\u0027s first do this, the dz integral, I mean."},{"Start":"12:02.020 ","End":"12:07.210","Text":"I\u0027ll do this highlighted bit over here."},{"Start":"12:07.210 ","End":"12:17.140","Text":"The integral of 2 minus z, dz is just 2z minus z squared over 2,"},{"Start":"12:17.140 ","End":"12:22.300","Text":"evaluated from 0-1."},{"Start":"12:22.300 ","End":"12:25.495","Text":"At 0, I get nothing."},{"Start":"12:25.495 ","End":"12:29.075","Text":"At 1, I get 2 minus 1/2,"},{"Start":"12:29.075 ","End":"12:32.430","Text":"which is 1 and a 1/2 or 3 over 2,"},{"Start":"12:32.430 ","End":"12:36.100","Text":"it doesn\u0027t really matter, I\u0027ll write it as 1 and a 1/2."},{"Start":"12:36.230 ","End":"12:38.580","Text":"Now I go back here,"},{"Start":"12:38.580 ","End":"12:41.355","Text":"make a note that this is 1 and a 1/2."},{"Start":"12:41.355 ","End":"12:44.380","Text":"Let\u0027s see what we get."},{"Start":"12:44.610 ","End":"12:49.554","Text":"The 1 and a 1/2 is a constant so I can bring it in front of the integral."},{"Start":"12:49.554 ","End":"12:57.865","Text":"So it\u0027s 1 and a 1/2 times the integral from 0-1 of just the dy,"},{"Start":"12:57.865 ","End":"13:01.040","Text":"but I\u0027ll write it as 1dy."},{"Start":"13:01.080 ","End":"13:04.900","Text":"Wherever we have the integral of 1 like this,"},{"Start":"13:04.900 ","End":"13:08.455","Text":"we just take the upper limit minus the lower limit gives us 1."},{"Start":"13:08.455 ","End":"13:11.870","Text":"This is 1 and a 1/2."},{"Start":"13:11.910 ","End":"13:18.715","Text":"I\u0027ll highlight it, and that\u0027s the S_4 surface."},{"Start":"13:18.715 ","End":"13:22.270","Text":"Now let\u0027s go on to S_3,"},{"Start":"13:22.270 ","End":"13:24.700","Text":"and just like I did with S_1 and S_2,"},{"Start":"13:24.700 ","End":"13:26.200","Text":"I\u0027d like to save a lot of this work."},{"Start":"13:26.200 ","End":"13:28.315","Text":"So let\u0027s just see what the difference will be."},{"Start":"13:28.315 ","End":"13:32.125","Text":"Instead of S_4, I\u0027m taking S_3."},{"Start":"13:32.125 ","End":"13:37.960","Text":"The first difference is that this function is not 1, it\u0027s 0."},{"Start":"13:37.960 ","End":"13:40.300","Text":"The other difference is that instead of i,"},{"Start":"13:40.300 ","End":"13:42.040","Text":"I have minus i."},{"Start":"13:42.040 ","End":"13:45.955","Text":"Remember for S_3, it\u0027s minus i."},{"Start":"13:45.955 ","End":"13:48.835","Text":"So what I would get here,"},{"Start":"13:48.835 ","End":"13:50.875","Text":"instead of 2x minus z,"},{"Start":"13:50.875 ","End":"13:55.420","Text":"it would be z minus 2x,"},{"Start":"13:55.420 ","End":"14:00.505","Text":"and so this would be z minus 2x."},{"Start":"14:00.505 ","End":"14:03.115","Text":"Then we would do the substitution."},{"Start":"14:03.115 ","End":"14:09.850","Text":"Previously, we did that x equals 1."},{"Start":"14:10.380 ","End":"14:16.705","Text":"That was the x equals 1 plane here, x equals 0."},{"Start":"14:16.705 ","End":"14:22.490","Text":"If x is 0, then all I get is z."},{"Start":"14:23.100 ","End":"14:27.355","Text":"So not this, but just z,"},{"Start":"14:27.355 ","End":"14:31.015","Text":"not this, just z."},{"Start":"14:31.015 ","End":"14:34.495","Text":"Instead of this, I get the integral of z,"},{"Start":"14:34.495 ","End":"14:41.470","Text":"which is just z squared over 2 between 0 and 1,"},{"Start":"14:41.470 ","End":"14:46.600","Text":"and this comes out to be 1/2."},{"Start":"14:46.600 ","End":"14:48.220","Text":"Instead of 1 and a 1/2,"},{"Start":"14:48.220 ","End":"14:50.995","Text":"I just get 1/2,"},{"Start":"14:50.995 ","End":"14:53.680","Text":"and this we already figured it was 1."},{"Start":"14:53.680 ","End":"15:01.360","Text":"For S_3, it comes out to be 1/2."},{"Start":"15:01.360 ","End":"15:05.755","Text":"I\u0027ll highlight it, and that\u0027s 4 out of the 6 done."},{"Start":"15:05.755 ","End":"15:09.100","Text":"Next, we\u0027ll do S_5,"},{"Start":"15:09.100 ","End":"15:12.880","Text":"we\u0027ll call this and S_6."},{"Start":"15:12.880 ","End":"15:17.935","Text":"I want to reuse some of the stuff so I\u0027ll just erase and replace."},{"Start":"15:17.935 ","End":"15:21.325","Text":"Here we are, this is the new picture."},{"Start":"15:21.325 ","End":"15:25.870","Text":"This time, y is the special one,"},{"Start":"15:25.870 ","End":"15:31.780","Text":"y will be a function of x and z."},{"Start":"15:31.780 ","End":"15:35.665","Text":"This will be x, this will be z,"},{"Start":"15:35.665 ","End":"15:40.975","Text":"and when we project either of these surfaces,"},{"Start":"15:40.975 ","End":"15:45.295","Text":"this one will be S_6,"},{"Start":"15:45.295 ","End":"15:48.890","Text":"where y equals 1."},{"Start":"15:50.130 ","End":"15:54.085","Text":"This one, S_5, where y equals 0."},{"Start":"15:54.085 ","End":"15:55.900","Text":"We\u0027re starting with S6,"},{"Start":"15:55.900 ","End":"16:00.429","Text":"y is g of x and z, which is 1."},{"Start":"16:00.429 ","End":"16:07.060","Text":"F.n is F. the vector j,"},{"Start":"16:07.060 ","End":"16:10.039","Text":"and when we take a dot product with the vector j,"},{"Start":"16:10.039 ","End":"16:17.805","Text":"we just need the second component of the vector field,"},{"Start":"16:17.805 ","End":"16:19.500","Text":"which is this one,"},{"Start":"16:19.500 ","End":"16:24.975","Text":"which is x squared times y."},{"Start":"16:24.975 ","End":"16:30.280","Text":"Then we need the surface integral"},{"Start":"16:30.280 ","End":"16:38.660","Text":"over S_6 of x squared y dS."},{"Start":"16:41.490 ","End":"16:49.480","Text":"I need to also give you the equivalent to this formula when this time,"},{"Start":"16:49.480 ","End":"16:55.450","Text":"y is the special variable,"},{"Start":"16:55.450 ","End":"16:57.805","Text":"y is expressed in terms of x and z."},{"Start":"16:57.805 ","End":"17:02.275","Text":"So we have here x and instead of y,"},{"Start":"17:02.275 ","End":"17:06.250","Text":"we have g of x, z."},{"Start":"17:06.250 ","End":"17:09.640","Text":"I\u0027m reusing the letter g a lot."},{"Start":"17:09.640 ","End":"17:12.400","Text":"Should have really used different letters each time,"},{"Start":"17:12.400 ","End":"17:14.840","Text":"but there\u0027s no confusion."},{"Start":"17:16.500 ","End":"17:19.210","Text":"Under the square root,"},{"Start":"17:19.210 ","End":"17:21.850","Text":"the odd one out is the middle 1,"},{"Start":"17:21.850 ","End":"17:26.035","Text":"we have g with respect 2x squared plus 1"},{"Start":"17:26.035 ","End":"17:30.910","Text":"plus partial derivative of g with respect to z squared."},{"Start":"17:30.910 ","End":"17:35.440","Text":"That\u0027s the formula, and applying it here,"},{"Start":"17:35.440 ","End":"17:40.990","Text":"we get the double integral over D."},{"Start":"17:40.990 ","End":"17:46.795","Text":"Now I have to replace y by 1,"},{"Start":"17:46.795 ","End":"17:51.760","Text":"so I just get x squared."},{"Start":"17:51.760 ","End":"17:57.250","Text":"I have the square root of something dA,"},{"Start":"17:57.250 ","End":"18:01.585","Text":"and that is g with respect to x."},{"Start":"18:01.585 ","End":"18:07.930","Text":"Well, yeah, with respect to x and with respect to z it\u0027s 0"},{"Start":"18:07.930 ","End":"18:08.770","Text":"because they\u0027re constant,"},{"Start":"18:08.770 ","End":"18:18.710","Text":"so I just get 0 squared plus 1 plus 0 squared from here, dA, this is just 1."},{"Start":"18:20.130 ","End":"18:23.530","Text":"I\u0027ll just put a line through it and write 1,"},{"Start":"18:23.530 ","End":"18:26.155","Text":"and 1 I don\u0027t need at all."},{"Start":"18:26.155 ","End":"18:29.320","Text":"If I do it in an iterated integral,"},{"Start":"18:29.320 ","End":"18:34.930","Text":"we get the integral from 0-1, integral from 0-1."},{"Start":"18:34.930 ","End":"18:37.030","Text":"It doesn\u0027t really matter."},{"Start":"18:37.030 ","End":"18:39.800","Text":"I\u0027ll do it as a dx dz."},{"Start":"18:44.460 ","End":"18:47.725","Text":"Then this means that x,"},{"Start":"18:47.725 ","End":"18:51.820","Text":"I\u0027ll just write x is the one that goes from 0-1 actually,"},{"Start":"18:51.820 ","End":"18:53.575","Text":"and so does z."},{"Start":"18:53.575 ","End":"18:56.695","Text":"We\u0027ll start with the inner one."},{"Start":"18:56.695 ","End":"18:59.455","Text":"I\u0027ll highlight that."},{"Start":"18:59.455 ","End":"19:06.820","Text":"I\u0027ll just do this quick calculation at the side x squared gives me x cubed over 3,"},{"Start":"19:06.820 ","End":"19:11.560","Text":"I have to evaluate this from 0-1,"},{"Start":"19:11.560 ","End":"19:18.655","Text":"and this comes out to be 1/3 minus 0, it\u0027s just 1/3."},{"Start":"19:18.655 ","End":"19:21.460","Text":"All this is 1/3,"},{"Start":"19:21.460 ","End":"19:24.140","Text":"and now I have to do the integral dz."},{"Start":"19:26.700 ","End":"19:30.430","Text":"Well, the third I can take in front,"},{"Start":"19:30.430 ","End":"19:39.020","Text":"and I\u0027ve just got the integral from 0-1 of just dz or 1dz."},{"Start":"19:39.360 ","End":"19:43.090","Text":"This integral we\u0027ve seen plenty of times before,"},{"Start":"19:43.090 ","End":"19:45.325","Text":"the answer comes out to be 1."},{"Start":"19:45.325 ","End":"19:49.480","Text":"This whole thing is 1/3,"},{"Start":"19:49.480 ","End":"19:51.745","Text":"and I\u0027ll highlight it."},{"Start":"19:51.745 ","End":"19:55.540","Text":"This was the S_6,"},{"Start":"19:55.540 ","End":"19:59.500","Text":"and now all we\u0027re left with is S_5."},{"Start":"19:59.500 ","End":"20:01.930","Text":"I won\u0027t do it all again."},{"Start":"20:01.930 ","End":"20:04.570","Text":"I\u0027ll just say where the changes are."},{"Start":"20:04.570 ","End":"20:12.160","Text":"If this is S_5, then this instead of 1 is going to be 0,"},{"Start":"20:12.160 ","End":"20:16.990","Text":"and also instead of j for the normal,"},{"Start":"20:16.990 ","End":"20:20.240","Text":"we\u0027re going to have minus j."},{"Start":"20:23.280 ","End":"20:29.845","Text":"This will be a minus x squared y minus x squared y."},{"Start":"20:29.845 ","End":"20:34.810","Text":"But when we make the substitution,"},{"Start":"20:34.810 ","End":"20:37.090","Text":"previously, y was 1,"},{"Start":"20:37.090 ","End":"20:38.649","Text":"so we got x squared,"},{"Start":"20:38.649 ","End":"20:41.965","Text":"but now y is 0,"},{"Start":"20:41.965 ","End":"20:44.350","Text":"so we get times 0 in here."},{"Start":"20:44.350 ","End":"20:47.260","Text":"This whole thing comes out to be 0."},{"Start":"20:47.260 ","End":"20:48.865","Text":"It\u0027s very important."},{"Start":"20:48.865 ","End":"20:52.030","Text":"Then after you can continue doing all this stuff from here,"},{"Start":"20:52.030 ","End":"20:58.075","Text":"we can straight away conclude that the S_5 bit is 0."},{"Start":"20:58.075 ","End":"21:00.010","Text":"I\u0027ll highlight that."},{"Start":"21:00.010 ","End":"21:06.415","Text":"Now we\u0027ve computed all 6 separate faces from S_1 through S_6."},{"Start":"21:06.415 ","End":"21:08.095","Text":"Let\u0027s see what we have."},{"Start":"21:08.095 ","End":"21:10.960","Text":"Now it\u0027s time to add them all up."},{"Start":"21:10.960 ","End":"21:15.730","Text":"As I remember, here we had 0,"},{"Start":"21:15.730 ","End":"21:18.040","Text":"here we had minus a 1/2,"},{"Start":"21:18.040 ","End":"21:20.920","Text":"here we had a 1/2,"},{"Start":"21:20.920 ","End":"21:23.379","Text":"here we had 1 and a 1/2,"},{"Start":"21:23.379 ","End":"21:30.380","Text":"and here we have 0 and 1/3."},{"Start":"21:31.020 ","End":"21:34.165","Text":"If we add all these together,"},{"Start":"21:34.165 ","End":"21:38.380","Text":"we get 1 and 5/6,"},{"Start":"21:38.380 ","End":"21:42.040","Text":"and that\u0027s our final answer."},{"Start":"21:42.040 ","End":"21:47.770","Text":"Just let me say a word that there is a shorter way to do this"},{"Start":"21:47.770 ","End":"21:51.070","Text":"using something that\u0027s called the divergence theorem,"},{"Start":"21:51.070 ","End":"21:53.035","Text":"which you may or may not have learned."},{"Start":"21:53.035 ","End":"21:55.510","Text":"But there will be a very quick way of doing this."},{"Start":"21:55.510 ","End":"21:57.009","Text":"Look how hard we worked."},{"Start":"21:57.009 ","End":"22:00.530","Text":"So I\u0027m done."}],"ID":8833},{"Watched":false,"Name":"Exercise 2 – Verified one direction","Duration":"4m 2s","ChapterTopicVideoID":8767,"CourseChapterTopicPlaylistID":4966,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.805","Text":"In this exercise, we\u0027re asked to verify the divergence theorem for a particular case."},{"Start":"00:05.805 ","End":"00:08.580","Text":"This is the divergence theorem in general,"},{"Start":"00:08.580 ","End":"00:12.900","Text":"where R is a region in 3D"},{"Start":"00:12.900 ","End":"00:17.490","Text":"and S is the surface which is the boundary of R."},{"Start":"00:17.490 ","End":"00:20.340","Text":"F is a vector field."},{"Start":"00:20.340 ","End":"00:23.010","Text":"In our case, this is what it is."},{"Start":"00:23.010 ","End":"00:31.305","Text":"We\u0027re also given that the region R is the unit ball given by this equation."},{"Start":"00:31.305 ","End":"00:34.725","Text":"Just remember that the difference between ball and a sphere,"},{"Start":"00:34.725 ","End":"00:38.460","Text":"sphere is just the surface that would be with equals 1."},{"Start":"00:38.460 ","End":"00:40.230","Text":"The ball is a solid object,"},{"Start":"00:40.230 ","End":"00:42.384","Text":"it includes the interior."},{"Start":"00:42.384 ","End":"00:45.620","Text":"I\u0027d like to start with the left-hand side."},{"Start":"00:45.620 ","End":"00:48.650","Text":"Let\u0027s compute this and later we\u0027ll compute the right-hand side"},{"Start":"00:48.650 ","End":"00:50.345","Text":"and see that they\u0027re equal."},{"Start":"00:50.345 ","End":"00:54.050","Text":"First thing I\u0027ll need to do is compute the divergence of F."},{"Start":"00:54.050 ","End":"00:56.980","Text":"Well, in general,"},{"Start":"00:56.980 ","End":"01:03.740","Text":"if F is equal to some function times i plus another function of x, y, z"},{"Start":"01:03.740 ","End":"01:08.000","Text":"times j plus another function times K."},{"Start":"01:08.000 ","End":"01:11.135","Text":"I like to use P, Q, R some people use F, G, H."},{"Start":"01:11.135 ","End":"01:17.640","Text":"Anyway, the divergence of F is the derivative of P partial"},{"Start":"01:17.640 ","End":"01:24.350","Text":"with respect to x plus Q with respect to y plus R with respect to z."},{"Start":"01:24.350 ","End":"01:31.620","Text":"In our case, this is P, this is Q, this is R,"},{"Start":"01:31.620 ","End":"01:34.888","Text":"and what we get here is,"},{"Start":"01:34.888 ","End":"01:38.300","Text":"P with respect to x is the constant 1,"},{"Start":"01:38.300 ","End":"01:42.520","Text":"Q with respect to y minus 2,"},{"Start":"01:42.520 ","End":"01:46.650","Text":"R with respect to z plus 3."},{"Start":"01:46.650 ","End":"01:50.940","Text":"This is just equal to 2, the constant."},{"Start":"01:50.940 ","End":"01:56.615","Text":"If the divergence is equal to 2 the constant,"},{"Start":"01:56.615 ","End":"02:00.260","Text":"then I can pull it out in front of the integral sign."},{"Start":"02:00.260 ","End":"02:05.165","Text":"What we get is twice the triple integral"},{"Start":"02:05.165 ","End":"02:13.215","Text":"over the unit ball R of 1 dv."},{"Start":"02:13.215 ","End":"02:17.035","Text":"Now the integral of 1 dv over a region,"},{"Start":"02:17.035 ","End":"02:19.715","Text":"as you recall, is just the volume,"},{"Start":"02:19.715 ","End":"02:31.680","Text":"it\u0027s twice the volume of R."},{"Start":"02:31.680 ","End":"02:37.415","Text":"Well, you should know the formula for a volume of a ball."},{"Start":"02:37.415 ","End":"02:44.060","Text":"Sometimes we say volume of a sphere is, from geometry,"},{"Start":"02:44.060 ","End":"02:54.195","Text":"the volume of a sphere is just 4/3 Pi r cubed,"},{"Start":"02:54.195 ","End":"02:57.390","Text":"where r is the radius."},{"Start":"02:57.390 ","End":"03:05.200","Text":"In our case, the radius is equal to 1 in our case."},{"Start":"03:06.110 ","End":"03:14.850","Text":"This is just equal to twice 4/3 Pi times 1 cubed."},{"Start":"03:14.850 ","End":"03:16.980","Text":"This is equal to what?"},{"Start":"03:16.980 ","End":"03:21.550","Text":"8 over 3 Pi."},{"Start":"03:21.740 ","End":"03:26.630","Text":"Now we\u0027ve computed the left-hand side."},{"Start":"03:26.630 ","End":"03:31.309","Text":"Now if you go to the chapter on surface integrals,"},{"Start":"03:31.309 ","End":"03:35.790","Text":"this exercise was actually solved there."},{"Start":"03:35.790 ","End":"03:39.980","Text":"The answer did indeed come out to be 8/3 Pi."},{"Start":"03:39.980 ","End":"03:44.435","Text":"But in case you can\u0027t find it or you don\u0027t have it."},{"Start":"03:44.435 ","End":"03:49.975","Text":"I\u0027m going to repeat it in the very next clip."},{"Start":"03:49.975 ","End":"03:52.560","Text":"There you will see all the calculation."},{"Start":"03:52.560 ","End":"03:57.515","Text":"This was extremely lengthy as composed to how short this was."},{"Start":"03:57.515 ","End":"03:59.615","Text":"But we did get the same answer."},{"Start":"03:59.615 ","End":"04:02.580","Text":"Okay. I\u0027m done here."}],"ID":8834},{"Watched":false,"Name":"Exercise 2 – Verified second direction","Duration":"24m 4s","ChapterTopicVideoID":8766,"CourseChapterTopicPlaylistID":4966,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.530","Text":"In this exercise, we have a type 2 surface integral."},{"Start":"00:05.540 ","End":"00:09.180","Text":"On a unit sphere."},{"Start":"00:09.180 ","End":"00:10.770","Text":"S is the unit sphere."},{"Start":"00:10.770 ","End":"00:14.270","Text":"I\u0027ve tried to illustrate it here."},{"Start":"00:14.270 ","End":"00:22.630","Text":"F is a vector field given in the i, j, k form."},{"Start":"00:26.150 ","End":"00:30.470","Text":"N is the outward unit normal."},{"Start":"00:30.470 ","End":"00:41.130","Text":"At any given point have a normal vector which is 1 in length and faces outwards."},{"Start":"00:41.170 ","End":"00:45.530","Text":"Here, for example, is the normal vector n."},{"Start":"00:45.530 ","End":"00:50.855","Text":"If I\u0027m here, 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":"00:58.750","Text":"the upper hemisphere and the lower hemisphere."},{"Start":"00:58.750 ","End":"01:01.320","Text":"If I want to call this,"},{"Start":"01:01.320 ","End":"01:07.360","Text":"say S_1 and the lower hemisphere S_2."},{"Start":"01:09.950 ","End":"01:17.425","Text":"The main reason is that then I can write z as a function of x and y."},{"Start":"01:17.425 ","End":"01:21.595","Text":"They\u0027ll have all kinds of theorems and formulas for that."},{"Start":"01:21.595 ","End":"01:28.140","Text":"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":"01:46.145","Text":"For S_1 we take the positive square root,"},{"Start":"01:46.145 ","End":"01:52.655","Text":"and for S_2, we would take the negative square root."},{"Start":"01:52.655 ","End":"01:59.225","Text":"That would be minus square root of 1 minus x squared minus y squared."},{"Start":"01:59.225 ","End":"02:03.080","Text":"Then the idea is to break the integral up over S."},{"Start":"02:03.080 ","End":"02:14.535","Text":"To say that it\u0027s the integral over S_1 plus the integral over S_2 and to do it in 2 bits."},{"Start":"02:14.535 ","End":"02:16.995","Text":"Let\u0027s begin with S_1."},{"Start":"02:16.995 ","End":"02:23.090","Text":"I can write this as z equals g of x, y,"},{"Start":"02:23.090 ","End":"02:26.445","Text":"where g of x, y is this here."},{"Start":"02:26.445 ","End":"02:29.600","Text":"When 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":"02:44.460","Text":"Let me write F as x, minus 2y, 3z."},{"Start":"02:44.460 ","End":"02:46.695","Text":"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":"02:55.820 ","End":"03:01.160","Text":"in this case it would be S_1, but in general,"},{"Start":"03:01.160 ","End":"03:16.040","Text":"the double integral over a surface of F.n dS is equal"},{"Start":"03:16.040 ","End":"03:21.050","Text":"to the double integral over R."