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[{"Name":"Stokes Theorem","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Stokes Theorem","Duration":"6m 50s","ChapterTopicVideoID":8752,"CourseChapterTopicPlaylistID":4973,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8752.jpeg","UploadDate":"2020-02-26T12:30:12.3030000","DurationForVideoObject":"PT6M50S","Description":null,"MetaTitle":"Stokes Theorem: Video + Workbook | Proprep","MetaDescription":"Stokes Theorem - Stokes 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/stokes-theorem/stokes-theorem/vid8820","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"We\u0027re still in the subject of surface integrals and now"},{"Start":"00:03.450 ","End":"00:06.840","Text":"we come to the Kelvin Stokes theorem,"},{"Start":"00:06.840 ","End":"00:09.390","Text":"also known as the Curl theorem,"},{"Start":"00:09.390 ","End":"00:13.620","Text":"and sometimes it\u0027s just called Kelvin."},{"Start":"00:13.620 ","End":"00:16.900","Text":"But then you\u0027d have to put an apostrophe here."},{"Start":"00:17.300 ","End":"00:23.685","Text":"Before I get into this I need to say something about orientation."},{"Start":"00:23.685 ","End":"00:26.370","Text":"I brought in a picture."},{"Start":"00:26.370 ","End":"00:32.840","Text":"What we have here is a surface which is oriented and we know what that is,"},{"Start":"00:32.840 ","End":"00:35.765","Text":"it has a smooth unit normal."},{"Start":"00:35.765 ","End":"00:40.275","Text":"For one thing it\u0027s a two-sided surface."},{"Start":"00:40.275 ","End":"00:42.980","Text":"But I\u0027m going to assume it\u0027s not closed,"},{"Start":"00:42.980 ","End":"00:49.295","Text":"which means that it does have a boundary and the boundary would be described by"},{"Start":"00:49.295 ","End":"00:58.445","Text":"a curve C. Now we also know about orientation of curves or direction."},{"Start":"00:58.445 ","End":"01:00.080","Text":"A curve could go in both ways."},{"Start":"01:00