},{"Start":"03:21.050 ","End":"03:24.590","Text":"I\u0027ll take a break to show you what I mean by R."},{"Start":"03:24.590 ","End":"03:28.820","Text":"R is the projection of the surface onto the xy plane,"},{"Start":"03:28.820 ","End":"03:31.800","Text":"or if you like, the domain."},{"Start":"03:31.870 ","End":"03:41.240","Text":"Here, the domain would be where x squared plus y squared is less than or equal to 1."},{"Start":"03:41.240 ","End":"03:44.270","Text":"You could see this if this is the unit sphere,"},{"Start":"03:44.270 ","End":"03:51.540","Text":"where this is 1, 1,"},{"Start":"03:51.540 ","End":"03:54.915","Text":"where x is 1, this is 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":"04:06.930","Text":"Here it is."},{"Start":"04:06.930 ","End":"04:08.720","Text":"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":"04:18.950","Text":"It\u0027s equal to 1 on the circumference of the circle."},{"Start":"04:18.950 ","End":"04:21.495","Text":"Back here."},{"Start":"04:21.495 ","End":"04:24.890","Text":"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":"04:48.545","Text":"partial derivative with respect to x comma minus partial derivative with respect to y_1."},{"Start":"04:48.545 ","End":"04:51.440","Text":"All this, dA."},{"Start":"04:51.440 ","End":"04:56.330","Text":"This is not quite precise as a condition"},{"Start":"04:56.330 ","End":"05:00.740","Text":"that is provided that the normal vector n."},{"Start":"05:00.740 ","End":"05:03.815","Text":"It has an upward component."},{"Start":"05:03.815 ","End":"05:06.600","Text":"It doesn\u0027t have to be exactly upward."},{"Start":"05:06.600 ","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":"05:18.135","Text":"has to be partially upward."},{"Start":"05:18.135 ","End":"05:20.265","Text":"That would work for S_1,"},{"Start":"05:20.265 ","End":"05:22.510","Text":"but it would not work for S_2."},{"Start":"05:22.510 ","End":"05:25.910","Text":"If the normal has a downward component,"},{"Start":"05:25.910 ","End":"05:29.590","Text":"then we have to replace all the signs here."},{"Start":"05:29.590 ","End":"05:31.580","Text":"This would be a plus,"},{"Start":"05:31.580 ","End":"05:32.870","Text":"this would be a plus."},{"Start":"05:32.870 ","End":"05:35.645","Text":"This would be a minus 1."},{"Start":"05:35.645 ","End":"05:38.060","Text":"In case we had a down with normal,"},{"Start":"05:38.060 ","End":"05:42.670","Text":"but for S_1, we\u0027re okay as it is here."},{"Start":"05:42.670 ","End":"05:46.880","Text":"When we get to the lower hemisphere and we take this n,"},{"Start":"05:46.880 ","End":"05:51.065","Text":"then we\u0027ll use the reverse signs."},{"Start":"05:51.065 ","End":"05:54.590","Text":"Now, here\u0027s the formula for g."},{"Start":"05:54.590 ","End":"05:59.855","Text":"Let\u0027s compute what are the partial derivatives."},{"Start":"05:59.855 ","End":"06:04.835","Text":"G with respect to x is equal to,"},{"Start":"06:04.835 ","End":"06:07.130","Text":"we have a square root."},{"Start":"06:07.130 ","End":"06:09.410","Text":"It\u0027s first of all,"},{"Start":"06:09.410 ","End":"06:13.970","Text":"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 2s cancel."},{"Start":"06:25.245 ","End":"06:27.270","Text":"Now for g_y."},{"Start":"06:27.270 ","End":"06:33.410","Text":"Well, same thing, the 2\u0027s going to cancel,"},{"Start":"06:33.410 ","End":"06:35.000","Text":"so I\u0027m not even going to bother writing it,"},{"Start":"06:35.000 ","End":"06:38.029","Text":"on the denominator, we\u0027re going to have the same thing."},{"Start":"06:38.029 ","End":"06:44.210","Text":"The only difference is that in the numerator we have a y instead of an x."},{"Start":"06:44.210 ","End":"06:46.640","Text":"That\u0027s this here."},{"Start":"06:46.640 ","End":"06:49.130","Text":"Now that we have these 2,"},{"Start":"06:49.130 ","End":"06:53.484","Text":"we can do the dot product."},{"Start":"06:53.484 ","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":"07:06.900","Text":"But I\u0027m going to use the g_x from here and here."},{"Start":"07:06.900 ","End":"07:09.450","Text":"Let\u0027s see what we get."},{"Start":"07:09.450 ","End":"07:15.260","Text":"I\u0027m continuing from here equals and I come out here."},{"Start":"07:15.260 ","End":"07:23.795","Text":"What I get is the double integral over R, this unit disk."},{"Start":"07:23.795 ","End":"07:26.300","Text":"This of course, will be the same when we get to S_2,"},{"Start":"07:26.300 ","End":"07:30.770","Text":"I\u0027m just mentioning it, let\u0027s see now,"},{"Start":"07:30.770 ","End":"07:33.350","Text":"we take the x component with the x component,"},{"Start":"07:33.350 ","End":"07:39.510","Text":"I need minus x, g_x, and g_x is this."},{"Start":"07:40.370 ","End":"07:44.090","Text":"We get the minus and the minus is the plus."},{"Start":"07:44.090 ","End":"07:50.720","Text":"We get x minus x times this will be x squared"},{"Start":"07:50.720 ","End":"07:58.860","Text":"over the square root of 1 minus x squared minus y squared,"},{"Start":"07:58.860 ","End":"08:04.150","Text":"and then I need minus 2y with minus g_y."},{"Start":"08:07.350 ","End":"08:10.885","Text":"There\u0027s 3 minuses; there\u0027s a minus here,"},{"Start":"08:10.885 ","End":"08:12.310","Text":"there\u0027s a minus here,"},{"Start":"08:12.310 ","End":"08:14.080","Text":"and there\u0027s a minus in the g_y."},{"Start":"08:14.080 ","End":"08:17.230","Text":"So minus, minus, minus will be minus,"},{"Start":"08:17.230 ","End":"08:21.880","Text":"and then it\u0027ll be 2y times y is 2y squared"},{"Start":"08:21.880 ","End":"08:28.165","Text":"over the square root of 1 minus x squared minus y squared."},{"Start":"08:28.165 ","End":"08:36.940","Text":"Lastly, with this, is just 3z so that will be plus 3z,"},{"Start":"08:36.940 ","End":"08:42.715","Text":"and I\u0027ll 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":"08:55.570","Text":"I write it as a function of x and y, and here it is."},{"Start":"08:55.570 ","End":"09:02.530","Text":"It\u0027s the square root of 1 minus x squared minus y squared."},{"Start":"09:02.530 ","End":"09:09.700","Text":"This is dA, 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":"09:19.210","Text":"First of all, the region R has a circular symmetry."},{"Start":"09:19.210 ","End":"09:23.890","Text":"It\u0027s a disk, also there are 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":"09:35.695","Text":"This is quite a mess really,"},{"Start":"09:35.695 ","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":"10:07.210","Text":"I could multiply by the square root"},{"Start":"10:07.210 ","End":"10:13.255","Text":"and get 3 times 1 minus x squared minus y squared over the square root."},{"Start":"10:13.255 ","End":"10:14.710","Text":"It\u0027s the same thing,"},{"Start":"10:14.710 ","End":"10:17.155","Text":"1 minus x squared minus y squared."},{"Start":"10:17.155 ","End":"10:20.840","Text":"Let\u0027s see what this comes out to."},{"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":"10:34.629","Text":"and I have minus 3x squared,"},{"Start":"10:34.629 ","End":"10:38.570","Text":"so that\u0027s minus 2x squared."},{"Start":"10:39.390 ","End":"10:42.175","Text":"As for y squared,"},{"Start":"10:42.175 ","End":"10:45.545","Text":"I have minus 2y squared,"},{"Start":"10:45.545 ","End":"10:50.370","Text":"and minus 3y squared,"},{"Start":"10:50.370 ","End":"10:56.865","Text":"so 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 one 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":"11:35.455","Text":"I have to put another minus 3y squared and then I\u0027ll be all right."},{"Start":"11:35.455 ","End":"11:46.450","Text":"It\u0027s still 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":"12:09.790","Text":"and we substitute for dA equals rdrd Theta."},{"Start":"12:09.790 ","End":"12:13.690","Text":"Then there\u0027s that extra equation that\u0027s very useful most of the times"},{"Start":"12:13.690 ","End":"12:17.320","Text":"is x squared plus y squared equals r squared."},{"Start":"12:17.320 ","End":"12:20.260","Text":"It\u0027s 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":"12:30.625","Text":"Remember we have R and Theta,"},{"Start":"12:30.625 ","End":"12:36.040","Text":"and Theta goes the whole way so Theta goes"},{"Start":"12:36.040 ","End":"12:45.110","Text":"from 0 all the way around to 2 Pi and R goes from 0-1."},{"Start":"12:46.320 ","End":"12:49.045","Text":"When I convert to polar,"},{"Start":"12:49.045 ","End":"13:00.430","Text":"I get Theta from 0-2 Pi, r from 0-1."},{"Start":"13:00.430 ","End":"13:03.925","Text":"What\u0027s written here, that goes up there."},{"Start":"13:03.925 ","End":"13:11.260","Text":"We have 3 minus 2r 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":"13:24.175","Text":"Now y squared from here is r squared sine squared Theta."},{"Start":"13:24.175 ","End":"13:29.335","Text":"So 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":"13:44.590","Text":"Again I\u0027m using the x squared plus y squared equals r squared."},{"Start":"13:44.590 ","End":"13:53.125","Text":"Then I need the dA to be rdrd Theta."},{"Start":"13:53.125 ","End":"13:55.510","Text":"I\u0027d like to do a slight rewrite."},{"Start":"13:55.510 ","End":"14:02.830","Text":"If I can take the minus out of 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":"14:14.170","Text":"and splitting it up to something minus something based on this minus."},{"Start":"14:14.170 ","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":"14:24.025","Text":"there\u0027s an r here,"},{"Start":"14:24.025 ","End":"14:31.735","Text":"so I get the double integral 0-2 Pi for Theta,"},{"Start":"14:31.735 ","End":"14:47.725","Text":"0-1 for r of r over the square root of 1 minus r squared drd Theta."},{"Start":"14:47.725 ","End":"14:49.600","Text":"That\u0027s the first bit."},{"Start":"14:49.600 ","End":"14:53.335","Text":"Then minus from this minus here."},{"Start":"14:53.335 ","End":"14:56.140","Text":"Now notice that I have here r squared,"},{"Start":"14:56.140 ","End":"14:57.220","Text":"and here r squared,"},{"Start":"14:57.220 ","End":"14:59.995","Text":"I should have really have 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, 0-2 Pi, 0-1,"},{"Start":"15:16.465 ","End":"15:27.290","Text":"and first of all, take the 2 plus 3 sine squared Theta,"},{"Start":"15:32.430 ","End":"15:36.400","Text":"and then the stuff with the r like I said,"},{"Start":"15:36.400 ","End":"15:53.170","Text":"it\u0027s r cubed over 1 minus r squared square root and then drd Theta."},{"Start":"15:53.170 ","End":"15:55.450","Text":"For this integral actually,"},{"Start":"15:55.450 ","End":"15:59.365","Text":"I could take the part with just Theta."},{"Start":"15:59.365 ","End":"16:04.615","Text":"I can imagine I\u0027ve put this here in front of the integral sign."},{"Start":"16:04.615 ","End":"16:09.970","Text":"So all I have to do for the first integral is the stuff with the r."},{"Start":"16:09.970 ","End":"16:17.150","Text":"What I\u0027m going to get is 2 side exercises."},{"Start":"16:17.190 ","End":"16:22.660","Text":"In both cases, both bits I do the dr integral first."},{"Start":"16:22.660 ","End":"16:28.180","Text":"So I need to know what is the integral from 0-1 of"},{"Start":"16:28.180 ","End":"16:33.580","Text":"r over the square root of 1 minus r squared dr."},{"Start":"16:33.580 ","End":"16:37.210","Text":"That\u0027s one exercise that will help me here."},{"Start":"16:37.210 ","End":"16:44.875","Text":"The other bit will be the integral from 0-1 of r cubed"},{"Start":"16:44.875 ","End":"16:49.450","Text":"over square root of 1 minus r squared dr."},{"Start":"16:49.450 ","End":"16:52.960","Text":"Now 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 second one,"},{"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":"17:21.655","Text":"1/3 square root of 1 minus r squared cubed 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":"17:40.600","Text":"The same substitution works in the first integral"},{"Start":"17:40.600 ","End":"17:47.020","Text":"and in this case, we 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":"18:00.290","Text":"When r is 1, 1 minus r squared is 0 so this whole thing comes out to be 0."},{"Start":"18:01.980 ","End":"18:05.725","Text":"If we let r equals 0,"},{"Start":"18:05.725 ","End":"18:07.870","Text":"then we had 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":"18:50.650","Text":"Now I\u0027ve got the main pieces that I need"},{"Start":"18:50.650 ","End":"18:53.755","Text":"and now I\u0027m going to go back here."},{"Start":"18:53.755 ","End":"18:56.990","Text":"I\u0027ll just scroll down a bit."},{"Start":"18:58.830 ","End":"19:02.440","Text":"Here I\u0027m going to continue."},{"Start":"19:02.440 ","End":"19:17.920","Text":"Over here, I\u0027ve got 3 times the integral from 0-2 Pi, d Theta"},{"Start":"19:17.920 ","End":"19:23.155","Text":"and we got this integral already from the first bit is 1,"},{"Start":"19:23.155 ","End":"19:27.535","Text":"so it\u0027s just 1 d Theta."},{"Start":"19:27.535 ","End":"19:31.480","Text":"The second bit minus,"},{"Start":"19:31.480 ","End":"19:37.360","Text":"I\u0027ve got the integral from 0-2 Pi,"},{"Start":"19:37.360 ","End":"19:40.105","Text":"I have to take this bit,"},{"Start":"19:40.105 ","End":"19:47.390","Text":"which is 2 plus 3 sine squared Theta."},{"Start":"19:47.430 ","End":"19:53.245","Text":"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":"20:30.400","Text":"3 minus 4/3 comes out to be 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":"20:42.070","Text":"Now I\u0027m going to quote a trigonometrical identity"},{"Start":"20:42.070 ","End":"20:50.740","Text":"that sine squared Theta is 1/2 of 1 minus cosine 2 Theta"},{"Start":"20:50.740 ","End":"20:56.210","Text":"and put this in here 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":"21:15.490","Text":"In other words, we get 0-2 Pi, 5/3 minus 1 is 2/3"},{"Start":"21:15.490 ","End":"21:16.373","Text":"and like we said,"},{"Start":"21:16.373 ","End":"21:23.530","Text":"we get the plus from the minus minus cosine 2 Theta d Theta"},{"Start":"21:23.530 ","End":"21:28.160","Text":"and this is equal to, on we go."},{"Start":"21:28.350 ","End":"21:32.215","Text":"2/3 gives me 2/3 Theta,"},{"Start":"21:32.215 ","End":"21:37.270","Text":"cosine 2 Theta is not quite sine 2 Theta."},{"Start":"21:37.270 ","End":"21:40.150","Text":"In the end we\u0027ll also have to divide by the 2"},{"Start":"21:40.150 ","End":"21:47.440","Text":"and this to take from 0-2 Pi."},{"Start":"21:47.440 ","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":"22:01.990","Text":"because sine of 0 is 0 and sine of 4 Pi is also 0."},{"Start":"22:01.990 ","End":"22:04.780","Text":"I 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":"22:27.010","Text":"2/3 times 2 Pi, which is 4 Pi over 3."},{"Start":"22:27.010 ","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 need to do all this work for the lower hemisphere?"},{"Start":"22:42.100 ","End":"22:43.900","Text":"I claim not."},{"Start":"22:43.900 ","End":"22:48.760","Text":"Because what happens is that"},{"Start":"22:48.760 ","End":"22:57.235","Text":"g this time is not the square root but minus the square root."},{"Start":"22:57.235 ","End":"23:02.545","Text":"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":"23:10.700","Text":"it\u0027s like putting a minus in front here."},{"Start":"23:10.710 ","End":"23:14.590","Text":"The 2 minuses cancel each other out"},{"Start":"23:14.590 ","End":"23:18.880","Text":"and I\u0027m going to get exactly the same answer in the end."},{"Start":"23:18.880 ","End":"23:21.070","Text":"If I go back here,"},{"Start":"23:21.070 ","End":"23:33.130","Text":"that was for S_1 and it\u0027s going to be the same for S_2"},{"Start":"23:33.130 ","End":"23:36.040","Text":"because of the minus minus."},{"Start":"23:36.040 ","End":"23:42.790","Text":"The final answer, 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":"23:55.280","Text":"which is 8 Pi over 3."},{"Start":"23:55.500 ","End":"24:00.880","Text":"The final answer is this, not this."},{"Start":"24:00.880 ","End":"24:04.790","Text":"We\u0027re finally done."}],"ID":8835},{"Watched":false,"Name":"Exercise 3 – Verified one direction","Duration":"18m 47s","ChapterTopicVideoID":8770,"CourseChapterTopicPlaylistID":4966,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.604","Text":"In this exercise, we have to verify the divergence theorem,"},{"Start":"00:04.604 ","End":"00:09.420","Text":"which is this for a particular example."},{"Start":"00:09.420 ","End":"00:16.530","Text":"In our case, we\u0027re given that F the vector field is as follows."},{"Start":"00:16.530 ","End":"00:24.480","Text":"R is the 3D region which is a pyramid and it\u0027s defined by 4 planes,"},{"Start":"00:24.480 ","End":"00:27.000","Text":"the 3 coordinate planes,"},{"Start":"00:27.000 ","End":"00:33.230","Text":"and this diagonal, if you want to call it the plane,"},{"Start":"00:33.230 ","End":"00:35.930","Text":"that\u0027s A, B, C in the picture."},{"Start":"00:35.930 ","End":"00:39.070","Text":"I\u0027ve labeled the points of the pyramid."},{"Start":"00:39.070 ","End":"00:41.840","Text":"S is the surface of the pyramid,"},{"Start":"00:41.840 ","End":"00:54.590","Text":"which is basically the union of 4 planes for triangles as in the picture."},{"Start":"00:54.590 ","End":"00:57.980","Text":"When we have an equality and want to verify it,"},{"Start":"00:57.980 ","End":"01:00.725","Text":"I\u0027ll start with one side first."},{"Start":"01:00.725 ","End":"01:02.675","Text":"Let\u0027s say, we start with the left side,"},{"Start":"01:02.675 ","End":"01:03.820","Text":"see what this equals,"},{"Start":"01:03.820 ","End":"01:07.555","Text":"and later we\u0027ll compute the right side and see what that equals."},{"Start":"01:07.555 ","End":"01:18.235","Text":"Now, to compute this integral over the solid R,"},{"Start":"01:18.235 ","End":"01:21.590","Text":"I have to label, I don\u0027t know where to put the R."},{"Start":"01:21.590 ","End":"01:25.050","Text":"R is just the whole pyramid, the solid pyramid."},{"Start":"01:25.570 ","End":"01:31.485","Text":"What I\u0027d like to do, though there\u0027s 2 main problems,"},{"Start":"01:31.485 ","End":"01:34.790","Text":"first of all, easy thing is to compute the divergence."},{"Start":"01:34.790 ","End":"01:37.400","Text":"Then we have to figure out how to describe this pyramid"},{"Start":"01:37.400 ","End":"01:39.680","Text":"in terms of x goes from something to something"},{"Start":"01:39.680 ","End":"01:44.230","Text":"and y goes from something to something as an iterated integral."},{"Start":"01:44.230 ","End":"01:51.515","Text":"First, I\u0027ll do the technical bit of figuring out what the divergence of F is equal to."},{"Start":"01:51.515 ","End":"01:54.235","Text":"Well, if in general F is,"},{"Start":"01:54.235 ","End":"01:55.770","Text":"let say this part,"},{"Start":"01:55.770 ","End":"01:57.435","Text":"this function is P,"},{"Start":"01:57.435 ","End":"01:59.010","Text":"this function is Q,"},{"Start":"01:59.010 ","End":"02:00.840","Text":"and this function is R,"},{"Start":"02:00.840 ","End":"02:04.210","Text":"then when we have Pi plus Qj plus Rk,"},{"Start":"02:04.210 ","End":"02:10.280","Text":"the divergence is just the derivative of P partial with respect to x"},{"Start":"02:10.280 ","End":"02:17.375","Text":"plus the derivative of Q with respect to y plus the derivative of R with respect to z."},{"Start":"02:17.375 ","End":"02:23.785","Text":"That will equal, P with respect to x gives us 2y here."},{"Start":"02:23.785 ","End":"02:29.110","Text":"Q with respect to y also gives us 2y,"},{"Start":"02:29.110 ","End":"02:32.960","Text":"and R with respect to z is just 0."},{"Start":"02:32.960 ","End":"02:35.190","Text":"There is no z here,"},{"Start":"02:35.540 ","End":"02:40.385","Text":"and so what we get is 4y."},{"Start":"02:40.385 ","End":"02:42.670","Text":"That\u0027s the divergence."},{"Start":"02:42.670 ","End":"02:44.800","Text":"Now how do I describe this region?"},{"Start":"02:44.800 ","End":"02:51.435","Text":"What I\u0027d like to do is write this as z,"},{"Start":"02:51.435 ","End":"02:53.775","Text":"as a function of x and y,"},{"Start":"02:53.775 ","End":"02:56.800","Text":"and then I\u0027ll have 2 functions of x and y."},{"Start":"02:56.800 ","End":"02:59.555","Text":"I\u0027ll have z equals 0, which is in the xy plane,"},{"Start":"02:59.555 ","End":"03:05.510","Text":"and the projection will be the triangle O, B, C,"},{"Start":"03:05.510 ","End":"03:07.665","Text":"and I\u0027ll return to that in a moment."},{"Start":"03:07.665 ","End":"03:12.610","Text":"Meanwhile let\u0027s just say that we want the triple integral"},{"Start":"03:12.610 ","End":"03:26.815","Text":"over the pyramid R of 4y, dV, and as a first step,"},{"Start":"03:26.815 ","End":"03:34.090","Text":"I\u0027m going to say that it\u0027s the double integral over this shaded part."},{"Start":"03:34.090 ","End":"03:35.740","Text":"You know what, I\u0027ll call that D."},{"Start":"03:35.740 ","End":"03:40.260","Text":"This triangle in the xy plane."},{"Start":"03:40.260 ","End":"03:43.839","Text":"Then the region is between 2 planes."},{"Start":"03:43.839 ","End":"03:45.700","Text":"I have the plane A, B, C,"},{"Start":"03:45.700 ","End":"03:47.995","Text":"which is given by this,"},{"Start":"03:47.995 ","End":"03:52.660","Text":"so this I have to write in terms of z equals,"},{"Start":"03:52.660 ","End":"03:54.820","Text":"I can just bring everything to the other side and say,"},{"Start":"03:54.820 ","End":"04:01.790","Text":"that this is z equals 6 minus 2x minus 2y,"},{"Start":"04:01.790 ","End":"04:06.089","Text":"and this is z equals 0."},{"Start":"04:06.089 ","End":"04:09.179","Text":"Let me just highlight these 2 functions."},{"Start":"04:09.179 ","End":"04:12.800","Text":"6 minus 2x minus 2y,"},{"Start":"04:12.800 ","End":"04:15.880","Text":"and z equals 0."},{"Start":"04:15.880 ","End":"04:17.410","Text":"This is the upper one,"},{"Start":"04:17.410 ","End":"04:19.390","Text":"this is the lower one,"},{"Start":"04:19.390 ","End":"04:31.490","Text":"so I can write the integral from 0-6 minus 2x minus 2y."},{"Start":"04:31.850 ","End":"04:40.500","Text":"These are the ones here of 4y dV."},{"Start":"04:40.500 ","End":"04:46.845","Text":"Well, really, this is dz on the inside."},{"Start":"04:46.845 ","End":"04:51.630","Text":"Then for d, I\u0027ll just leave it as dA,"},{"Start":"04:51.630 ","End":"04:56.430","Text":"which will be dx dy or dy dx, we\u0027ll decide in a moment."},{"Start":"04:56.430 ","End":"05:01.120","Text":"Now, I have to figure out what is this triangle D."},{"Start":"05:01.120 ","End":"05:04.630","Text":"I made another sketch."},{"Start":"05:04.630 ","End":"05:09.730","Text":"It might help of just the region D in the xy plane."},{"Start":"05:09.730 ","End":"05:13.570","Text":"I think I\u0027ll take it as a type 1 region,"},{"Start":"05:13.570 ","End":"05:15.975","Text":"meaning we\u0027ll take what x goes from and to."},{"Start":"05:15.975 ","End":"05:19.530","Text":"Then we\u0027ll pick vertical slices,"},{"Start":"05:19.530 ","End":"05:26.110","Text":"so that if I have a given x, this is 0,"},{"Start":"05:26.110 ","End":"05:31.015","Text":"but what is this and what is the point C?"},{"Start":"05:31.015 ","End":"05:34.930","Text":"I can easily compute the equation of this line,"},{"Start":"05:34.930 ","End":"05:38.440","Text":"which is actually the same as this line here,"},{"Start":"05:38.440 ","End":"05:45.385","Text":"by letting z equals 0 in the equation of this diagonal plane."},{"Start":"05:45.385 ","End":"05:47.575","Text":"If I let z equals 0 here,"},{"Start":"05:47.575 ","End":"05:53.920","Text":"I get 0 equals 6 minus 2x minus 2y."},{"Start":"05:53.920 ","End":"05:56.050","Text":"Bring the 2y over,"},{"Start":"05:56.050 ","End":"05:58.960","Text":"it\u0027s equal to 6 minus 2x."},{"Start":"05:58.960 ","End":"06:00.475","Text":"I could divide by 2,"},{"Start":"06:00.475 ","End":"06:04.490","Text":"y equals 3 minus x."},{"Start":"06:04.490 ","End":"06:07.735","Text":"If this is y equals 3 minus x,"},{"Start":"06:07.735 ","End":"06:12.890","Text":"if I let y equals 0,"},{"Start":"06:12.890 ","End":"06:16.910","Text":"that will give me 3 minus x is 0,"},{"Start":"06:16.910 ","End":"06:19.445","Text":"will give me that x equals 3."},{"Start":"06:19.445 ","End":"06:20.960","Text":"Now I have everything."},{"Start":"06:20.960 ","End":"06:25.080","Text":"This is 0 and this is 3,"},{"Start":"06:25.700 ","End":"06:29.730","Text":"and I\u0027ve got this equation."},{"Start":"06:29.730 ","End":"06:32.565","Text":"Like I said, here it is,"},{"Start":"06:32.565 ","End":"06:36.150","Text":"y equals 3 minus x."},{"Start":"06:36.150 ","End":"06:38.370","Text":"This turns out to also be 3,"},{"Start":"06:38.370 ","End":"06:39.885","Text":"although I don\u0027t need that."},{"Start":"06:39.885 ","End":"06:45.945","Text":"Now I know that when x goes from 0-3,"},{"Start":"06:45.945 ","End":"06:48.690","Text":"y goes from 0-3 minus x,"},{"Start":"06:48.690 ","End":"06:53.255","Text":"so I can now write this as a fully iterated integral."},{"Start":"06:53.255 ","End":"06:57.860","Text":"We say x goes from 0-3,"},{"Start":"06:57.860 ","End":"07:04.140","Text":"y goes from 0-3 minus x,"},{"Start":"07:06.170 ","End":"07:13.260","Text":"and z goes from what we said here,"},{"Start":"07:13.260 ","End":"07:18.630","Text":"0-6 minus 2x minus 2y."},{"Start":"07:18.630 ","End":"07:21.975","Text":"All this, 4y dz,"},{"Start":"07:21.975 ","End":"07:28.200","Text":"and now we can write dA as dy dx."},{"Start":"07:28.200 ","End":"07:32.110","Text":"This is now purely technical."},{"Start":"07:33.650 ","End":"07:36.120","Text":"The first integral is dz,"},{"Start":"07:36.120 ","End":"07:38.550","Text":"but 4y doesn\u0027t contain z."},{"Start":"07:38.550 ","End":"07:41.555","Text":"I like to pull things out in front that are not needed."},{"Start":"07:41.555 ","End":"07:43.775","Text":"I\u0027ll put this in front here."},{"Start":"07:43.775 ","End":"07:46.615","Text":"Now I have just the integral of dz."},{"Start":"07:46.615 ","End":"07:53.235","Text":"What we get is the integral from 0-3,"},{"Start":"07:53.235 ","End":"08:03.250","Text":"the integral from 0-3 minus x of 4y."},{"Start":"08:03.800 ","End":"08:10.425","Text":"Now this integral is just going to be integral of 1."},{"Start":"08:10.425 ","End":"08:14.710","Text":"I pull it in front, it\u0027s like I left a 1 there of 1 dz."},{"Start":"08:16.550 ","End":"08:23.670","Text":"I meant to emphasize that I\u0027m doing the dz integral first."},{"Start":"08:23.670 ","End":"08:26.340","Text":"Only it\u0027s not 4y it\u0027s 1,"},{"Start":"08:26.340 ","End":"08:31.340","Text":"and the integral of 1 is always the upper limit minus the lower limit,"},{"Start":"08:31.340 ","End":"08:37.770","Text":"so it\u0027s just 6 minus 2x minus 2y minus 0,"},{"Start":"08:37.770 ","End":"08:39.135","Text":"which I don\u0027t need,"},{"Start":"08:39.135 ","End":"08:45.195","Text":"and this is going to be now dy dx."},{"Start":"08:45.195 ","End":"08:47.250","Text":"There\u0027s no more z."},{"Start":"08:47.250 ","End":"08:53.920","Text":"Now the inner integral is the dy integral."},{"Start":"08:55.250 ","End":"09:00.105","Text":"I\u0027d like to do this one as a side calculation over here."},{"Start":"09:00.105 ","End":"09:06.645","Text":"What I have is the integral from 0-3 minus x."},{"Start":"09:06.645 ","End":"09:09.495","Text":"Let me open the brackets."},{"Start":"09:09.495 ","End":"09:17.055","Text":"I\u0027ve got 24,"},{"Start":"09:17.055 ","End":"09:25.350","Text":"4 times 6, y minus 4 times 2 is 8xy,"},{"Start":"09:25.350 ","End":"09:33.890","Text":"and minus 8y squared dy."},{"Start":"09:33.890 ","End":"09:43.500","Text":"Let\u0027s see, y squared over 2 times 24 so this is 12y squared,"},{"Start":"09:43.500 ","End":"09:46.635","Text":"y gives me y squared over 2,"},{"Start":"09:46.635 ","End":"09:52.005","Text":"so I\u0027ve got minus 4xy squared."},{"Start":"09:52.005 ","End":"09:53.955","Text":"From here, y cubed over 3,"},{"Start":"09:53.955 ","End":"09:58.005","Text":"so minus 8/3 y cubed."},{"Start":"09:58.005 ","End":"10:02.965","Text":"All this from 0-3 minus x."},{"Start":"10:02.965 ","End":"10:10.955","Text":"This gives me 12 times 3 minus x squared."},{"Start":"10:10.955 ","End":"10:18.800","Text":"Then minus 4x,"},{"Start":"10:18.800 ","End":"10:22.280","Text":"3 minus x squared,"},{"Start":"10:22.280 ","End":"10:30.580","Text":"minus 8/3, 3 minus x cubed."},{"Start":"10:30.580 ","End":"10:34.650","Text":"Now, I\u0027m going to put this back in here,"},{"Start":"10:34.650 ","End":"10:49.275","Text":"and so we get the integral from 0-3 of 12 times 3 minus x squared,"},{"Start":"10:49.275 ","End":"10:51.270","Text":"you know what, I want to change the order a bit."},{"Start":"10:51.270 ","End":"10:54.900","Text":"I want to take the bits with only 3 minus x in them first,"},{"Start":"10:54.900 ","End":"10:59.310","Text":"minus 8/3, 3 minus x cubed."},{"Start":"10:59.310 ","End":"11:01.230","Text":"Maybe I\u0027ll substitute 3 minus x,"},{"Start":"11:01.230 ","End":"11:02.655","Text":"maybe I\u0027ll do another trick."},{"Start":"11:02.655 ","End":"11:05.280","Text":"In fact, I\u0027m going to split it up into 2 integrals,"},{"Start":"11:05.280 ","End":"11:08.490","Text":"dx, that\u0027s this bit and this bit,"},{"Start":"11:08.490 ","End":"11:18.100","Text":"and then I\u0027ll take away the integral from 0-3 of this bit dx,"},{"Start":"11:18.100 ","End":"11:29.620","Text":"so 4x, 3 minus x squared dx."},{"Start":"11:30.890 ","End":"11:34.380","Text":"Now, I need more space and it\u0027s all technical,"},{"Start":"11:34.380 ","End":"11:36.120","Text":"I don\u0027t need the pictures."},{"Start":"11:36.120 ","End":"11:38.880","Text":"I don\u0027t care if I scroll out."},{"Start":"11:38.880 ","End":"11:42.585","Text":"Let\u0027s do the first one."},{"Start":"11:42.585 ","End":"11:49.590","Text":"Let me call this asterisk and this one double asterisk."},{"Start":"11:49.590 ","End":"11:51.900","Text":"First of all, the asterisk."},{"Start":"11:51.900 ","End":"11:57.130","Text":"That is equal to the integral."},{"Start":"11:57.770 ","End":"12:01.725","Text":"Well, let me do the integral already."},{"Start":"12:01.725 ","End":"12:07.229","Text":"Now, I look at 3 minus x as if it was x,"},{"Start":"12:07.229 ","End":"12:08.610","Text":"but it\u0027s not quite."},{"Start":"12:08.610 ","End":"12:12.210","Text":"If it was x, I would get x cubed over 3."},{"Start":"12:12.210 ","End":"12:15.180","Text":"I would get 4,"},{"Start":"12:15.180 ","End":"12:21.910","Text":"and then I would get 3 minus x cubed."},{"Start":"12:22.220 ","End":"12:28.050","Text":"Again, I raise the power by 1 is 3 divided by that but it isn\u0027t x,"},{"Start":"12:28.050 ","End":"12:31.350","Text":"it\u0027s 3 minus x is an inner derivative of minus 1."},{"Start":"12:31.350 ","End":"12:33.945","Text":"You have to divide by minus 1."},{"Start":"12:33.945 ","End":"12:37.305","Text":"I\u0027m just going to put a minus in front."},{"Start":"12:37.305 ","End":"12:41.100","Text":"That will take care of the fact that it was 3 minus x."},{"Start":"12:41.100 ","End":"12:46.304","Text":"Similarly here, I\u0027m going to start off with minus,"},{"Start":"12:46.304 ","End":"12:49.440","Text":"now 3 minus x^4,"},{"Start":"12:49.440 ","End":"12:51.495","Text":"and I\u0027m divide by the 4,"},{"Start":"12:51.495 ","End":"12:56.265","Text":"so I\u0027ll only get 2/3."},{"Start":"12:56.265 ","End":"12:58.410","Text":"Again there\u0027s a master of the minus"},{"Start":"12:58.410 ","End":"13:02.430","Text":"so this time minus will make this into a plus."},{"Start":"13:02.430 ","End":"13:10.840","Text":"This, I\u0027ll take between 0 and 3."},{"Start":"13:12.350 ","End":"13:18.750","Text":"If I plug in x equals 3,"},{"Start":"13:18.750 ","End":"13:22.815","Text":"everything becomes 0 because 3 minus x is 0."},{"Start":"13:22.815 ","End":"13:25.305","Text":"All I\u0027m left with is the 0."},{"Start":"13:25.305 ","End":"13:28.090","Text":"When I put in x equals 0,"},{"Start":"13:28.090 ","End":"13:31.745","Text":"I get minus 4,"},{"Start":"13:31.745 ","End":"13:34.100","Text":"3 minus 0 cubed,"},{"Start":"13:34.100 ","End":"13:44.600","Text":"which is just 3 cubed, plus 2/3, 3^4,"},{"Start":"13:44.600 ","End":"13:48.689","Text":"and this becomes, let\u0027s see,"},{"Start":"13:49.630 ","End":"13:59.170","Text":"minus 4 times 3 cubed is minus 4 times 27 is minus 108."},{"Start":"13:59.570 ","End":"14:05.355","Text":"Here, 3 goes into 3^4, 3 cubed times,"},{"Start":"14:05.355 ","End":"14:10.785","Text":"so it\u0027s 27 times 2 is 54,"},{"Start":"14:10.785 ","End":"14:14.565","Text":"so altogether I have minus 54."},{"Start":"14:14.565 ","End":"14:19.155","Text":"That\u0027s from the asterisk and now the double asterisk,"},{"Start":"14:19.155 ","End":"14:22.005","Text":"which is the second part,"},{"Start":"14:22.005 ","End":"14:30.840","Text":"is the integral from 0-3."},{"Start":"14:30.840 ","End":"14:34.605","Text":"I\u0027m going to do this computation as a side exercise,"},{"Start":"14:34.605 ","End":"14:40.695","Text":"what this equals and I\u0027ll do that over here."},{"Start":"14:40.695 ","End":"14:48.150","Text":"I\u0027ve got 4x, now 3 minus x squared using special binomial expansion"},{"Start":"14:48.150 ","End":"14:55.920","Text":"is 3 squared is 9 minus twice 3 times x plus x squared,"},{"Start":"14:55.920 ","End":"15:10.300","Text":"and that is equal to 36x minus 24x squared plus 4x cubed."},{"Start":"15:12.020 ","End":"15:18.210","Text":"Well, I\u0027ll just write that here again, I copy paste"},{"Start":"15:18.210 ","End":"15:26.175","Text":"and that\u0027s dx and that is equal to, let\u0027s see,"},{"Start":"15:26.175 ","End":"15:28.320","Text":"x squared over 2,"},{"Start":"15:28.320 ","End":"15:33.630","Text":"so I\u0027ve got 18x squared and then x cubed over 3,"},{"Start":"15:33.630 ","End":"15:37.275","Text":"so minus 8x cubed,"},{"Start":"15:37.275 ","End":"15:38.730","Text":"and then x^4 over 4,"},{"Start":"15:38.730 ","End":"15:45.149","Text":"it\u0027s just x^4 between 0 and 3"},{"Start":"15:45.149 ","End":"15:53.324","Text":"and this is equal to when x is 0, I get nothing."},{"Start":"15:53.324 ","End":"16:02.265","Text":"When x is 3, I\u0027ve got 18 times 3 squared"},{"Start":"16:02.265 ","End":"16:14.745","Text":"minus 8 times 3 cubed plus 3^4 and that gives me,"},{"Start":"16:14.745 ","End":"16:25.540","Text":"let\u0027s see, 18 times 9, which is 162,"},{"Start":"16:25.760 ","End":"16:42.030","Text":"8 times 27 is 216, plus 81, and I make that 27."},{"Start":"16:42.030 ","End":"16:46.140","Text":"At this point, I realize I made a small mistake."},{"Start":"16:46.140 ","End":"16:51.030","Text":"When I substituted here the 0,"},{"Start":"16:51.030 ","End":"16:54.900","Text":"I should have subtracted the 0."},{"Start":"16:54.900 ","End":"16:59.805","Text":"All this has to be a minus,"},{"Start":"16:59.805 ","End":"17:03.390","Text":"same thing here this should be a minus"},{"Start":"17:03.390 ","End":"17:08.910","Text":"and so the answer is plus 54 in the case of the asterisk."},{"Start":"17:08.910 ","End":"17:11.070","Text":"Sorry about that."},{"Start":"17:11.070 ","End":"17:13.440","Text":"Now, we just have to collect together."},{"Start":"17:13.440 ","End":"17:14.910","Text":"This was the asterisk,"},{"Start":"17:14.910 ","End":"17:16.544","Text":"this was a double asterisk,"},{"Start":"17:16.544 ","End":"17:19.755","Text":"we computed the asterisk, came out 54."},{"Start":"17:19.755 ","End":"17:22.785","Text":"Double asterisk came out 27."},{"Start":"17:22.785 ","End":"17:34.905","Text":"We want to subtract asterisk minus double asterisk is equal to 54 minus 27,"},{"Start":"17:34.905 ","End":"17:40.605","Text":"which equals 27 and that is the answer."},{"Start":"17:40.605 ","End":"17:43.050","Text":"At least that\u0027s half the answer."},{"Start":"17:43.050 ","End":"17:44.760","Text":"That is the answer."},{"Start":"17:44.760 ","End":"17:50.295","Text":"I\u0027m going to scroll back up to this part here."},{"Start":"17:50.295 ","End":"17:54.195","Text":"The left-hand side is 27."},{"Start":"17:54.195 ","End":"17:57.470","Text":"Now, we need to compute the right-hand side,"},{"Start":"17:57.470 ","End":"18:01.475","Text":"the surface integral but"},{"Start":"18:01.475 ","End":"18:08.165","Text":"this was already computed in the chapter on surface integrals."},{"Start":"18:08.165 ","End":"18:11.630","Text":"You can go and check one of the exercises in the chapter there,"},{"Start":"18:11.630 ","End":"18:14.215","Text":"had exactly the same thing."},{"Start":"18:14.215 ","End":"18:20.330","Text":"The answer did indeed come out to 27 also"},{"Start":"18:20.330 ","End":"18:25.355","Text":"but in case you don\u0027t have it for whatever reason or you can\u0027t find it,"},{"Start":"18:25.355 ","End":"18:31.205","Text":"then I\u0027m going to do this in the very next clip."},{"Start":"18:31.205 ","End":"18:34.130","Text":"Take a look at the next clip"},{"Start":"18:34.130 ","End":"18:38.170","Text":"and there\u0027s where I do the calculation for this."},{"Start":"18:38.170 ","End":"18:41.420","Text":"Assuming that\u0027s taken care of,"},{"Start":"18:41.420 ","End":"18:42.590","Text":"I am now done"},{"Start":"18:42.590 ","End":"18:47.069","Text":"and we have verified the divergence theorem in this example."}],"ID":8836},{"Watched":false,"Name":"Exercise 3 – Verified second direction","Duration":"21m 44s","ChapterTopicVideoID":8768,"CourseChapterTopicPlaylistID":4966,"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":"00:16.875","Text":"We\u0027re told that S is the surface of the pyramid given by 4 planes,"},{"Start":"00:16.875 ","End":"00:22.460","Text":"3 of the planes are simply the coordinate planes."},{"Start":"00:22.460 ","End":"00:26.745","Text":"When x is 0, it\u0027s the yz plane and so on."},{"Start":"00:26.745 ","End":"00:29.120","Text":"This 1 is the more interesting 1."},{"Start":"00:29.120 ","End":"00:36.550","Text":"That\u0027s this skew plane and it cuts all 3 of the coordinate axis"},{"Start":"00:36.550 ","End":"00:39.010","Text":"at points A, B, and C."},{"Start":"00:39.010 ","End":"00:41.245","Text":"This is the origin."},{"Start":"00:41.245 ","End":"00:46.230","Text":"I just want to compute the coordinates of A, B, and C."},{"Start":"00:46.230 ","End":"00:52.805","Text":"When I let x and y equals 0,"},{"Start":"00:52.805 ","End":"00:55.100","Text":"then I just get z equals 6."},{"Start":"00:55.100 ","End":"01:01.365","Text":"I\u0027m just going to write the z comp1nt because x and y are 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 will let y and z equals 0,"},{"Start":"01:16.740 ","End":"01:21.450","Text":"2x equals 6, x equals 3."},{"Start":"01:21.450 ","End":"01:25.070","Text":"I have these points this of course is a point 0, 0, 0"},{"Start":"01:25.070 ","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":"01:42.380 ","End":"01:45.200","Text":"I\u0027ll tell you which order I\u0027ll do them in."},{"Start":"01:45.200 ","End":"01:47.870","Text":"Let me, first of all, do the difficult 1,"},{"Start":"01:47.870 ","End":"01:49.550","Text":"which is the skew 1."},{"Start":"01:49.550 ","End":"01:57.945","Text":"Which will be the ABC triangle plus we\u0027ll take the double integral over."},{"Start":"01:57.945 ","End":"01:59.520","Text":"I want This 1."},{"Start":"01:59.520 ","End":"02:07.985","Text":"The 1 in the xy plane that will be OBC and then the other 2,"},{"Start":"02:07.985 ","End":"02:15.430","Text":"I\u0027ll take the double integral of the 1 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 1 here,"},{"Start":"02:25.435 ","End":"02:28.190","Text":"which will be OAC."},{"Start":"02:28.190 ","End":"02:32.355","Text":"That\u0027s the general idea to do 4 separate calculations."},{"Start":"02:32.355 ","End":"02:35.140","Text":"As I said, I\u0027m going to start with the difficult 1,"},{"Start":"02:35.140 ","End":"02:37.730","Text":"which is the ABC."},{"Start":"02:39.650 ","End":"02:47.830","Text":"For this, I would like to rewrite this plane in the terms of z"},{"Start":"02:47.830 ","End":"02:49.660","Text":"as a function of x and y."},{"Start":"02:49.660 ","End":"02:56.989","Text":"I\u0027ll write from here 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 d1 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":"03:21.980","Text":"this triangle here onto the xy plane is this here which I\u0027ll shade."},{"Start":"03:21.980 ","End":"03:24.620","Text":"I\u0027m going to label this as R."},{"Start":"03:24.620 ","End":"03:31.610","Text":"I want to bring in a formula which works"},{"Start":"03:31.610 ","End":"03:34.999","Text":"when we have z extracted as a function of x and y"},{"Start":"03:34.999 ","End":"03:40.400","Text":"and this theorem or formula says that"},{"Start":"03:40.400 ","End":"03:47.220","Text":"we can calculate the double integral over some s."},{"Start":"03:47.220 ","End":"03:50.685","Text":"Not the same s as the pyramid just,"},{"Start":"03:50.685 ","End":"03:54.465","Text":"this s will be 1 of these planes."},{"Start":"03:54.465 ","End":"04:05.320","Text":"This is in general 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":"04:23.700","Text":"which is the projection of that surface onto the xy plane of F dot,"},{"Start":"04:23.700 ","End":"04:28.590","Text":"and here I write 1 of 2 things."},{"Start":"04:28.590 ","End":"04:31.195","Text":"Let me write 1 of them."},{"Start":"04:31.195 ","End":"04:34.385","Text":"I\u0027ll use the angular bracket form."},{"Start":"04:34.385 ","End":"04:36.155","Text":"It\u0027ll be easier for me."},{"Start":"04:36.155 ","End":"04:41.360","Text":"What we have is minus g with respect to x,"},{"Start":"04:41.360 ","End":"04:46.670","Text":"where g is the function that defines z over the region R."},{"Start":"04:46.670 ","End":"04:58.000","Text":"Partial derivative minus g with respect to y and then 1 and this is dA."},{"Start":"04:59.360 ","End":"05:02.160","Text":"It\u0027s amusing angular brackets here,"},{"Start":"05:02.160 ","End":"05:11.270","Text":"let me rewrite F as in angular bracket form as 2xy plus z in the first comp1nt."},{"Start":"05:11.270 ","End":"05:13.820","Text":"Y squared in the second comp1nt"},{"Start":"05:13.820 ","End":"05:23.040","Text":"and this is a minus x plus 3y in the third comp1nt."},{"Start":"05:23.040 ","End":"05:26.445","Text":"It\u0027ll be easier rather than i, j, k."},{"Start":"05:26.445 ","End":"05:30.845","Text":"Here, when we substitute x, y, and z,"},{"Start":"05:30.845 ","End":"05:35.785","Text":"instead of z, we have to put in g of xy, we\u0027ll see this."},{"Start":"05:35.785 ","End":"05:38.670","Text":"I was saying that these 2 cases,"},{"Start":"05:38.670 ","End":"05:41.540","Text":"it all depends on whether the normal vector"},{"Start":"05:41.540 ","End":"05:46.100","Text":"has an upward comp1nt or a downward comp1nt."},{"Start":"05:46.100 ","End":"05:47.990","Text":"In the case of this plane,"},{"Start":"05:47.990 ","End":"05:53.320","Text":"if I took a point on this plane, ABC, the normal."},{"Start":"05:53.320 ","End":"05:57.320","Text":"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 comp1nt."},{"Start":"05:59.270 ","End":"06:01.190","Text":"As we go outward,"},{"Start":"06:01.190 ","End":"06:03.815","Text":"we also go somewhat higher."},{"Start":"06:03.815 ","End":"06:07.265","Text":"This has an upward normal comp1nt."},{"Start":"06:07.265 ","End":"06:11.960","Text":"Whereas when we get to the triangle OBC,"},{"Start":"06:11.960 ","End":"06:13.970","Text":"which is this and I take a point,"},{"Start":"06:13.970 ","End":"06:18.004","Text":"its normal vector is actually strictly downwards,"},{"Start":"06:18.004 ","End":"06:19.910","Text":"but it has a downward comp1nt."},{"Start":"06:19.910 ","End":"06:28.440","Text":"This is the formula that works when the normal goes tilts upwards"},{"Start":"06:28.440 ","End":"06:33.770","Text":"and there\u0027s another formula for when the normal tilts downwards."},{"Start":"06:33.770 ","End":"06:38.910","Text":"I copy-pasted this and now let me just make a small change."},{"Start":"06:38.910 ","End":"06:45.350","Text":"For the downward case is that I just take the minus of this by reversing the signs,"},{"Start":"06:45.350 ","End":"06:50.720","Text":"make this minus into a plus, this into a plus, and this to a minus."},{"Start":"06:50.720 ","End":"06:53.410","Text":"Actually, I\u0027ll erase the pluses."},{"Start":"06:53.410 ","End":"06:59.875","Text":"For ABC, since this normal has a positive z,"},{"Start":"06:59.875 ","End":"07:05.520","Text":"a positive like a k-comp1nt will be using this formula here."},{"Start":"07:05.520 ","End":"07:10.690","Text":"Later when we get to OBC we\u0027ll be using this formula here."},{"Start":"07:11.150 ","End":"07:20.090","Text":"Let\u0027s see then we need g_x and g_y and those are easy."},{"Start":"07:20.090 ","End":"07:25.430","Text":"G with respect to x is just a constant minus 2,"},{"Start":"07:25.430 ","End":"07:31.830","Text":"and g with respect to y is also the constant minus 2."},{"Start":"07:34.070 ","End":"07:43.380","Text":"What I\u0027m saying is that the S in our case is just the ABC triangle"},{"Start":"07:43.700 ","End":"07:49.320","Text":"of F.ndS is equal by the formula."},{"Start":"07:49.320 ","End":"07:54.440","Text":"I\u0027m using the top 1 is equal to the double integral over the projection,"},{"Start":"07:54.440 ","End":"08:01.950","Text":"which is the region R of F dot"},{"Start":"08:01.950 ","End":"08:09.600","Text":"and I have this vector already g_x is minus 2."},{"Start":"08:09.600 ","End":"08:12.449","Text":"Sorry, I\u0027m using the top formula."},{"Start":"08:12.449 ","End":"08:16.140","Text":"It\u0027s going to be plus 2 because it\u0027s minus g_x."},{"Start":"08:16.140 ","End":"08:22.780","Text":"Then minus g_y will also be 2 and here we just have a 1 dA."},{"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":"08:37.250 ","End":"08:43.430","Text":"I get the double integral and I\u0027m not writing R here"},{"Start":"08:43.430 ","End":"08:46.865","Text":"because I\u0027m going to replace this by an iterated integral in a moment."},{"Start":"08:46.865 ","End":"08:48.725","Text":"Let\u0027s just do the dot product."},{"Start":"08:48.725 ","End":"08:56.045","Text":"I\u0027ve got 2 times this, 2xy plus z."},{"Start":"08:56.045 ","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":"09:05.040 ","End":"09:08.180","Text":"It wouldn\u0027t hurt to highlight this also,"},{"Start":"09:08.180 ","End":"09:10.650","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 comp1nt, this with this."},{"Start":"09:19.485 ","End":"09:24.120","Text":"Now 2 with y squared is just 2y squared."},{"Start":"09:24.120 ","End":"09:33.490","Text":"Then 1 with this gives me minus x plus 3y."},{"Start":"09:35.120 ","End":"09:40.720","Text":"All this is dA."},{"Start":"09:41.920 ","End":"09:43.430","Text":"You know what?"},{"Start":"09:43.430 ","End":"09:46.355","Text":"I\u0027ll write it first of all, like this."},{"Start":"09:46.355 ","End":"09:49.040","Text":"Next step I\u0027ll simplify this,"},{"Start":"09:49.040 ","End":"09:52.640","Text":"but I also want to change this integral over the triangle"},{"Start":"09:52.640 ","End":"09:56.460","Text":"to an iterated integral, something dx dy."},{"Start":"09:56.680 ","End":"10:01.159","Text":"I\u0027m thinking I\u0027ll bring in an extra sketch for R over here."},{"Start":"10:01.159 ","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":"10:10.470","Text":"and we already compute it."},{"Start":"10:10.470 ","End":"10:13.060","Text":"That this was 3 and this was 3."},{"Start":"10:13.060 ","End":"10:17.205","Text":"What I need is the equation of this line here."},{"Start":"10:17.205 ","End":"10:22.420","Text":"This is just where this plane cuts the xy plane."},{"Start":"10:22.420 ","End":"10:26.600","Text":"If I set z equals 0 here and divide by 2,"},{"Start":"10:26.600 ","End":"10:29.030","Text":"I get x plus y equals 3."},{"Start":"10:29.030 ","End":"10:33.484","Text":"This line is x plus y equals 3."},{"Start":"10:33.484 ","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":"10:42.020","Text":"Let\u0027s make it y in terms of x."},{"Start":"10:42.020 ","End":"10:48.625","Text":"I\u0027ll write it as y equals 3 minus x."},{"Start":"10:48.625 ","End":"10:54.630","Text":"Now we can take this region as a type 1 region,"},{"Start":"10:54.630 ","End":"11:00.800","Text":"meaning take vertical slices and different color."},{"Start":"11:00.800 ","End":"11:04.445","Text":"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":"11:11.999","Text":"This is just the x-axis,"},{"Start":"11:11.999 ","End":"11:14.130","Text":"which is y equals 0."},{"Start":"11:14.130 ","End":"11:16.980","Text":"We notice that x goes from 0-3,"},{"Start":"11:16.980 ","End":"11:19.500","Text":"y goes from 0-3 minus x."},{"Start":"11:19.500 ","End":"11:23.915","Text":"I\u0027m going to rewrite the integral"},{"Start":"11:23.915 ","End":"11:33.100","Text":"as the integral where x goes from 0-3."},{"Start":"11:33.100 ","End":"11:47.590","Text":"Then inside that, y goes from 0-3 minus x of all this dy dx."},{"Start":"11:49.260 ","End":"11:53.600","Text":"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":"12:30.580","Text":"that will be minus 7y,"},{"Start":"12:30.580 ","End":"12:36.820","Text":"and finally, plus 2y squared."},{"Start":"12:36.820 ","End":"12:43.525","Text":"This is the integral we have to compute to get the 1 of 4,"},{"Start":"12:43.525 ","End":"12:46.435","Text":"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":"13:01.250","Text":"This is equal to the integral from x equals 0-3."},{"Start":"13:03.000 ","End":"13:06.490","Text":"Now let\u0027s see, with respect to y, we\u0027re doing this."},{"Start":"13:06.490 ","End":"13:11.110","Text":"So with respect to y, x is a constant."},{"Start":"13:11.110 ","End":"13:16.180","Text":"We get 4xy squared over 2,"},{"Start":"13:16.180 ","End":"13:19.990","Text":"which is 2xy squared,"},{"Start":"13:19.990 ","End":"13:25.280","Text":"then plus 12y minus 5xy."},{"Start":"13:25.680 ","End":"13:34.240","Text":"Then for this we get minus 7 over 2y squared"},{"Start":"13:34.240 ","End":"13:40.460","Text":"and here, plus 2/3 y cubed."},{"Start":"13:41.550 ","End":"13:43.990","Text":"Sorry, that\u0027s not the integral."},{"Start":"13:43.990 ","End":"13:45.670","Text":"I mean, that really is the integral."},{"Start":"13:45.670 ","End":"13:49.360","Text":"Forget that."},{"Start":"13:49.360 ","End":"14:03.160","Text":"This bit has to be taken from 0-3 minus x, for y,"},{"Start":"14:03.160 ","End":"14:07.670","Text":"and then you still have the integral dx."},{"Start":"14:07.890 ","End":"14:12.430","Text":"I should have d1 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":"14:20.395","Text":"but we still have to plug in 3 minus x."},{"Start":"14:20.395 ","End":"14:31.255","Text":"We get the integral from 0-3 of 2x 3 minus x squared,"},{"Start":"14:31.255 ","End":"14:33.745","Text":"need another brackets here,"},{"Start":"14:33.745 ","End":"14:38.470","Text":"plus 12 times 3 minus x,"},{"Start":"14:38.470 ","End":"14:45.340","Text":"minus 5x 3 minus x, minus 7 over 2,"},{"Start":"14:45.340 ","End":"14:49.930","Text":"3 minus x squared"},{"Start":"14:49.930 ","End":"15:01.510","Text":"and then plus 2/3 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":"15:05.320","Text":"Let\u0027s see."},{"Start":"15:05.320 ","End":"15:07.900","Text":"Let\u0027s continue here."},{"Start":"15:07.900 ","End":"15:13.390","Text":"We get the integral from 0-3"},{"Start":"15:13.390 ","End":"15:20.050","Text":"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":"15:35.455","Text":"This 1 is going to be 2x times,"},{"Start":"15:35.455 ","End":"15:39.940","Text":"using the special binomial expansion,"},{"Start":"15:39.940 ","End":"15:48.980","Text":"this is going to be 3 squared is 9 minus twice 3 times x is 6x plus x squared."},{"Start":"15:49.230 ","End":"16:08.635","Text":"This is equal to 2x times 9 is 18x minus 12x squared plus 2x cubed."},{"Start":"16:08.635 ","End":"16:09.810","Text":"That This 1."},{"Start":"16:09.810 ","End":"16:28.820","Text":"This 1 will give me minus 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":"16:41.400","Text":"18x minus 15x is 3x,"},{"Start":"16:41.400 ","End":"16:48.860","Text":"minus 12x squared plus 5x squared is minus 7x squared"},{"Start":"16:48.860 ","End":"16:52.590","Text":"and then plus 2x cubed."},{"Start":"16:52.590 ","End":"16:56.710","Text":"Then the rest of them 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":"17:03.760","Text":"just copying the rest,"},{"Start":"17:03.760 ","End":"17:14.350","Text":"7 over 2, 3 minus x squared plus 2/3, 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 as usual,"},{"Start":"17:20.740 ","End":"17:27.265","Text":"3 over 2x squared minus 7 over 3x cubed."},{"Start":"17:27.265 ","End":"17:28.780","Text":"Here, 2 over 4,"},{"Start":"17:28.780 ","End":"17:31.640","Text":"I can write as 1/2x^4."},{"Start":"17:32.940 ","End":"17:35.350","Text":"Now, in these remaining bits,"},{"Start":"17:35.350 ","End":"17:37.840","Text":"I have 3 minus x instead of x."},{"Start":"17:37.840 ","End":"17:40.450","Text":"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":"17:47.155","Text":"What I\u0027ll get is,"},{"Start":"17:47.155 ","End":"17:54.010","Text":"I\u0027ll take the 3 minus x squared over 2,"},{"Start":"17:54.010 ","End":"17:56.095","Text":"which will give me just 6,"},{"Start":"17:56.095 ","End":"18:00.430","Text":"but instead of a plus I\u0027ll write a minus because of that inner derivative."},{"Start":"18:00.430 ","End":"18:04.270","Text":"Similarly here, the minus is going to become a plus."},{"Start":"18:04.270 ","End":"18:06.160","Text":"That takes care of the minus."},{"Start":"18:06.160 ","End":"18:10.465","Text":"Raise the power by 1, that\u0027s 3, divide by 3."},{"Start":"18:10.465 ","End":"18:17.860","Text":"So it\u0027s 7 over 2 times 3, 7 over 6, 3 minus x cubed."},{"Start":"18:17.860 ","End":"18:19.855","Text":"Similar idea here."},{"Start":"18:19.855 ","End":"18:21.790","Text":"Going to become a minus."},{"Start":"18:21.790 ","End":"18:24.710","Text":"There\u0027s going to be a 4 here."},{"Start":"18:24.720 ","End":"18:28.210","Text":"I\u0027m going to divide by the 4."},{"Start":"18:28.210 ","End":"18:36.505","Text":"I\u0027ll end up with 2 over 4 times 3 is just going to be 2 over 12 is 1/6."},{"Start":"18:36.505 ","End":"18:38.440","Text":"That\u0027s the integral,"},{"Start":"18:38.440 ","End":"18:49.135","Text":"and now I 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":"19:10.165","Text":"I\u0027ll get 3 over 2 times 3 squared is 27 over 2 is 13 and a 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 a 1/2."},{"Start":"19:26.095 ","End":"19:30.280","Text":"All the rest of them came 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, these 3 come out 0."},{"Start":"19:39.220 ","End":"19:43.795","Text":"Now here, when x is 0,"},{"Start":"19:43.795 ","End":"19:57.500","Text":"I got 3 squared is 9 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":"20:09.745","Text":"here I make it 31 and a 1/2."},{"Start":"20:09.745 ","End":"20:19.000","Text":"Here I\u0027ll take 81 over 6,"},{"Start":"20:19.000 ","End":"20:29.575","Text":"which is 27 over 2 is 13 and a 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 a 1/2 plus, 40 and a 1/2 is, I make that 54."},{"Start":"20:40.180 ","End":"20:48.380","Text":"54 minus 63, that would be minus 9."},{"Start":"20:49.530 ","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 a 1/2."},{"Start":"21:02.830 ","End":"21:10.360","Text":"31 and a 1/2 minus 67 and a 1/2 is minus 36."},{"Start":"21:10.360 ","End":"21:13.480","Text":"What do I get altogether?"},{"Start":"21:13.480 ","End":"21:16.510","Text":"Plus 36 minus 9,"},{"Start":"21:16.510 ","End":"21:19.585","Text":"I make that 27,"},{"Start":"21:19.585 ","End":"21:22.540","Text":"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 of those, the most difficult 1."},{"Start":"21:29.740 ","End":"21:32.630","Text":"Now, let\u0027s scroll back up."},{"Start":"21:37.770 ","End":"21:41.635","Text":"I\u0027ll record this as 27,"},{"Start":"21:41.635 ","End":"21:44.600","Text":"and now we\u0027ll tackle the other 3."}],"ID":8837},{"Watched":false,"Name":"Exercise 3 – Verified second direction (continued)","Duration":"19m 54s","ChapterTopicVideoID":8769,"CourseChapterTopicPlaylistID":4966,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.425","Text":"Now we\u0027ll go and do the OBC 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":"00:16.845","Text":"The equation of this plane is z equals 0."},{"Start":"00:16.845 ","End":"00:20.825","Text":"Our g of x, 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":"00:32.060","Text":"The other difference is that here the outward normal has a downward component."},{"Start":"00:32.060 ","End":"00:34.445","Text":"Instead of this formula,"},{"Start":"00:34.445 ","End":"00:37.850","Text":"we\u0027ll be using this formula."},{"Start":"00:37.850 ","End":"00:44.705","Text":"Then, what we get is F dot g_x,"},{"Start":"00:44.705 ","End":"00:53.160","Text":"g_y minus 1 is 0, 0, minus 1."},{"Start":"00:53.160 ","End":"01:01.660","Text":"Here\u0027s my F, and when I do the dot product,"},{"Start":"01:02.060 ","End":"01:06.210","Text":"we will get the double integral and it\u0027s going to be"},{"Start":"01:06.210 ","End":"01:12.270","Text":"the same R. This projection is the thing itself."},{"Start":"01:12.270 ","End":"01:13.830","Text":"One other small thing,"},{"Start":"01:13.830 ","End":"01:16.740","Text":"this is not an A now 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":"01:26.880","Text":"That\u0027s our piece of surface S. We"},{"Start":"01:26.880 ","End":"01:33.520","Text":"just take the minus 1 of the last component of F. It\u0027s going to obliterate this minus,"},{"Start":"01:33.520 ","End":"01:40.040","Text":"and we\u0027ll just get x plus 3y,"},{"Start":"01:40.040 ","End":"01:47.300","Text":"the minus 1 times this is this and the other is a 0, dA."},{"Start":"01:50.030 ","End":"01:52.470","Text":"It\u0027s the same R,"},{"Start":"01:52.470 ","End":"01:59.120","Text":"so we can break it up the same way into a double integral."},{"Start":"01:59.510 ","End":"02:03.745","Text":"If I do it iterated will be the integral,"},{"Start":"02:03.745 ","End":"02:08.445","Text":"x goes from 0 to 3."},{"Start":"02:08.445 ","End":"02:10.840","Text":"Then for each such x,"},{"Start":"02:10.840 ","End":"02:15.565","Text":"y goes from 0 to 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":"02:31.030","Text":"We\u0027ll start from the inside."},{"Start":"02:31.030 ","End":"02:35.040","Text":"I\u0027m going to get some more space here."},{"Start":"02:35.040 ","End":"02:37.950","Text":"Let\u0027s continue here."},{"Start":"02:37.950 ","End":"02:41.980","Text":"We\u0027ve got the double integral,"},{"Start":"02:43.010 ","End":"02:46.455","Text":"x goes from 0 to 3."},{"Start":"02:46.455 ","End":"02:53.140","Text":"Now I\u0027m going to evaluate this one, the dy_1."},{"Start":"02:55.160 ","End":"03:06.375","Text":"What I have is xy plus or 3/2 y squared,"},{"Start":"03:06.375 ","End":"03:13.755","Text":"this taken, that is for y,"},{"Start":"03:13.755 ","End":"03:17.295","Text":"from 0 to 3 minus x,"},{"Start":"03:17.295 ","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":"03:31.750","Text":"If I put y equals 0,"},{"Start":"03:31.750 ","End":"03:40.430","Text":"I don\u0027t get anything so it\u0027s just the 3 minus x. I get x times 3 minus x"},{"Start":"03:40.430 ","End":"03:45.430","Text":"plus 3/2 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":"04:06.045","Text":"I think we\u0027ll just go ahead and do it here, get some space."},{"Start":"04:06.045 ","End":"04:09.770","Text":"We got an integral from 0 to 3."},{"Start":"04:09.770 ","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 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":"04:47.210","Text":"Now I\u0027ll do the actual integral,"},{"Start":"04:47.210 ","End":"04:49.445","Text":"so here I\u0027ve got"},{"Start":"04:49.445 ","End":"04:58.105","Text":"3/2x squared minus 1/3x cubed."},{"Start":"04:58.105 ","End":"05:03.740","Text":"Here I\u0027ll look at it as a function of 3 minus x."},{"Start":"05:05.000 ","End":"05:10.440","Text":"What I do is I take the 3 minus x and instead of 2"},{"Start":"05:10.440 ","End":"05:15.920","Text":"make it to the power of 3 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":"05:21.490","Text":"I can just cancel with this 3."},{"Start":"05:21.490 ","End":"05:23.955","Text":"This just becomes 1.5."},{"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, 3 minus x. I need to also divide by minus 1,"},{"Start":"05:31.580 ","End":"05:34.590","Text":"so I put a minus here."},{"Start":"05:36.890 ","End":"05:39.660","Text":"That\u0027s the integral already."},{"Start":"05:39.660 ","End":"05:47.340","Text":"I have to just evaluate this from 0 to 3 for x."},{"Start":"05:47.340 ","End":"05:50.345","Text":"What we get is,"},{"Start":"05:50.345 ","End":"05:52.940","Text":"if we put in 3,"},{"Start":"05:52.940 ","End":"05:59.180","Text":"we get 3/2 times 3 squared is 27 over 2."},{"Start":"05:59.180 ","End":"06:02.000","Text":"I\u0027ll write it as 13.5."},{"Start":"06:02.000 ","End":"06:04.415","Text":"Here. If x is 3,"},{"Start":"06:04.415 ","End":"06:06.925","Text":"x cubed over 3,"},{"Start":"06:06.925 ","End":"06:10.200","Text":"just 3 cubed over 3 is 3 squared,"},{"Start":"06:10.200 ","End":"06:13.605","Text":"that\u0027s 9, and that\u0027s a minus."},{"Start":"06:13.605 ","End":"06:18.735","Text":"Here when x is 3, it\u0027s just 0."},{"Start":"06:18.735 ","End":"06:23.870","Text":"This is the 3 part 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":"06:31.845","Text":"this is 0, 3 minus 0 is 3."},{"Start":"06:31.845 ","End":"06:35.970","Text":"It\u0027s minus 1/2, 3 cubed,"},{"Start":"06:35.970 ","End":"06:44.130","Text":"minus 27/2 minus 13.5,"},{"Start":"06:44.130 ","End":"06:46.680","Text":"and altogether, what do I get?"},{"Start":"06:46.680 ","End":"06:50.885","Text":"It\u0027s 13.5 plus 13.5 is 27."},{"Start":"06:50.885 ","End":"06:55.900","Text":"27 minus 9 is 18."},{"Start":"06:55.900 ","End":"07:03.750","Text":"This would be the answer for the OBC triangle,"},{"Start":"07:03.750 ","End":"07:06.045","Text":"I\u0027ll just highlight that,"},{"Start":"07:06.045 ","End":"07:10.095","Text":"and that\u0027s the second of 4."},{"Start":"07:10.095 ","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":"07:37.340","Text":"Let me erase what I don\u0027t need. Here we are."},{"Start":"07:37.340 ","End":"07:41.750","Text":"I erased some, I replaced some this time we\u0027re on this phase here,"},{"Start":"07:41.750 ","End":"07:46.620","Text":"which is the x equals 0 phase."},{"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":"07:59.790","Text":"This time we have x equals some function,"},{"Start":"07:59.790 ","End":"08:07.125","Text":"I\u0027ll also call it g of y and z and that function is just 0."},{"Start":"08:07.125 ","End":"08:10.170","Text":"Here\u0027s the picture in the z,"},{"Start":"08:10.170 ","End":"08:13.759","Text":"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":"08:35.535","Text":"which we also call g. Slight differences,"},{"Start":"08:35.535 ","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 2 do we need?"},{"Start":"08:40.670 ","End":"08:49.560","Text":"Well, the first formula applies for when the normal has a component in the positive,"},{"Start":"08:57.910 ","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":"09:13.825","Text":"It\u0027s going to be going against the direction of x."},{"Start":"09:13.825 ","End":"09:18.190","Text":"It\u0027s a negative x and so we\u0027re going to use this formula."},{"Start":"09:18.190 ","End":"09:20.450","Text":"The arrow was there 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 g_y and g_z."},{"Start":"09:26.930 ","End":"09:30.890","Text":"Obviously since this is the 0 function,"},{"Start":"09:30.890 ","End":"09:37.460","Text":"the derivatives are also g_y and g_z are both 0,"},{"Start":"09:37.460 ","End":"09:41.035","Text":"so that this vector that I\u0027m going to be using,"},{"Start":"09:41.035 ","End":"09:44.310","Text":"in our case is going to be minus 1,0,0."},{"Start":"09:45.830 ","End":"09:50.020","Text":"If I do a dot product,"},{"Start":"09:50.570 ","End":"09:59.820","Text":"the integral over OAB of F dot n dS for"},{"Start":"09:59.820 ","End":"10:08.705","Text":"our case where x is a function of y and z and the normal is in the negative x direction."},{"Start":"10:08.705 ","End":"10:13.975","Text":"This is equal to the double integral over"},{"Start":"10:13.975 ","End":"10:20.330","Text":"R. The R is actually the same as the side of the pyramid,"},{"Start":"10:20.330 ","End":"10:28.035","Text":"but it\u0027s R as in this picture, of the dot-product."},{"Start":"10:28.035 ","End":"10:30.890","Text":"Since there is only a minus 1 and these 2 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":"10:47.660","Text":"which is minus 1"},{"Start":"10:47.660 ","End":"10:57.735","Text":"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":"11:05.995","Text":"I want what x is equal to,"},{"Start":"11:05.995 ","End":"11:10.790","Text":"and x is equal to 0."},{"Start":"11:12.360 ","End":"11:20.260","Text":"I\u0027ll leave it like this for the moment, I replace x,"},{"Start":"11:20.260 ","End":"11:24.640","Text":"because it\u0027s just going to be a function of y and z, by 0,"},{"Start":"11:24.640 ","End":"11:26.545","Text":"so I get z with the minus,"},{"Start":"11:26.545 ","End":"11:33.410","Text":"it\u0027s the double integral over r of minus z dA."},{"Start":"11:33.780 ","End":"11:39.475","Text":"To do this as an iterated integral I\u0027ll be needing the equation of this line here,"},{"Start":"11:39.475 ","End":"11:41.575","Text":"I can get this from here,"},{"Start":"11:41.575 ","End":"11:44.770","Text":"by letting x equal 0."},{"Start":"11:44.770 ","End":"11:46.675","Text":"If I let x equals 0,"},{"Start":"11:46.675 ","End":"11:51.145","Text":"I get that z equals 6 minus 2y so this line here,"},{"Start":"11:51.145 ","End":"11:58.780","Text":"is given by z equals 6 minus 2y."},{"Start":"11:58.780 ","End":"12:03.295","Text":"As before, this is not y equals 0,"},{"Start":"12:03.295 ","End":"12:07.850","Text":"this is z equals 0 in the zy 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":"12:29.650","Text":"the outer loop on y going from 0-3,"},{"Start":"12:29.650 ","End":"12:39.670","Text":"and for each y, z goes from 0-6 minus 2y,"},{"Start":"12:39.670 ","End":"12:49.310","Text":"and 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":"13:01.390","Text":"and now let\u0027s move down here,"},{"Start":"13:01.390 ","End":"13:03.865","Text":"and let\u0027s see what it\u0027s equal to."},{"Start":"13:03.865 ","End":"13:12.940","Text":"We\u0027ll do the inner one first, this bit,"},{"Start":"13:14.160 ","End":"13:18.129","Text":"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":"13:29.200","Text":"the integral of z dz is one-half z squared,"},{"Start":"13:29.200 ","End":"13:37.165","Text":"which I have to evaluate from 0-6 minus 2y,"},{"Start":"13:37.165 ","End":"13:41.710","Text":"and then that\u0027s going to be dy."},{"Start":"13:41.710 ","End":"13:50.125","Text":"This will equal minus 1/2,"},{"Start":"13:50.125 ","End":"13:55.210","Text":"I can also take the 1/2 out of just z"},{"Start":"13:55.210 ","End":"14:03.415","Text":"squared the integral from 0-3,"},{"Start":"14:03.415 ","End":"14:07.750","Text":"and 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 0 squared doesn\u0027t matter,"},{"Start":"14:17.380 ","End":"14:20.170","Text":"this is basically what we\u0027re left with,"},{"Start":"14:20.170 ","End":"14:25.390","Text":"dy, and let\u0027s see,"},{"Start":"14:25.390 ","End":"14:28.600","Text":"I can do this integral right away."},{"Start":"14:28.600 ","End":"14:30.850","Text":"It could do a substitution,"},{"Start":"14:30.850 ","End":"14:32.830","Text":"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":"14:49.660","Text":"But then we would say, it wasn\u0027t,"},{"Start":"14:49.660 ","End":"14:53.335","Text":"6 minus 2y where the inner derivative is minus 2."},{"Start":"14:53.335 ","End":"14:56.860","Text":"I have to divide that by minus 2,"},{"Start":"14:56.860 ","End":"15:04.150","Text":"that\u0027s like minus 1/2 and then I also have a minus 1/2 here."},{"Start":"15:04.150 ","End":"15:09.490","Text":"It\u0027s minus 1/2, minus 1/2 times 1/3 times 6 minus"},{"Start":"15:09.490 ","End":"15:16.070","Text":"2y cubed and all this from 0-3."},{"Start":"15:17.010 ","End":"15:26.310","Text":"This is now easy to evaluate when we put in 3,"},{"Start":"15:26.310 ","End":"15:31.455","Text":"6 minus twice 3 is 0 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":"15:39.445","Text":"If I put in 0, I\u0027ve got,"},{"Start":"15:39.445 ","End":"15:41.170","Text":"well, I\u0027ll just write it."},{"Start":"15:41.170 ","End":"15:44.275","Text":"It\u0027s 1/2 times a half."},{"Start":"15:44.275 ","End":"15:52.190","Text":"The minus will cancel with the minus 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":"16:09.510","Text":"One of the 6s will cancel with 2 and 3 so it\u0027s just 6 squared over 2, 36/2."},{"Start":"16:09.510 ","End":"16:17.420","Text":"I make it minus 18 and I\u0027ll highlight it."},{"Start":"16:17.420 ","End":"16:21.655","Text":"This was the answer for OAB."},{"Start":"16:21.655 ","End":"16:28.630","Text":"I\u0027m going to go back up and mark that as what was this,"},{"Start":"16:28.630 ","End":"16:32.680","Text":"this was OAB, 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":"16:48.940","Text":"Now some replacements, I\u0027ll change the B to a C. I\u0027ll also change"},{"Start":"16:48.940 ","End":"16:56.485","Text":"this because now y is 0 and y is a function of x and z."},{"Start":"16:56.485 ","End":"17:01.060","Text":"Here y, here x and of"},{"Start":"17:01.060 ","End":"17:05.980","Text":"course we\u0027ll need the partial derivative of g with respect to x is 0,"},{"Start":"17:05.980 ","End":"17:09.550","Text":"and g with respect to z equals 0."},{"Start":"17:09.550 ","End":"17:15.580","Text":"What else? 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":"17:27.655","Text":"but this one has to be z and this one has to be"},{"Start":"17:27.655 ","End":"17:35.630","Text":"x. I have to change 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":"17:43.945","Text":"We have the g_x and the g_z and 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":"17:49.840","Text":"We use the top formula when the normal has"},{"Start":"17:49.840 ","End":"17:54.115","Text":"a positive y middle component"},{"Start":"17:54.115 ","End":"17:59.499","Text":"and we use this one when it has a negative y component or j component."},{"Start":"17:59.499 ","End":"18:04.870","Text":"In our case, if we take a point on this face,"},{"Start":"18:04.870 ","End":"18:12.295","Text":"this plane, the normal goes exactly in the opposite direction as the y-direction."},{"Start":"18:12.295 ","End":"18:13.780","Text":"This is correct."},{"Start":"18:13.780 ","End":"18:17.425","Text":"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 and 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 and dot-product with this,"},{"Start":"18:55.000 ","End":"18:57.640","Text":"we\u0027ll just 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 is 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":"19:12.160","Text":"But y equals 0,"},{"Start":"19:12.160 ","End":"19:16.225","Text":"so since y equals 0,"},{"Start":"19:16.225 ","End":"19:18.730","Text":"this is just equal to 0."},{"Start":"19:18.730 ","End":"19:21.595","Text":"I\u0027ll just 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":"19:29.395","Text":"One of them came out easy and I can just say that for"},{"Start":"19:29.395 ","End":"19:35.730","Text":"this one was for OAC."},{"Start":"19:35.730 ","End":"19:37.380","Text":"We\u0027ve done the fourth one out of 4,"},{"Start":"19:37.380 ","End":"19:39.435","Text":"this one\u0027s a 0."},{"Start":"19:39.435 ","End":"19:43.140","Text":"The grand total, adding all these up,"},{"Start":"19:43.140 ","End":"19:46.290","Text":"this is 0, these two cancel each other out."},{"Start":"19:46.290 ","End":"19:53.950","Text":"We just get 27 and that is our final answer."}],"ID":8838},{"Watched":false,"Name":"Exercise 4","Duration":"9m 52s","ChapterTopicVideoID":8771,"CourseChapterTopicPlaylistID":4966,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.534","Text":"In this exercise, we\u0027re given the surface of a body,"},{"Start":"00:05.534 ","End":"00:08.205","Text":"it\u0027s bounded by the cylinder,"},{"Start":"00:08.205 ","End":"00:10.980","Text":"this one and 2 planes."},{"Start":"00:10.980 ","End":"00:14.760","Text":"Maybe I\u0027ll bring a picture in just a second."},{"Start":"00:14.760 ","End":"00:16.920","Text":"Now, we have to compute the flux,"},{"Start":"00:16.920 ","End":"00:18.360","Text":"this is a question from physics,"},{"Start":"00:18.360 ","End":"00:24.000","Text":"but don\u0027t worry of the vector field so and so, through S."},{"Start":"00:24.000 ","End":"00:27.270","Text":"We don\u0027t have to know physics because we\u0027re given the formula,"},{"Start":"00:27.270 ","End":"00:28.485","Text":"what we have to compute,"},{"Start":"00:28.485 ","End":"00:36.915","Text":"the flux is just the surface integral type 2 of F dot the normal vector ds."},{"Start":"00:36.915 ","End":"00:43.150","Text":"As usual, n is unless stated otherwise would be the outward unit normal."},{"Start":"00:43.640 ","End":"00:47.500","Text":"Now, if I didn\u0027t have the divergence theorem,"},{"Start":"00:47.500 ","End":"00:53.665","Text":"this might be a difficult task because the surface S would be broken up into 3 parts,"},{"Start":"00:53.665 ","End":"00:56.480","Text":"S_1 and S_2 and S_3."},{"Start":"00:56.480 ","End":"01:00.340","Text":"There\u0027s the sides of the cylinder, the cylinder itself,"},{"Start":"01:00.340 ","End":"01:02.320","Text":"and then there\u0027s 2 bases and each one of them"},{"Start":"01:02.320 ","End":"01:06.530","Text":"has a normal going outwards and it would be quite difficult."},{"Start":"01:06.530 ","End":"01:13.210","Text":"Instead, if we call the whole solid cylinder R for 3D region,"},{"Start":"01:13.210 ","End":"01:17.830","Text":"then we can use the divergence theorem to say that"},{"Start":"01:17.830 ","End":"01:23.165","Text":"the double integral over the surface of the cylinder of"},{"Start":"01:23.165 ","End":"01:30.530","Text":"F dot n ds is equal to the triple integral"},{"Start":"01:30.530 ","End":"01:39.785","Text":"over the 3D region R of the divergence of F dV."},{"Start":"01:39.785 ","End":"01:42.020","Text":"Let me just annotate this a bit,"},{"Start":"01:42.020 ","End":"01:46.160","Text":"this would be the origin and this is the xy plane."},{"Start":"01:46.160 ","End":"01:50.955","Text":"The cylinder goes from 0 up to 2."},{"Start":"01:50.955 ","End":"01:55.709","Text":"This would be the plane z equals 2 where it cuts the cylinder"},{"Start":"01:55.709 ","End":"01:59.520","Text":"and the part in the xy plane,"},{"Start":"01:59.520 ","End":"02:02.205","Text":"I\u0027ll just highlight it,"},{"Start":"02:02.205 ","End":"02:10.205","Text":"this part here would be just the circle or the disk rather of radius 3,"},{"Start":"02:10.205 ","End":"02:14.060","Text":"because 9 is 3 squared."},{"Start":"02:14.060 ","End":"02:16.730","Text":"This would be 3 and 3,"},{"Start":"02:16.730 ","End":"02:18.875","Text":"disk of radius 3."},{"Start":"02:18.875 ","End":"02:23.870","Text":"That would be the base of the cylinder."},{"Start":"02:23.870 ","End":"02:26.540","Text":"I\u0027m going to give that a name,"},{"Start":"02:26.540 ","End":"02:29.285","Text":"I\u0027ll call this D,"},{"Start":"02:29.285 ","End":"02:31.220","Text":"I\u0027ve used up the letter R,"},{"Start":"02:31.220 ","End":"02:35.120","Text":"so D for 2D domain in the xy plane,"},{"Start":"02:35.120 ","End":"02:39.495","Text":"3 by 3 disk, 3, 3."},{"Start":"02:39.495 ","End":"02:40.860","Text":"Back here."},{"Start":"02:40.860 ","End":"02:44.180","Text":"Now, let\u0027s first of all compute the divergence"},{"Start":"02:44.180 ","End":"02:45.815","Text":"and then we\u0027ll worry about doing the integral."},{"Start":"02:45.815 ","End":"02:51.245","Text":"The divergence of F is,"},{"Start":"02:51.245 ","End":"02:54.740","Text":"well, before I say what it is, let\u0027s give some notation."},{"Start":"02:54.740 ","End":"02:56.390","Text":"The first component of F,"},{"Start":"02:56.390 ","End":"03:01.985","Text":"I\u0027ll call this one, P, I\u0027ll call this Q, and I\u0027ll call this R,"},{"Start":"03:01.985 ","End":"03:04.415","Text":"this bit, this bit, and this bit."},{"Start":"03:04.415 ","End":"03:07.445","Text":"Sometimes I use F, G, H, or whatever,"},{"Start":"03:07.445 ","End":"03:12.680","Text":"anyway it\u0027s equal to the partial derivative of P with respect to x"},{"Start":"03:12.680 ","End":"03:16.460","Text":"plus the partial derivative of Q with respect to y,"},{"Start":"03:16.460 ","End":"03:19.765","Text":"plus the partial derivative of R with respect to z."},{"Start":"03:19.765 ","End":"03:25.760","Text":"Just the 1st, 2nd, and 3rd components of the function with respect to x, y, and z."},{"Start":"03:25.760 ","End":"03:28.070","Text":"It doesn\u0027t matter what letters you use."},{"Start":"03:28.070 ","End":"03:34.905","Text":"In our case, it will be this with respect to x will be 3x squared,"},{"Start":"03:34.905 ","End":"03:41.985","Text":"this with respect to y will be 3y squared and 2z,"},{"Start":"03:41.985 ","End":"03:44.490","Text":"so that\u0027s the divergence."},{"Start":"03:44.490 ","End":"03:51.045","Text":"Now, let me write the region."},{"Start":"03:51.045 ","End":"03:56.089","Text":"In the inside, I\u0027ll do the integration with respect to z first."},{"Start":"03:56.089 ","End":"04:09.950","Text":"I get the integral with respect to z from 0-2 of what I wrote here,"},{"Start":"04:09.950 ","End":"04:17.215","Text":"the divergence 3x squared plus 3y squared plus 2z dz."},{"Start":"04:17.215 ","End":"04:25.730","Text":"Then we\u0027ll do the integral over the disk D,"},{"Start":"04:25.730 ","End":"04:30.230","Text":"the radius 3 disk D, dA,"},{"Start":"04:30.230 ","End":"04:33.210","Text":"afterwards we\u0027ll decide whether it\u0027s dx, dy"},{"Start":"04:33.210 ","End":"04:36.020","Text":"or maybe we\u0027ll do it by polar coordinates."},{"Start":"04:36.020 ","End":"04:47.360","Text":"Let\u0027s start with this and then we\u0027ll see how we open up the x and y plane."},{"Start":"04:47.360 ","End":"04:56.629","Text":"Let\u0027s see, I\u0027ll do this one at the side."},{"Start":"04:56.629 ","End":"05:01.730","Text":"I\u0027ll highlight it so you see what I mean, the integral dz."},{"Start":"05:01.800 ","End":"05:06.025","Text":"What I will get remembering that x and y are like constants"},{"Start":"05:06.025 ","End":"05:13.780","Text":"is 3x squared z plus 3y squared z,"},{"Start":"05:13.780 ","End":"05:17.295","Text":"2z gives me z squared."},{"Start":"05:17.295 ","End":"05:23.335","Text":"All this has to be evaluated from z equals 0-2,"},{"Start":"05:23.335 ","End":"05:25.795","Text":"so this is equal 2."},{"Start":"05:25.795 ","End":"05:27.670","Text":"Now, when z equals 0,"},{"Start":"05:27.670 ","End":"05:29.500","Text":"I\u0027ll just get 0\u0027s everywhere,"},{"Start":"05:29.500 ","End":"05:31.875","Text":"so I can discount that."},{"Start":"05:31.875 ","End":"05:34.800","Text":"Just put in z equals 2,"},{"Start":"05:34.800 ","End":"05:39.330","Text":"we get 2 times 3, 6x squared."},{"Start":"05:39.330 ","End":"05:41.475","Text":"From here, 2 times 3,"},{"Start":"05:41.475 ","End":"05:49.510","Text":"also 6y squared and z squared is 4."},{"Start":"05:49.820 ","End":"05:54.980","Text":"I\u0027ll just copy this here and I\u0027ll take 6 outside the brackets"},{"Start":"05:54.980 ","End":"06:01.739","Text":"and write x squared plus y squared plus 4,"},{"Start":"06:01.739 ","End":"06:04.410","Text":"that\u0027s the highlighted bit."},{"Start":"06:04.410 ","End":"06:08.220","Text":"I\u0027m definitely going to go for polar coordinates."},{"Start":"06:08.390 ","End":"06:10.695","Text":"Here I have some space."},{"Start":"06:10.695 ","End":"06:14.360","Text":"For polar coordinates, there are several equations."},{"Start":"06:14.360 ","End":"06:17.060","Text":"There\u0027s an equation what x equals and what y equals,"},{"Start":"06:17.060 ","End":"06:18.620","Text":"but I won\u0027t need that."},{"Start":"06:18.620 ","End":"06:26.590","Text":"I\u0027ll just need the equation that dA is equal to rdrd Theta."},{"Start":"06:26.590 ","End":"06:30.650","Text":"I\u0027ll also use the optional 4th equation,"},{"Start":"06:30.650 ","End":"06:36.310","Text":"that x squared plus y squared equals r squared."},{"Start":"06:36.310 ","End":"06:41.735","Text":"Those are the equations, I also have to describe this region in polar terms."},{"Start":"06:41.735 ","End":"06:45.080","Text":"We\u0027ve already seen circles and disks many times."},{"Start":"06:45.080 ","End":"06:47.185","Text":"A disk of radius 3,"},{"Start":"06:47.185 ","End":"06:51.740","Text":"the D in polar will be 1 complete circle,"},{"Start":"06:51.740 ","End":"06:57.390","Text":"so Theta between 0 and 2 Pi, it\u0027s 360 degrees."},{"Start":"06:57.390 ","End":"07:04.530","Text":"The radius from the center to the perimeter is 0-3 for r."},{"Start":"07:04.530 ","End":"07:06.945","Text":"Putting all this together,"},{"Start":"07:06.945 ","End":"07:20.440","Text":"we can get the integral Theta from 0-2 Pi, r from 0-3."},{"Start":"07:20.440 ","End":"07:23.420","Text":"Here I make a substitution."},{"Start":"07:23.420 ","End":"07:26.780","Text":"I\u0027ve got 6r squared because"},{"Start":"07:26.780 ","End":"07:30.170","Text":"x squared plus y squared is r squared plus 4"},{"Start":"07:30.170 ","End":"07:35.070","Text":"and dA is rdrd Theta."},{"Start":"07:35.860 ","End":"07:39.230","Text":"Once again, I\u0027ll do the middle bit,"},{"Start":"07:39.230 ","End":"07:42.115","Text":"which is this one at the side."},{"Start":"07:42.115 ","End":"07:45.060","Text":"But I\u0027ll need a bit more space."},{"Start":"07:45.060 ","End":"07:49.970","Text":"This one, I\u0027ll do it over here."},{"Start":"07:49.970 ","End":"07:56.685","Text":"I have the integral from 0-3 of 6,"},{"Start":"07:56.685 ","End":"07:58.860","Text":"I multiply the r inside,"},{"Start":"07:58.860 ","End":"08:06.870","Text":"so it\u0027s 6r cubed plus 4r, dr."},{"Start":"08:06.870 ","End":"08:08.510","Text":"This equals, let\u0027s see,"},{"Start":"08:08.510 ","End":"08:10.835","Text":"raise the power by 1, it\u0027s 4,"},{"Start":"08:10.835 ","End":"08:20.250","Text":"6 over 4 is like 3 over 2 to the 4th plus, let\u0027s see,"},{"Start":"08:20.250 ","End":"08:23.010","Text":"raise the power of 1, r squared divided by 2,"},{"Start":"08:23.010 ","End":"08:28.920","Text":"2r squared from 0-3,"},{"Start":"08:28.920 ","End":"08:31.905","Text":"0 gives me nothing,"},{"Start":"08:31.905 ","End":"08:41.610","Text":"3 gives me 3 times 81 is 243 over 2,"},{"Start":"08:41.610 ","End":"08:52.455","Text":"243 over 2 plus 2 times 3 squared is 2 times 9 is 18."},{"Start":"08:52.455 ","End":"08:56.100","Text":"This equals, let\u0027s see."},{"Start":"08:56.100 ","End":"08:58.695","Text":"This is 121 and 1/2,"},{"Start":"08:58.695 ","End":"09:06.030","Text":"121 plus 18 is a 139 and 1/2,"},{"Start":"09:06.030 ","End":"09:10.890","Text":"139 and 1/2 here."},{"Start":"09:10.890 ","End":"09:12.360","Text":"Now, let\u0027s continue."},{"Start":"09:12.360 ","End":"09:14.910","Text":"This is a constant,"},{"Start":"09:15.650 ","End":"09:20.510","Text":"139 and 1/2, so I\u0027m taking that out,"},{"Start":"09:20.510 ","End":"09:25.365","Text":"integral from 0-2 Pi of d Theta,"},{"Start":"09:25.365 ","End":"09:26.880","Text":"it\u0027s just 1 d Theta."},{"Start":"09:26.880 ","End":"09:28.140","Text":"When we have the integral of 1,"},{"Start":"09:28.140 ","End":"09:31.645","Text":"it\u0027s just this limit minus this limit, which is 2 Pi,"},{"Start":"09:31.645 ","End":"09:37.100","Text":"so I get 139 and 1/2 times 2 Pi."},{"Start":"09:37.100 ","End":"09:48.930","Text":"2 times this is 279 Pi."},{"Start":"09:48.930 ","End":"09:53.820","Text":"Just highlight that because that\u0027s the final answer and we\u0027re done."}],"ID":8839},{"Watched":false,"Name":"Exercise 5","Duration":"10m 38s","ChapterTopicVideoID":8772,"CourseChapterTopicPlaylistID":4966,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.895","Text":"In this exercise, we\u0027re given a Type 2 surface integral to calculate."},{"Start":"00:05.895 ","End":"00:15.360","Text":"We\u0027re given the vector field F as follows and let me call the components P, Q,"},{"Start":"00:15.360 ","End":"00:21.930","Text":"and R. S is the surface of the body,"},{"Start":"00:21.930 ","End":"00:26.730","Text":"it\u0027s given by 4 different surfaces."},{"Start":"00:26.730 ","End":"00:35.370","Text":"The 2 planes, x equals naught and x equals 3, and then z."},{"Start":"00:35.370 ","End":"00:39.275","Text":"Well, this is a parabolic cylinder. You know what?"},{"Start":"00:39.275 ","End":"00:46.450","Text":"I\u0027ll try and sketch this at least in 2 dimensions."},{"Start":"00:46.450 ","End":"00:49.680","Text":"If I look at these 2 equations,"},{"Start":"00:49.680 ","End":"00:52.575","Text":"the last 2 in the yz-plane,"},{"Start":"00:52.575 ","End":"00:55.120","Text":"I\u0027ve got 2 functions,"},{"Start":"00:55.120 ","End":"01:00.440","Text":"z equals 4 minus y squared is an inverted parabola."},{"Start":"01:00.440 ","End":"01:03.350","Text":"If I look at where it cuts the axes,"},{"Start":"01:03.350 ","End":"01:07.205","Text":"when y is 0, then z is 4."},{"Start":"01:07.205 ","End":"01:09.860","Text":"When z is 0,"},{"Start":"01:09.860 ","End":"01:11.600","Text":"then I get y squared is 4,"},{"Start":"01:11.600 ","End":"01:17.210","Text":"so y is plus or minus 2."},{"Start":"01:17.210 ","End":"01:25.515","Text":"The parabola looks like this, something like this."},{"Start":"01:25.515 ","End":"01:29.390","Text":"But because it\u0027s bounded below by z equals 0,"},{"Start":"01:29.390 ","End":"01:32.690","Text":"it doesn\u0027t continue on, it closes here."},{"Start":"01:32.690 ","End":"01:37.020","Text":"This is what it looks like in the zy-plane."},{"Start":"01:37.060 ","End":"01:45.290","Text":"In fact, it would be the same intersection in any plane where x is constant."},{"Start":"01:45.290 ","End":"01:47.480","Text":"This happens to be where x is 0."},{"Start":"01:47.480 ","End":"01:50.340","Text":"X goes from here out."},{"Start":"01:50.350 ","End":"01:58.770","Text":"It goes above out of the paper, so to speak."},{"Start":"01:59.120 ","End":"02:01.430","Text":"It has a thickness of 3,"},{"Start":"02:01.430 ","End":"02:03.710","Text":"but it looks the same everywhere."},{"Start":"02:03.710 ","End":"02:08.670","Text":"Perhaps I\u0027ll show you what it might look like."},{"Start":"02:09.010 ","End":"02:12.750","Text":"Here\u0027s what a parabolic cylinder looks like."},{"Start":"02:12.750 ","End":"02:16.459","Text":"This one seems to be made of wood from a different angle,"},{"Start":"02:16.459 ","End":"02:19.145","Text":"but if we rotated this,"},{"Start":"02:19.145 ","End":"02:21.620","Text":"then this would be maybe x equals 0."},{"Start":"02:21.620 ","End":"02:23.090","Text":"This would be x equals 3."},{"Start":"02:23.090 ","End":"02:24.935","Text":"The x would be upwards,"},{"Start":"02:24.935 ","End":"02:30.380","Text":"the flow would be the zy-plane and it would give an impression like this."},{"Start":"02:30.380 ","End":"02:35.645","Text":"Let\u0027s also draw one more from the side, from the xy-plane."},{"Start":"02:35.645 ","End":"02:39.050","Text":"Now I\u0027m looking from above, from the z-direction,"},{"Start":"02:39.050 ","End":"02:45.660","Text":"and this just looks like a line from minus 2-2,"},{"Start":"02:45.660 ","End":"02:49.395","Text":"but I know that x goes from 0-3."},{"Start":"02:49.395 ","End":"02:52.574","Text":"Instead of coming upwards,"},{"Start":"02:52.574 ","End":"02:54.090","Text":"I just go like this,"},{"Start":"02:54.090 ","End":"03:02.710","Text":"3 units, so I\u0027ll get something like this, not quite straight."},{"Start":"03:02.780 ","End":"03:08.870","Text":"Let me tell you right away that we\u0027re going to do it"},{"Start":"03:08.870 ","End":"03:14.225","Text":"with the divergence theorem because this surface is made up of 4 separate surfaces."},{"Start":"03:14.225 ","End":"03:16.240","Text":"I have the cylindrical,"},{"Start":"03:16.240 ","End":"03:18.695","Text":"no, the parabolic cylinder."},{"Start":"03:18.695 ","End":"03:20.420","Text":"I have this rectangle,"},{"Start":"03:20.420 ","End":"03:22.230","Text":"which would be this base here,"},{"Start":"03:22.230 ","End":"03:24.965","Text":"and I have 2 of these that look like this."},{"Start":"03:24.965 ","End":"03:27.800","Text":"1 at x is 0 and 1at x equals 3,"},{"Start":"03:27.800 ","End":"03:33.460","Text":"and it\u0027s going to be messy to do it over the surface."},{"Start":"03:34.880 ","End":"03:38.290","Text":"Let me move this to the side."},{"Start":"03:38.290 ","End":"03:42.170","Text":"Let\u0027s just write the divergence theorem that we\u0027re going to use is that"},{"Start":"03:42.170 ","End":"03:46.640","Text":"this double integral over S of"},{"Start":"03:46.640 ","End":"03:52.550","Text":"F.n ds will equal"},{"Start":"03:52.550 ","End":"03:57.200","Text":"the triple integral over the region."},{"Start":"03:57.200 ","End":"04:02.190","Text":"Let\u0027s call this region the parabolic cylinder."},{"Start":"04:02.190 ","End":"04:05.065","Text":"The solid one, we\u0027ll call it R,"},{"Start":"04:05.065 ","End":"04:10.895","Text":"which is defined by these equations of"},{"Start":"04:10.895 ","End":"04:20.550","Text":"the divergence of F and that\u0027s going to be dv."},{"Start":"04:20.550 ","End":"04:30.990","Text":"Now the divergence is just equal to P with respect to x,"},{"Start":"04:30.990 ","End":"04:33.660","Text":"plus Q with respect to y,"},{"Start":"04:33.660 ","End":"04:37.180","Text":"plus R with respect to z."},{"Start":"04:37.370 ","End":"04:43.265","Text":"This is equal to P with respect to x is minus 1,"},{"Start":"04:43.265 ","End":"04:50.209","Text":"Q with respect to y is minus x,"},{"Start":"04:50.209 ","End":"04:55.800","Text":"and R with respect to z is just 3."},{"Start":"04:55.800 ","End":"05:00.165","Text":"Altogether, this is equal to 3 plus 1,"},{"Start":"05:00.165 ","End":"05:04.305","Text":"2 minus x, that\u0027s the divergence."},{"Start":"05:04.305 ","End":"05:05.750","Text":"I want the integral."},{"Start":"05:05.750 ","End":"05:08.780","Text":"Now, what I\u0027ll do is I\u0027ll first of all, the outer integral,"},{"Start":"05:08.780 ","End":"05:13.790","Text":"I\u0027ll take x from 0-3."},{"Start":"05:15.340 ","End":"05:18.065","Text":"This is the limit for x."},{"Start":"05:18.065 ","End":"05:21.569","Text":"Now for each x between 0 and 3,"},{"Start":"05:21.569 ","End":"05:25.870","Text":"y is just going to go from minus 2-2,"},{"Start":"05:25.870 ","End":"05:29.710","Text":"that\u0027s a constant, and for each such y,"},{"Start":"05:29.710 ","End":"05:33.295","Text":"wherever I slice it at any x,"},{"Start":"05:33.295 ","End":"05:36.265","Text":"I\u0027m still going to get the same cross section."},{"Start":"05:36.265 ","End":"05:42.530","Text":"Z is going to go from 0."},{"Start":"05:43.350 ","End":"05:47.690","Text":"Here, z equals 0"},{"Start":"05:49.380 ","End":"05:56.830","Text":"and here z is equal to, what was it?"},{"Start":"05:58.580 ","End":"06:04.120","Text":"There it is, 4 minus y squared from here to here."},{"Start":"06:05.240 ","End":"06:13.285","Text":"That\u0027s y, and now z from 0-4 minus y squared,"},{"Start":"06:13.285 ","End":"06:16.060","Text":"then I need the divergence, which is this,"},{"Start":"06:16.060 ","End":"06:21.740","Text":"just 2 minus x, and then dzdydx."},{"Start":"06:22.290 ","End":"06:25.795","Text":"From this point on, it\u0027s just purely technical."},{"Start":"06:25.795 ","End":"06:28.480","Text":"Don\u0027t need to relate to the pictures."},{"Start":"06:28.480 ","End":"06:34.270","Text":"I personally like to pull constants as far front as they\u0027ll go."},{"Start":"06:34.270 ","End":"06:38.365","Text":"For example here, 2 minus x doesn\u0027t depend on z,"},{"Start":"06:38.365 ","End":"06:47.610","Text":"doesn\u0027t depend on y. I can actually write this as the integral from 0-3dx of 2 minus x,"},{"Start":"06:47.610 ","End":"06:49.340","Text":"and then the rest of it,"},{"Start":"06:49.340 ","End":"06:52.710","Text":"y from minus 2-2,"},{"Start":"06:52.710 ","End":"06:59.650","Text":"z from 0-4 minus y squared is just 1dzdydx."},{"Start":"07:02.990 ","End":"07:05.120","Text":"It\u0027s not something you have to do,"},{"Start":"07:05.120 ","End":"07:09.845","Text":"but I find that it comes out neater if you pull stuff to the front."},{"Start":"07:09.845 ","End":"07:12.530","Text":"I start from the inner integral,"},{"Start":"07:12.530 ","End":"07:17.490","Text":"the inner integral, the dz1."},{"Start":"07:17.490 ","End":"07:21.890","Text":"When you have the integral of 1, it\u0027s just the upper limit minus the lower limit."},{"Start":"07:21.890 ","End":"07:27.295","Text":"This whole thing comes out to be 4 minus y squared,"},{"Start":"07:27.295 ","End":"07:29.965","Text":"and then I have here dydx,"},{"Start":"07:29.965 ","End":"07:35.565","Text":"here the integral from minus 2-2,"},{"Start":"07:35.565 ","End":"07:38.865","Text":"and here the integral from 0-3."},{"Start":"07:38.865 ","End":"07:41.565","Text":"This is y, I emphasize,"},{"Start":"07:41.565 ","End":"07:48.820","Text":"this is x. I almost forgot to copy the 2 minus x here."},{"Start":"07:48.860 ","End":"07:52.575","Text":"Now we need to do this integral,"},{"Start":"07:52.575 ","End":"07:56.530","Text":"the dy integral, I mean."},{"Start":"07:57.350 ","End":"07:59.825","Text":"I\u0027ll do that at the side."},{"Start":"07:59.825 ","End":"08:07.385","Text":"What we get here is 4y minus y cubed over 3,"},{"Start":"08:07.385 ","End":"08:11.290","Text":"evaluated from minus 2-2."},{"Start":"08:12.650 ","End":"08:15.515","Text":"If I put in 2,"},{"Start":"08:15.515 ","End":"08:20.540","Text":"I\u0027ve got 2 times 4 is 8,"},{"Start":"08:20.540 ","End":"08:24.630","Text":"and 2 cubed over 3 is 8 over"},{"Start":"08:24.630 ","End":"08:31.439","Text":"3 is 2 and 2/3,"},{"Start":"08:31.439 ","End":"08:32.845","Text":"the same thing with minus 2."},{"Start":"08:32.845 ","End":"08:35.465","Text":"Just a second. Let me just compute this one."},{"Start":"08:35.465 ","End":"08:38.495","Text":"This comes out to be 5 and 1/3."},{"Start":"08:38.495 ","End":"08:40.775","Text":"Now, for minus 2,"},{"Start":"08:40.775 ","End":"08:42.620","Text":"everything just comes out negative,"},{"Start":"08:42.620 ","End":"08:44.119","Text":"they\u0027re all odd powers."},{"Start":"08:44.119 ","End":"08:48.550","Text":"It\u0027s going to be minus 5 and 1/3."},{"Start":"08:49.220 ","End":"08:53.215","Text":"I just have to multiply twice 5 and 1/3."},{"Start":"08:53.215 ","End":"09:00.075","Text":"That comes out to be 10 and 2/3."},{"Start":"09:00.075 ","End":"09:03.225","Text":"This whole bit comes out 10 and 2/3."},{"Start":"09:03.225 ","End":"09:09.960","Text":"Now I can pull this. What was it?"},{"Start":"09:09.960 ","End":"09:15.050","Text":"10 and 2/3 to the front and I like pulling things to the front."},{"Start":"09:15.050 ","End":"09:19.830","Text":"I\u0027ve got 10 and 2/3 times just the"},{"Start":"09:19.830 ","End":"09:27.520","Text":"integral from 0-3 of 2 minus x dx."},{"Start":"09:28.300 ","End":"09:33.770","Text":"I\u0027ll do this one at the side also."},{"Start":"09:33.770 ","End":"09:38.490","Text":"What I get is, let\u0027s see,"},{"Start":"09:38.490 ","End":"09:40.680","Text":"if 2 gives me 2x,"},{"Start":"09:40.680 ","End":"09:45.170","Text":"minus x gives me minus x squared over 2."},{"Start":"09:45.170 ","End":"09:49.880","Text":"I have to take this from 0-3,"},{"Start":"09:49.880 ","End":"09:54.535","Text":"0 gives me nothing but 3 gives me,"},{"Start":"09:54.535 ","End":"10:03.825","Text":"let\u0027s see, 6 minus 3 squared over 2 is 9 over 2 is 4 and 1/2,"},{"Start":"10:03.825 ","End":"10:07.210","Text":"which is 1 and 1/2."},{"Start":"10:07.210 ","End":"10:09.080","Text":"Now going back here,"},{"Start":"10:09.080 ","End":"10:15.255","Text":"I have 10 and 2/3 times 1 and 1/2."},{"Start":"10:15.255 ","End":"10:16.800","Text":"Let\u0027s do it in fractions."},{"Start":"10:16.800 ","End":"10:24.090","Text":"This is 32 over 3 times 3 over 2,"},{"Start":"10:24.090 ","End":"10:26.520","Text":"3 cancels with 3,"},{"Start":"10:26.520 ","End":"10:33.030","Text":"2 goes into 32 6 times and so the answer is 16,"},{"Start":"10:33.030 ","End":"10:38.950","Text":"and I\u0027ll highlight that and that\u0027s the answer."}],"ID":8840},{"Watched":false,"Name":"Exercise 6","Duration":"10m 33s","ChapterTopicVideoID":8773,"CourseChapterTopicPlaylistID":4966,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.285","Text":"In this exercise, we have to compute the following type to surface integral."},{"Start":"00:06.285 ","End":"00:14.350","Text":"N is the outward unit normal and the vector field F is given here."},{"Start":"00:14.360 ","End":"00:19.095","Text":"We have to compute it over the surface S,"},{"Start":"00:19.095 ","End":"00:23.895","Text":"which is the outer wrapper, if you like,"},{"Start":"00:23.895 ","End":"00:27.840","Text":"of the body bounded by,"},{"Start":"00:27.840 ","End":"00:30.420","Text":"well here the equations."},{"Start":"00:30.420 ","End":"00:33.770","Text":"Let me call the body B."},{"Start":"00:33.770 ","End":"00:39.080","Text":"B is like the 3D region and S is just the outer."},{"Start":"00:39.080 ","End":"00:42.335","Text":"It\u0027s a hemisphere and a disk at the bottom."},{"Start":"00:42.335 ","End":"00:44.600","Text":"Why did I say it\u0027s a hemisphere?"},{"Start":"00:44.600 ","End":"00:47.990","Text":"Because this is equivalent to,"},{"Start":"00:47.990 ","End":"00:52.700","Text":"if you square both sides and move stuff over,"},{"Start":"00:52.700 ","End":"00:56.195","Text":"you get x squared plus y squared,"},{"Start":"00:56.195 ","End":"01:00.295","Text":"plus z squared equals a squared,"},{"Start":"01:00.295 ","End":"01:05.150","Text":"or rather you get this from this by extracting z but only taking"},{"Start":"01:05.150 ","End":"01:09.630","Text":"the positive square root leaves us just with the top part."},{"Start":"01:09.630 ","End":"01:11.775","Text":"Anyway, this is the body,"},{"Start":"01:11.775 ","End":"01:16.060","Text":"and because we\u0027re in the chapter on divergence theorem,"},{"Start":"01:16.060 ","End":"01:19.225","Text":"you can guess what we\u0027re not going to compute the surface integral,"},{"Start":"01:19.225 ","End":"01:24.940","Text":"we\u0027re going to call in the theorem and compute the triple integral over B."},{"Start":"01:24.940 ","End":"01:31.855","Text":"In fact, the divergence theorem here says that the integral above the double integral"},{"Start":"01:31.855 ","End":"01:39.370","Text":"of F.ndS is equal to the triple integral."},{"Start":"01:39.370 ","End":"01:41.515","Text":"S is the border,"},{"Start":"01:41.515 ","End":"01:46.795","Text":"the boundary of body B of"},{"Start":"01:46.795 ","End":"01:56.270","Text":"the divergence dv for 3D integrals."},{"Start":"01:56.270 ","End":"01:59.225","Text":"Usually we label each of the functions."},{"Start":"01:59.225 ","End":"02:03.785","Text":"This one, this one and this one."},{"Start":"02:03.785 ","End":"02:05.870","Text":"Some like to use [inaudible] ,"},{"Start":"02:05.870 ","End":"02:09.755","Text":"I like to use p, q, and r,"},{"Start":"02:09.755 ","End":"02:18.440","Text":"and in the event the divergence of F would be this thing with respect to x,"},{"Start":"02:18.440 ","End":"02:23.330","Text":"partial derivative plus second one with respect to y,"},{"Start":"02:23.330 ","End":"02:24.575","Text":"plus the third one,"},{"Start":"02:24.575 ","End":"02:28.140","Text":"in our case r with respect to z."},{"Start":"02:28.140 ","End":"02:31.400","Text":"In our case, what do we get?"},{"Start":"02:31.400 ","End":"02:38.070","Text":"P with respect to x is just z squared,"},{"Start":"02:38.830 ","End":"02:44.810","Text":"q with respect to y. I\u0027d like to apologize,"},{"Start":"02:44.810 ","End":"02:46.310","Text":"I miss copied the question."},{"Start":"02:46.310 ","End":"02:50.610","Text":"The two should go here like this, sorry about that."},{"Start":"02:50.610 ","End":"02:55.100","Text":"The derivative with respect to y is x squared and the"},{"Start":"02:55.100 ","End":"03:00.595","Text":"derivative with respect to z will be just y squared."},{"Start":"03:00.595 ","End":"03:09.630","Text":"What we have to compute now is the triple integral over the upper half ball"},{"Start":"03:09.630 ","End":"03:15.230","Text":"B. I can change the order to suite me to write it as x"},{"Start":"03:15.230 ","End":"03:22.320","Text":"squared plus y squared plus z squared dv."},{"Start":"03:22.320 ","End":"03:27.800","Text":"The way I\u0027m going to do this is to use spherical coordinates, remember those."},{"Start":"03:27.800 ","End":"03:30.940","Text":"I\u0027ll just write the word spherical."},{"Start":"03:30.940 ","End":"03:35.615","Text":"We have 3 coordinates that we use in 3D."},{"Start":"03:35.615 ","End":"03:37.924","Text":"Cartesian cylindrical, spherical."},{"Start":"03:37.924 ","End":"03:42.560","Text":"We have a sphere shape or part of a sphere centered at the origin."},{"Start":"03:42.560 ","End":"03:47.150","Text":"Also, x squared plus y squared plus z squared is very good for spherical."},{"Start":"03:47.150 ","End":"03:49.550","Text":"I\u0027ll just write some of the equations."},{"Start":"03:49.550 ","End":"03:52.520","Text":"We need an iterative equation for x,"},{"Start":"03:52.520 ","End":"03:58.890","Text":"for y, for z, and for dv."},{"Start":"03:59.550 ","End":"04:04.000","Text":"Now actually, I\u0027m not even going to use this equation."},{"Start":"04:04.000 ","End":"04:06.190","Text":"I\u0027ll leave this if I need it,"},{"Start":"04:06.190 ","End":"04:08.305","Text":"I\u0027ll go back and tell you what it is."},{"Start":"04:08.305 ","End":"04:10.255","Text":"I do need dv,"},{"Start":"04:10.255 ","End":"04:17.815","Text":"which is r squared sine of"},{"Start":"04:17.815 ","End":"04:27.510","Text":"Pi d r d Pi d Theta."},{"Start":"04:27.510 ","End":"04:33.130","Text":"Usually there\u0027s an extra equation that goes with these that it turns out that x squared"},{"Start":"04:33.130 ","End":"04:39.380","Text":"plus y squared plus z squared equals r squared."},{"Start":"04:39.380 ","End":"04:44.470","Text":"What we get here, it is as follows."},{"Start":"04:44.470 ","End":"04:49.840","Text":"We have to convert the region b,"},{"Start":"04:49.840 ","End":"04:52.195","Text":"the upper half bull,"},{"Start":"04:52.195 ","End":"04:58.645","Text":"and see what happens to each of the 3 new variables,"},{"Start":"04:58.645 ","End":"05:00.980","Text":"Theta, where does it go from?"},{"Start":"05:00.980 ","End":"05:02.910","Text":"Pi? Where does it go?"},{"Start":"05:02.910 ","End":"05:04.770","Text":"R, where does it go?"},{"Start":"05:04.770 ","End":"05:11.070","Text":"Well, Theta is like the longitude."},{"Start":"05:11.070 ","End":"05:17.080","Text":"Actually we start from the x-axis and go around,"},{"Start":"05:17.420 ","End":"05:20.860","Text":"and we get one complete circle around."},{"Start":"05:20.860 ","End":"05:24.800","Text":"We go from 0 to 2Pi."},{"Start":"05:26.070 ","End":"05:30.020","Text":"Right here, 0 to 2Pi."},{"Start":"05:30.020 ","End":"05:34.000","Text":"Now, Pi is like latitude only."},{"Start":"05:34.000 ","End":"05:35.500","Text":"We don\u0027t start from the equator,"},{"Start":"05:35.500 ","End":"05:37.960","Text":"we start from the North Pole."},{"Start":"05:37.960 ","End":"05:41.350","Text":"Here it\u0027s Pi is 0,"},{"Start":"05:41.350 ","End":"05:46.900","Text":"and then it goes all the way"},{"Start":"05:46.900 ","End":"05:52.585","Text":"down to the negative z-axis, the South Pole."},{"Start":"05:52.585 ","End":"05:57.535","Text":"Here it\u0027s a 180 degrees or Pi,"},{"Start":"05:57.535 ","End":"06:01.915","Text":"but we\u0027re only getting half the way around 90 degrees around the world."},{"Start":"06:01.915 ","End":"06:11.530","Text":"This is Pi over 2 and Pi goes from 0 to Pi over 2."},{"Start":"06:14.420 ","End":"06:18.920","Text":"R is just the distance from the origin."},{"Start":"06:18.920 ","End":"06:24.755","Text":"It goes from the origin to the sphere."},{"Start":"06:24.755 ","End":"06:27.185","Text":"It goes from 0 to a."},{"Start":"06:27.185 ","End":"06:32.720","Text":"This is the r, 0 to a."},{"Start":"06:32.720 ","End":"06:41.405","Text":"Now, x squared plus y squared plus z squared from this formula is r squared and dv"},{"Start":"06:41.405 ","End":"06:44.975","Text":"from here is"},{"Start":"06:44.975 ","End":"06:52.430","Text":"r squared sine Pi"},{"Start":"06:52.430 ","End":"06:57.970","Text":"d r d Pi d Theta."},{"Start":"06:57.970 ","End":"07:00.665","Text":"Well, you weren\u0027t looking at, I filled an x, y, and z,"},{"Start":"07:00.665 ","End":"07:04.210","Text":"I didn\u0027t feel right about not leaving them out, but we didn\u0027t use them."},{"Start":"07:04.210 ","End":"07:10.990","Text":"Now back here, I\u0027d like to make some minor changes."},{"Start":"07:10.990 ","End":"07:15.340","Text":"What I\u0027d like to do is, well, two things."},{"Start":"07:15.340 ","End":"07:16.795","Text":"I\u0027d like to combine,"},{"Start":"07:16.795 ","End":"07:25.345","Text":"let me just write these and wanted to leave a bit more space here. There we go."},{"Start":"07:25.345 ","End":"07:28.990","Text":"I want to put sine Pi in front."},{"Start":"07:28.990 ","End":"07:33.250","Text":"I like to pull things to the front because this doesn\u0027t depend on r,"},{"Start":"07:33.250 ","End":"07:37.505","Text":"and then r squared with r squared is r to the fourth,"},{"Start":"07:37.505 ","End":"07:42.660","Text":"and then I just have d r d Pi d Theta."},{"Start":"07:42.660 ","End":"07:46.605","Text":"I need to still write the limits 0 to 2Pi,"},{"Start":"07:46.605 ","End":"07:52.305","Text":"0 to Pi over 2, 0 to a."},{"Start":"07:52.305 ","End":"07:57.550","Text":"We start from the innermost, here."},{"Start":"07:58.760 ","End":"08:01.725","Text":"I\u0027ll do this one on the side."},{"Start":"08:01.725 ","End":"08:06.605","Text":"What I get is r to the fifth, just doing this one,"},{"Start":"08:06.605 ","End":"08:14.465","Text":"r to the fifth over 5 from 0 to a,"},{"Start":"08:14.465 ","End":"08:23.280","Text":"and that is equal to a to the fifth over 5 minus 0,"},{"Start":"08:23.280 ","End":"08:26.265","Text":"and this is a constant."},{"Start":"08:26.265 ","End":"08:34.550","Text":"I can actually bring this right in front a to the fifth over 5."},{"Start":"08:34.550 ","End":"08:36.225","Text":"There\u0027s nothing left here."},{"Start":"08:36.225 ","End":"08:38.960","Text":"I have integral 0 to 2Pi,"},{"Start":"08:38.960 ","End":"08:43.520","Text":"integral from 0 to Pi over 2 of"},{"Start":"08:43.520 ","End":"08:50.015","Text":"just sine Pi d Pi d Theta."},{"Start":"08:50.015 ","End":"08:55.380","Text":"Now, the innermost one is this one."},{"Start":"08:56.090 ","End":"08:59.580","Text":"I\u0027ll do that at the side also."},{"Start":"08:59.580 ","End":"09:09.050","Text":"Over here what we get is the integral of sine Pi is minus cosine Pi,"},{"Start":"09:09.050 ","End":"09:17.370","Text":"and this we want to take from 0 to Pi over 2."},{"Start":"09:17.370 ","End":"09:24.435","Text":"This equals cosine of Pi over 2 is 0."},{"Start":"09:24.435 ","End":"09:26.555","Text":"I have minus 0,"},{"Start":"09:26.555 ","End":"09:34.570","Text":"minus and cosine of 0 is 1 minus minus 1,"},{"Start":"09:34.570 ","End":"09:39.485","Text":"so this is just equal to 1."},{"Start":"09:39.485 ","End":"09:49.535","Text":"If this is 1, then all I\u0027m left with now is a to the fifth"},{"Start":"09:49.535 ","End":"09:55.430","Text":"over 5 times the"},{"Start":"09:55.430 ","End":"10:02.630","Text":"integral from 0 to 2 Pi of just 1d Theta."},{"Start":"10:02.630 ","End":"10:10.385","Text":"Now the integral of one is always the upper limit minus the lower limit is 2 Pi."},{"Start":"10:10.385 ","End":"10:21.235","Text":"What I get is 2 Pi a to the fifth over 5,"},{"Start":"10:21.235 ","End":"10:25.655","Text":"because like I said, this bit here came out to be just 2Pi,"},{"Start":"10:25.655 ","End":"10:27.320","Text":"I multiplied by this,"},{"Start":"10:27.320 ","End":"10:34.080","Text":"and this is the answer to the question. We\u0027re done."}],"ID":8841},{"Watched":false,"Name":"Exercise 7","Duration":"17m 28s","ChapterTopicVideoID":8774,"CourseChapterTopicPlaylistID":4966,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:09.615","Text":"In this exercise, we\u0027re given an open surface described by y between 0 and 4,"},{"Start":"00:09.615 ","End":"00:12.450","Text":"and x squared plus z squared is 16."},{"Start":"00:12.450 ","End":"00:16.875","Text":"This turns out to be a cylinder without the bases."},{"Start":"00:16.875 ","End":"00:20.385","Text":"The picture here shows the bases I\u0027m talking about,"},{"Start":"00:20.385 ","End":"00:24.795","Text":"just the sides of the cylinder."},{"Start":"00:24.795 ","End":"00:27.270","Text":"Let\u0027s see it is labeled."},{"Start":"00:27.270 ","End":"00:28.694","Text":"This is the origin."},{"Start":"00:28.694 ","End":"00:37.200","Text":"From here to here would be 4 because 16 is 4 squared and in the x,"},{"Start":"00:37.200 ","End":"00:42.890","Text":"z plane, this is just a circle with radius 4."},{"Start":"00:42.890 ","End":"00:45.860","Text":"But of course, if we allow y to vary,"},{"Start":"00:45.860 ","End":"00:47.915","Text":"then we get a cylinder."},{"Start":"00:47.915 ","End":"00:55.585","Text":"This also would be 4 for y so everything is 4 here."},{"Start":"00:55.585 ","End":"00:59.630","Text":"We have to compute the flux, don\u0027t worry,"},{"Start":"00:59.630 ","End":"01:04.400","Text":"if you don\u0027t know physics because it\u0027s"},{"Start":"01:04.400 ","End":"01:11.690","Text":"explained here what we mean a flux of a vector field through S. In other words,"},{"Start":"01:11.690 ","End":"01:19.695","Text":"we just have to compute this type 2 surface integral F.ndS."},{"Start":"01:19.695 ","End":"01:23.940","Text":"As usual, n is the outward normal."},{"Start":"01:23.940 ","End":"01:29.290","Text":"Now, we\u0027re in the chapter on the divergence theorem"},{"Start":"01:29.290 ","End":"01:35.470","Text":"and so there\u0027s some difficulty computing the integral over S,"},{"Start":"01:35.470 ","End":"01:39.620","Text":"which as I said in our picture is S1."},{"Start":"01:40.730 ","End":"01:44.440","Text":"We want to use the divergence theorem and for that,"},{"Start":"01:44.440 ","End":"01:50.920","Text":"we need a closed surface that bounds a region and that\u0027s why we\u0027re going to"},{"Start":"01:50.920 ","End":"01:57.955","Text":"add the 2 caps or the bases of the cylinder."},{"Start":"01:57.955 ","End":"02:06.990","Text":"This one is going to be S2 and that one was S3 and if I add these 2,"},{"Start":"02:06.990 ","End":"02:10.920","Text":"then I can use divergence theorem but at the end,"},{"Start":"02:10.920 ","End":"02:12.670","Text":"I\u0027m going to have to subtract."},{"Start":"02:12.670 ","End":"02:17.800","Text":"Let me just show you more detail what the strategy is."},{"Start":"02:17.800 ","End":"02:26.430","Text":"The divergence theorem says that the double integral over S and you know what?"},{"Start":"02:26.430 ","End":"02:29.080","Text":"To be consistent with the picture,"},{"Start":"02:29.080 ","End":"02:33.730","Text":"let\u0027s call this S1 and let\u0027s take the integral over"},{"Start":"02:33.730 ","End":"02:41.760","Text":"S1 and I\u0027ll let S be the whole surface,"},{"Start":"02:41.760 ","End":"02:46.455","Text":"will be S1 plus S2 plus S3."},{"Start":"02:46.455 ","End":"02:48.040","Text":"Then the integral of S,"},{"Start":"02:48.040 ","End":"02:53.260","Text":"which is the total S surface of the cylinder"},{"Start":"02:53.260 ","End":"03:02.150","Text":"of F.ndS will equal"},{"Start":"03:02.150 ","End":"03:05.525","Text":"the integral over the region."},{"Start":"03:05.525 ","End":"03:10.774","Text":"Let\u0027s call the cylinder the solid body,"},{"Start":"03:10.774 ","End":"03:13.835","Text":"let\u0027s call it say, B for body."},{"Start":"03:13.835 ","End":"03:22.025","Text":"The triple integral over B of the divergence of the vector field"},{"Start":"03:22.025 ","End":"03:25.880","Text":"F dv and then what we\u0027ll"},{"Start":"03:25.880 ","End":"03:31.415","Text":"do afterwards is we\u0027ll say if the surface area is made up of 3 parts,"},{"Start":"03:31.415 ","End":"03:42.125","Text":"then what I can say afterwards is that the triple integral over B,"},{"Start":"03:42.125 ","End":"03:48.650","Text":"which is equal to this,"},{"Start":"03:48.650 ","End":"03:54.500","Text":"is the double integral over S1 plus double integral over"},{"Start":"03:54.500 ","End":"04:01.290","Text":"S2 plus surface integral rather over S3."},{"Start":"04:01.290 ","End":"04:09.355","Text":"Then I\u0027ll just subtract and what we want is just this bit so I\u0027ll take this."},{"Start":"04:09.355 ","End":"04:15.380","Text":"This one that we want S1 will be the triple integral over B"},{"Start":"04:15.380 ","End":"04:24.330","Text":"minus surface integral over S2 minus surface integral over S3."},{"Start":"04:24.330 ","End":"04:27.540","Text":"We\u0027ll do the 3 calculations,"},{"Start":"04:27.540 ","End":"04:34.415","Text":"a triple integral here to surface integrals over the 2 bases and then a subtraction."},{"Start":"04:34.415 ","End":"04:36.005","Text":"That\u0027s the strategy."},{"Start":"04:36.005 ","End":"04:39.360","Text":"Let\u0027s start with the triple integral."},{"Start":"04:39.710 ","End":"04:43.450","Text":"The justification for all these is that it turns out that this"},{"Start":"04:43.450 ","End":"04:46.720","Text":"triple integral is easy to compute and so are"},{"Start":"04:46.720 ","End":"04:53.430","Text":"these 2 surface integrals whereas this is difficult to compute on its own."},{"Start":"04:53.430 ","End":"05:00.660","Text":"It\u0027s worthwhile doing 3 computations and subtraction,"},{"Start":"05:00.660 ","End":"05:03.515","Text":"would still be easier than doing this directly."},{"Start":"05:03.515 ","End":"05:10.405","Text":"Now, the divergence of F will need if we\u0027re going to compute this bit."},{"Start":"05:10.405 ","End":"05:16.419","Text":"Usually, we just label these component functions,"},{"Start":"05:16.419 ","End":"05:19.410","Text":"let\u0027s say P, Q,"},{"Start":"05:19.410 ","End":"05:25.130","Text":"and R and the divergence of F,"},{"Start":"05:25.130 ","End":"05:30.770","Text":"the formula is just P with respect to x plus Q with respect to"},{"Start":"05:30.770 ","End":"05:37.765","Text":"y plus R with respect to z. P with respect to x is 0,"},{"Start":"05:37.765 ","End":"05:41.624","Text":"Q with respect to y is the constant 5,"},{"Start":"05:41.624 ","End":"05:45.840","Text":"R with respect to z is 0,"},{"Start":"05:45.840 ","End":"05:47.760","Text":"which is the constant 5."},{"Start":"05:47.760 ","End":"05:49.395","Text":"That\u0027s already good."},{"Start":"05:49.395 ","End":"06:00.630","Text":"This triple integral of the divergence of"},{"Start":"06:00.630 ","End":"06:06.000","Text":"F dv is just the triple"},{"Start":"06:06.000 ","End":"06:14.345","Text":"integral of 5 dv but I can take the 5 out front and it\u0027s just dv,"},{"Start":"06:14.345 ","End":"06:19.255","Text":"or better to write it as 1 dv over B."},{"Start":"06:19.255 ","End":"06:27.850","Text":"Now, the triple integral of 1 over a 3 dimensional body,"},{"Start":"06:27.850 ","End":"06:30.500","Text":"we know is just the volume of B,"},{"Start":"06:30.500 ","End":"06:38.280","Text":"so it\u0027s 5 times the volume of B. I\u0027m just writing"},{"Start":"06:38.280 ","End":"06:46.865","Text":"this mathematical volume of B but we know the formula for the volume of a cylinder."},{"Start":"06:46.865 ","End":"06:50.120","Text":"In general, for cylinder has a radius r,"},{"Start":"06:50.120 ","End":"06:53.165","Text":"its base and the height is h,"},{"Start":"06:53.165 ","End":"06:59.355","Text":"then the volume is Pi r squared h, is the formula."},{"Start":"06:59.355 ","End":"07:06.915","Text":"Here the radius is 4 and the height is 4 so we get"},{"Start":"07:06.915 ","End":"07:16.020","Text":"5 times Pi times 4 squared times 4."},{"Start":"07:16.020 ","End":"07:17.850","Text":"If you compute that well,"},{"Start":"07:17.850 ","End":"07:20.145","Text":"4 squared times 4 is 64."},{"Start":"07:20.145 ","End":"07:30.270","Text":"64 times 5 is 320 so this is 320 Pi."},{"Start":"07:30.270 ","End":"07:33.945","Text":"That\u0027s this first bit."},{"Start":"07:33.945 ","End":"07:44.120","Text":"We\u0027ve got 320 Pi and now let\u0027s go for the other 2 bits."},{"Start":"07:44.120 ","End":"07:50.275","Text":"Let\u0027s go for, let\u0027s say the S2."},{"Start":"07:50.275 ","End":"07:54.215","Text":"Well, before that, let me just give an overview."},{"Start":"07:54.215 ","End":"07:58.805","Text":"Both of these surfaces, S2 and S3,"},{"Start":"07:58.805 ","End":"08:03.380","Text":"can be described as functions where y is"},{"Start":"08:03.380 ","End":"08:08.345","Text":"a function of x and z happens to be a constant function."},{"Start":"08:08.345 ","End":"08:17.480","Text":"This is the surface y equals 4 and this is the surface y equals"},{"Start":"08:17.480 ","End":"08:22.160","Text":"0 and both of them have"},{"Start":"08:22.160 ","End":"08:28.700","Text":"a projection of this part here,"},{"Start":"08:28.700 ","End":"08:31.400","Text":"which I drew over here,"},{"Start":"08:31.400 ","End":"08:35.970","Text":"let\u0027s call it D. This is what D looks like."},{"Start":"08:36.490 ","End":"08:43.740","Text":"Let\u0027s see, this is z and this is x, z and x."},{"Start":"08:43.740 ","End":"08:48.500","Text":"This would be the domain for both these functions,"},{"Start":"08:48.500 ","End":"08:50.345","Text":"which are actually functions."},{"Start":"08:50.345 ","End":"08:54.330","Text":"Although they\u0027re constant, I look at them as functions of x and z."},{"Start":"08:54.330 ","End":"08:57.880","Text":"Because I have a formula that when I have"},{"Start":"08:57.880 ","End":"09:05.335","Text":"y equals g of say,"},{"Start":"09:05.335 ","End":"09:07.895","Text":"z and x or x and z,"},{"Start":"09:07.895 ","End":"09:21.775","Text":"then the surface integral"},{"Start":"09:21.775 ","End":"09:24.475","Text":"over a surface S,"},{"Start":"09:24.475 ","End":"09:30.610","Text":"in this case it would be S2 or S3 of F.n."},{"Start":"09:35.160 ","End":"09:45.250","Text":"Ds is equal to the regular double"},{"Start":"09:45.250 ","End":"09:55.180","Text":"integral over D of f dot,"},{"Start":"09:55.180 ","End":"10:04.090","Text":"and I\u0027ll use the angular bracket notation for vector for the i, j,"},{"Start":"10:04.090 ","End":"10:12.820","Text":"k of minus g_x,"},{"Start":"10:12.820 ","End":"10:19.195","Text":"1, minus g_z, where these are partial derivatives."},{"Start":"10:19.195 ","End":"10:22.435","Text":"This was just a regular da."},{"Start":"10:22.435 ","End":"10:25.765","Text":"In this case, da would be dx, dz or dz,"},{"Start":"10:25.765 ","End":"10:31.030","Text":"dx because that\u0027s where the domain is in the xc plane."},{"Start":"10:31.030 ","End":"10:35.260","Text":"But not quite, there\u0027s actually 2 separate formulas,"},{"Start":"10:35.260 ","End":"10:42.550","Text":"1 for when the normal is pointing in the positive y-direction."},{"Start":"10:42.550 ","End":"10:46.240","Text":"For example, in the case of S3,"},{"Start":"10:46.240 ","End":"10:49.210","Text":"this normal would go in the direction of"},{"Start":"10:49.210 ","End":"10:53.500","Text":"the positive y or at least have a component in that direction,"},{"Start":"10:53.500 ","End":"10:57.465","Text":"at least go partially upwards."},{"Start":"10:57.465 ","End":"10:59.100","Text":"Then in the case of S2,"},{"Start":"10:59.100 ","End":"11:02.235","Text":"it would be in the negative y-direction."},{"Start":"11:02.235 ","End":"11:08.450","Text":"Actually, I can already say that in our case,"},{"Start":"11:08.450 ","End":"11:13.675","Text":"for S3 I\u0027ve got the positive y-direction,"},{"Start":"11:13.675 ","End":"11:23.860","Text":"but for S2 this relates to the normal."},{"Start":"11:23.860 ","End":"11:28.030","Text":"The normal goes in the negative, it goes downwards."},{"Start":"11:28.030 ","End":"11:29.680","Text":"Here it goes totally downwards,"},{"Start":"11:29.680 ","End":"11:39.625","Text":"but even if it went partially downwards it has to have a negative y component."},{"Start":"11:39.625 ","End":"11:50.260","Text":"In that case, the same integral will equal double integral of D of F dot,"},{"Start":"11:50.260 ","End":"11:53.170","Text":"the reverse of this vector the negative,"},{"Start":"11:53.170 ","End":"11:57.970","Text":"it would be plus g_x, minus 1,"},{"Start":"11:57.970 ","End":"12:02.290","Text":"and then plus just g_z plane,"},{"Start":"12:02.290 ","End":"12:03.910","Text":"da like I said,"},{"Start":"12:03.910 ","End":"12:09.980","Text":"that\u0027s when the normal has a negative y component."},{"Start":"12:10.590 ","End":"12:15.280","Text":"We have 2 different g\u0027s in this case."},{"Start":"12:15.280 ","End":"12:16.885","Text":"In the first case,"},{"Start":"12:16.885 ","End":"12:19.700","Text":"we\u0027ll get that g of x."},{"Start":"12:20.580 ","End":"12:27.265","Text":"Here g will equal for S3 is equal to 4,"},{"Start":"12:27.265 ","End":"12:32.980","Text":"and for S2 g will equal the constant function 0."},{"Start":"12:32.980 ","End":"12:40.735","Text":"But in both cases, it\u0027s the same D. Let\u0027s get some space and continue."},{"Start":"12:40.735 ","End":"12:42.760","Text":"Now in both cases,"},{"Start":"12:42.760 ","End":"12:46.210","Text":"g is a constant so all the partial derivatives are 0."},{"Start":"12:46.210 ","End":"12:48.985","Text":"This will be 0, this will be 0,"},{"Start":"12:48.985 ","End":"12:52.880","Text":"this will be 0, and this will be 0."},{"Start":"12:53.370 ","End":"13:00.655","Text":"I\u0027ve lost my f, I\u0027ll just maybe write it again over here,"},{"Start":"13:00.655 ","End":"13:02.155","Text":"copy it from here."},{"Start":"13:02.155 ","End":"13:05.665","Text":"That f is equal to,"},{"Start":"13:05.665 ","End":"13:08.875","Text":"and I\u0027ll write it with the angular bracket notation,"},{"Start":"13:08.875 ","End":"13:20.410","Text":"z squared, 5y, x^5."},{"Start":"13:20.410 ","End":"13:25.420","Text":"Now I need just do the dot product of this with 0."},{"Start":"13:25.420 ","End":"13:28.510","Text":"Well, I\u0027ll just copy this thing over again."},{"Start":"13:28.510 ","End":"13:30.340","Text":"This becomes 0, 1,"},{"Start":"13:30.340 ","End":"13:36.475","Text":"0 and here I\u0027ve got 0, minus 1, 0."},{"Start":"13:36.475 ","End":"13:41.900","Text":"For the case of S3,"},{"Start":"13:42.450 ","End":"13:53.725","Text":"what I get is the double integral over the radius for disk D of,"},{"Start":"13:53.725 ","End":"13:58.015","Text":"I need to take this dot product with this,"},{"Start":"13:58.015 ","End":"14:02.240","Text":"it just leaves me the middle component, 5yda."},{"Start":"14:05.370 ","End":"14:13.420","Text":"In the case of S2,"},{"Start":"14:13.420 ","End":"14:21.460","Text":"what I have is the double integral over the same D. Because of the minus 1,"},{"Start":"14:21.460 ","End":"14:24.710","Text":"I just get minus 5yda."},{"Start":"14:28.230 ","End":"14:36.100","Text":"But we don\u0027t leave y as is because y is g of x,"},{"Start":"14:36.100 ","End":"14:41.800","Text":"so I need to substitute what it is in each case."},{"Start":"14:41.800 ","End":"14:45.115","Text":"For S3 y equals 4."},{"Start":"14:45.115 ","End":"14:53.530","Text":"This actually equals the double integral over D,"},{"Start":"14:53.530 ","End":"14:56.990","Text":"y is 4, of 20da."},{"Start":"14:57.930 ","End":"15:01.450","Text":"Whereas in the case of S2,"},{"Start":"15:01.450 ","End":"15:08.170","Text":"y is 0 so it\u0027s just the double integral over D of 0da."},{"Start":"15:08.170 ","End":"15:10.390","Text":"That 1 we can do already,"},{"Start":"15:10.390 ","End":"15:12.950","Text":"that\u0027s equal to 0."},{"Start":"15:13.410 ","End":"15:18.550","Text":"Actually, this is not hard to do either because if"},{"Start":"15:18.550 ","End":"15:23.094","Text":"I take the 20 outside the integral sign,"},{"Start":"15:23.094 ","End":"15:29.575","Text":"I get 20 times double integral over D of 1da."},{"Start":"15:29.575 ","End":"15:32.845","Text":"The integral of 1da,"},{"Start":"15:32.845 ","End":"15:36.040","Text":"let me just continue over here where I have some room,"},{"Start":"15:36.040 ","End":"15:44.845","Text":"is 20 times double integral of 1 is just the area of"},{"Start":"15:44.845 ","End":"15:49.450","Text":"D. Now the area of"},{"Start":"15:49.450 ","End":"15:57.020","Text":"D using the formula for the area of a circle being Pi r squared."},{"Start":"16:00.960 ","End":"16:04.450","Text":"In our case, the radius is 4,"},{"Start":"16:04.450 ","End":"16:11.965","Text":"is 20 times Pi times 4 squared,"},{"Start":"16:11.965 ","End":"16:20.990","Text":"which equals 20 times 4 squared is 20 times 16, 320Pi."},{"Start":"16:22.890 ","End":"16:28.130","Text":"Let\u0027s go back to this line which was our summary line."},{"Start":"16:29.070 ","End":"16:35.440","Text":"This 320Pi was here and I\u0027ve written it."},{"Start":"16:35.440 ","End":"16:43.585","Text":"This 0 is the surface integral of S2 that was 0."},{"Start":"16:43.585 ","End":"16:50.360","Text":"This surface integral of S3 came out to be 320Pi."},{"Start":"16:52.170 ","End":"16:56.110","Text":"I\u0027ve got this minus this,"},{"Start":"16:56.110 ","End":"16:58.825","Text":"minus this, sorry it\u0027s a bit crowded"},{"Start":"16:58.825 ","End":"17:03.500","Text":"but altogether this comes out to be 0."},{"Start":"17:04.050 ","End":"17:08.030","Text":"In fact, this is our final answer."},{"Start":"17:09.000 ","End":"17:18.970","Text":"The surface integral over the cylinder just the side,"},{"Start":"17:18.970 ","End":"17:28.760","Text":"comes out to be a neat 0 and that\u0027s our final answer and we are done."}],"ID":8842},{"Watched":false,"Name":"Exercise 8","Duration":"18m 59s","ChapterTopicVideoID":8775,"CourseChapterTopicPlaylistID":4966,"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":"00:10.155","Text":"F.ndS over the surface S,"},{"Start":"00:10.155 ","End":"00:13.740","Text":"n is as usual the outward unit normal on"},{"Start":"00:13.740 ","End":"00:22.245","Text":"the surface S. 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":"00:38.295","Text":"We\u0027re given that S is the open surface defined by this function z of x and y."},{"Start":"00:38.295 ","End":"00:44.240","Text":"It\u0027s a paraboloid, an upside down paraboloid and z"},{"Start":"00:44.240 ","End":"00:50.015","Text":"bigger or equal to 0 limits it to the xy-plane and above."},{"Start":"00:50.015 ","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 xy-plane."},{"Start":"01:13.700 ","End":"01:15.934","Text":"If I let z equals 0,"},{"Start":"01:15.934 ","End":"01:21.220","Text":"what I would get would be that the part below,"},{"Start":"01:21.220 ","End":"01:24.305","Text":"I\u0027ll give it a name D for disk."},{"Start":"01:24.305 ","End":"01:26.735","Text":"It\u0027s a disk of radius 2,"},{"Start":"01:26.735 ","End":"01:33.275","Text":"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":"01:44.160","Text":"but we want the interior also so less than or equal to 4 and 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 xy is the origin,"},{"Start":"01:52.550 ","End":"01:54.530","Text":"then z is equal to 4,"},{"Start":"01:54.530 ","End":"01:56.075","Text":"4 minus 0 minus 0."},{"Start":"01:56.075 ","End":"01:59.360","Text":"This is the right picture and I\u0027d"},{"Start":"01:59.360 ","End":"02:03.550","Text":"actually like to also put it in an extra picture of just this D."},{"Start":"02:03.550 ","End":"02:12.680","Text":"Here\u0027s the disk D in the xy-plane and 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":"02:24.230","Text":"The reason I\u0027m concerned with this is I want to use"},{"Start":"02:24.230 ","End":"02:27.920","Text":"the divergence theorem and I want to take the whole surface of"},{"Start":"02:27.920 ","End":"02:34.920","Text":"a 3D body which will be the solid paraboloid."},{"Start":"02:36.050 ","End":"02:42.170","Text":"What I\u0027m going to do is use the divergence theorem and say that the double"},{"Start":"02:42.170 ","End":"02:49.440","Text":"integral over S plus D,"},{"Start":"02:49.440 ","End":"02:56.320","Text":"you could use the symbol plus sometimes I\u0027d use the union symbol."},{"Start":"02:56.320 ","End":"02:58.455","Text":"Perhaps better to use."},{"Start":"02:58.455 ","End":"03:02.250","Text":"This is not a u, this is the union."},{"Start":"03:02.250 ","End":"03:09.740","Text":"But they don\u0027t have any overlap except at the circle itself and that\u0027s why the"},{"Start":"03:09.740 ","End":"03:17.270","Text":"double integral will be the sum of the integral over S plus the integral over D. Well,"},{"Start":"03:17.270 ","End":"03:19.610","Text":"we\u0027ll get to that but first of all,"},{"Start":"03:19.610 ","End":"03:28.240","Text":"this is going to of F.ndS will be,"},{"Start":"03:28.240 ","End":"03:33.930","Text":"by the divergence theorem, the triple integral."},{"Start":"03:34.000 ","End":"03:39.380","Text":"Let\u0027s give the 3D body,"},{"Start":"03:39.380 ","End":"03:45.275","Text":"meaning the volume trapped inside between S and D,"},{"Start":"03:45.275 ","End":"03:49.600","Text":"I\u0027ll call that B for body."},{"Start":"03:49.700 ","End":"03:58.810","Text":"This solid object B of the divergence of FdV."},{"Start":"03:59.830 ","End":"04:03.830","Text":"That\u0027s what the divergence theorem will give us."},{"Start":"04:03.830 ","End":"04:08.200","Text":"The reason that this will help us is that,"},{"Start":"04:08.200 ","End":"04:10.380","Text":"I\u0027ll just show you at the side,"},{"Start":"04:10.380 ","End":"04:20.530","Text":"this surface integral will equal the surface integral over S"},{"Start":"04:20.530 ","End":"04:28.030","Text":"of whatever it is plus the surface integral over the disk D. Because"},{"Start":"04:28.030 ","End":"04:31.630","Text":"when we have this union since they don\u0027t really overlap except"},{"Start":"04:31.630 ","End":"04:36.585","Text":"on the circle itself 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":"04:59.250","Text":"that this will be this minus this at the end."},{"Start":"04:59.250 ","End":"05:02.880","Text":"I\u0027ll be able to say, I\u0027ll just write that down,"},{"Start":"05:02.880 ","End":"05:07.105","Text":"that the integral that we want over S"},{"Start":"05:07.105 ","End":"05:11.825","Text":"will be the triple integral over B, it\u0027s 1 computation,"},{"Start":"05:11.825 ","End":"05:17.890","Text":"minus the surface integral over the cap"},{"Start":"05:17.890 ","End":"05:24.380","Text":"at the bottom D. Now these 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":"05:40.050","Text":"Let\u0027s get started."},{"Start":"05:40.050 ","End":"05:43.750","Text":"I\u0027ll do the triple integral first."},{"Start":"05:43.910 ","End":"05:48.380","Text":"The first step in computing the triple integral is to first see what"},{"Start":"05:48.380 ","End":"05:52.595","Text":"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 1st component P,"},{"Start":"06:01.875 ","End":"06:06.855","Text":"the 2nd one I\u0027ll call this one here Q,"},{"Start":"06:06.855 ","End":"06:11.550","Text":"that\u0027s just what comes before the i,"},{"Start":"06:11.550 ","End":"06:20.120","Text":"what comes before the j and R is what goes with the k. In general,"},{"Start":"06:20.120 ","End":"06:26.640","Text":"the divergence of such an F is just the 1st component,"},{"Start":"06:26.640 ","End":"06:29.260","Text":"in our case P with respect to x,"},{"Start":"06:29.260 ","End":"06:32.990","Text":"plus 2nd component with respect to y,"},{"Start":"06:32.990 ","End":"06:36.925","Text":"plus the 3rd 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":"06:49.275","Text":"First of all, let\u0027s start with P. With respect to x,"},{"Start":"06:49.275 ","End":"06:52.280","Text":"since y is a constant,"},{"Start":"06:52.280 ","End":"06:57.440","Text":"this would be just differentiating x squared and getting 2x and the rest stays."},{"Start":"06:57.440 ","End":"07:04.590","Text":"This is 2xy over 1 plus y squared,"},{"Start":"07:04.590 ","End":"07:07.845","Text":"and 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":"07:23.240","Text":"The derivative of arctangent y is"},{"Start":"07:23.240 ","End":"07:32.055","Text":"1 over 1 plus y squared and the 2x which is a constant just sticks."},{"Start":"07:32.055 ","End":"07:36.090","Text":"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":"07:50.300","Text":"The denominator is a constant as far as z goes and I can just leave"},{"Start":"07:50.300 ","End":"07:56.970","Text":"it there and 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 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 a constant, goes to 0."},{"Start":"08:18.680 ","End":"08:21.200","Text":"This is the divergence."},{"Start":"08:21.200 ","End":"08:27.780","Text":"Notice that all 3 denominators here are the same."},{"Start":"08:29.240 ","End":"08:32.280","Text":"Let\u0027s simplify."},{"Start":"08:32.280 ","End":"08:38.220","Text":"We have the denominator 1 plus y squared and then"},{"Start":"08:38.220 ","End":"08:48.460","Text":"the numerator is 2xy plus 2x and let\u0027s multiply this out."},{"Start":"08:48.530 ","End":"08:52.500","Text":"I just noticed, this here is a minus,"},{"Start":"08:52.500 ","End":"08:55.930","Text":"fixed it just in time."},{"Start":"08:56.240 ","End":"09:02.925","Text":"I get minus 2x, minus 2xy."},{"Start":"09:02.925 ","End":"09:08.180","Text":"Look, this cancels with this and this cancels"},{"Start":"09:08.180 ","End":"09:12.780","Text":"with this and so this just comes out to be 0."},{"Start":"09:12.780 ","End":"09:14.595","Text":"Isn\u0027t that great."},{"Start":"09:14.595 ","End":"09:18.390","Text":"I don\u0027t really need to compute the integral."},{"Start":"09:18.390 ","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":"09:37.535","Text":"Next we\u0027re going to compute this part here."},{"Start":"09:37.535 ","End":"09:42.020","Text":"When I have it, 0 minus it will give me the answer I want."},{"Start":"09:42.020 ","End":"09:44.790","Text":"This is the bit that I want."},{"Start":"09:46.700 ","End":"09:52.520","Text":"Get some space here and just copy from here."},{"Start":"09:52.520 ","End":"09:59.125","Text":"I want the double integral but just over D of,"},{"Start":"09:59.125 ","End":"10:00.710","Text":"there should be a dot there,"},{"Start":"10:00.710 ","End":"10:06.570","Text":"it\u0027s a dot product, F.ndS."},{"Start":"10:09.880 ","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":"10:25.410","Text":"If I take a point on the disk here,"},{"Start":"10:25.430 ","End":"10:29.639","Text":"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 1 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":"10:53.330","Text":"Even if it went downward at an angle as long as it\u0027s downward."},{"Start":"10:53.330 ","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":"11:12.720 ","End":"11:18.505","Text":"Minus g with respect to x."},{"Start":"11:18.505 ","End":"11:25.570","Text":"No, it is gxi plus"},{"Start":"11:25.570 ","End":"11:34.575","Text":"gyj minus k dA."},{"Start":"11:34.575 ","End":"11:41.485","Text":"Now, I need to explain several things I want to say."},{"Start":"11:41.485 ","End":"11:46.195","Text":"First of all, usually you see it not so much in this form."},{"Start":"11:46.195 ","End":"11:52.270","Text":"As with the brackets form gx,"},{"Start":"11:52.270 ","End":"12:00.670","Text":"gy, -1."},{"Start":"12:00.670 ","End":"12:04.795","Text":"I\u0027ve just used the i, j, k to be consistent with the i, j, k here."},{"Start":"12:04.795 ","End":"12:08.170","Text":"The second thing, I haven\u0027t even told you what g is,"},{"Start":"12:08.170 ","End":"12:10.330","Text":"and 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":"12:20.830","Text":"the d part, the base,"},{"Start":"12:20.830 ","End":"12:24.805","Text":"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":"12:30.970","Text":"which is g of x,"},{"Start":"12:30.970 ","End":"12:36.340","Text":"y 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":"12:51.160","Text":"which is d itself."},{"Start":"12:51.160 ","End":"12:54.340","Text":"Well, technically you could say this is not the same thing,"},{"Start":"12:54.340 ","End":"12:56.450","Text":"this is the disk,"},{"Start":"12:57.480 ","End":"13:01.060","Text":"and this is the same disk only in 2 space."},{"Start":"13:01.060 ","End":"13:03.955","Text":"Maybe I\u0027ll put this one, I don\u0027t know,"},{"Start":"13:03.955 ","End":"13:07.960","Text":"D_0, just to be more precise."},{"Start":"13:07.960 ","End":"13:11.500","Text":"It\u0027s the same as d but just as considered only in"},{"Start":"13:11.500 ","End":"13:17.860","Text":"2-dimensional space without the naught in three-dimensional space."},{"Start":"13:17.860 ","End":"13:20.390","Text":"It\u0027s a technicality."},{"Start":"13:22.890 ","End":"13:27.070","Text":"That\u0027s basically it."},{"Start":"13:27.070 ","End":"13:33.505","Text":"We\u0027ll 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":"13:46.545","Text":"Of course, before I do the dot-product I have to compute the partial derivatives."},{"Start":"13:46.545 ","End":"13:49.335","Text":"Well, if g of x y is 0,"},{"Start":"13:49.335 ","End":"13:59.245","Text":"then gx is also equal to 0 and gy is also equal to 0."},{"Start":"13:59.245 ","End":"14:02.094","Text":"Just one 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":"14:15.250","Text":"then there\u0027d be a minus here,"},{"Start":"14:15.250 ","End":"14:17.920","Text":"a minus here, and a plus here."},{"Start":"14:17.920 ","End":"14:20.320","Text":"I\u0027m just putting it in context,"},{"Start":"14:20.320 ","End":"14:22.420","Text":"but since ours faces down,"},{"Start":"14:22.420 ","End":"14:25.210","Text":"then the plus, plus,"},{"Start":"14:25.210 ","End":"14:28.670","Text":"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 if it\u0027s scrolled up,"},{"Start":"14:43.015 ","End":"14:52.900","Text":"but I remember that it was something i plus something j plus something k,"},{"Start":"14:52.900 ","End":"14:56.199","Text":"afterwards, we\u0027ll scroll back up and see what it was,"},{"Start":"14:56.199 ","End":"15:02.875","Text":"dot and this vector gx we said is 0,"},{"Start":"15:02.875 ","End":"15:11.890","Text":"gy is 0, so all I\u0027m left with is minus k dA."},{"Start":"15:11.890 ","End":"15:17.575","Text":"Now, when I do the dot product with minus k,"},{"Start":"15:17.575 ","End":"15:22.855","Text":"all I\u0027m going to get is the double integral over"},{"Start":"15:22.855 ","End":"15:32.890","Text":"D_0 of just minus r dA."},{"Start":"15:32.890 ","End":"15:35.950","Text":"I went and peeked 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":"15:49.450","Text":"It was equal to 2xz,"},{"Start":"15:49.450 ","End":"15:59.020","Text":"1 plus y plus 1 plus y"},{"Start":"15:59.020 ","End":"16:04.330","Text":"squared 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":"16:15.379","Text":"Notice though that in our case,"},{"Start":"16:17.040 ","End":"16:23.335","Text":"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":"16:31.810","Text":"Clearly, the disc is in the x,"},{"Start":"16:31.810 ","End":"16:34.450","Text":"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, that cancels."},{"Start":"17:00.865 ","End":"17:02.650","Text":"If I go back here,"},{"Start":"17:02.650 ","End":"17:06.625","Text":"let me just clear a bit more space,"},{"Start":"17:06.625 ","End":"17:12.220","Text":"then we have the double integral over the disk,"},{"Start":"17:12.220 ","End":"17:16.060","Text":"D_0, 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":"17:42.460","Text":"This is just the area of D_0."},{"Start":"17:42.460 ","End":"17:49.540","Text":"But D_0 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 of the solid body B."},{"Start":"18:23.305 ","End":"18:30.460","Text":"This surface integral over the base D. We\u0027ve just computed it as"},{"Start":"18:30.460 ","End":"18:39.340","Text":"4Pi and 0 minus 4Pi is minus 4Pi."},{"Start":"18:39.340 ","End":"18:42.055","Text":"That\u0027s the bit that we wanted,"},{"Start":"18:42.055 ","End":"18:49.210","Text":"the surface integral over just S. The final answer"},{"Start":"18:49.210 ","End":"19:00.140","Text":"is minus 4Pi, and we\u0027re done."}],"ID":8843}],"Thumbnail":null,"ID":4966}]

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1.1

1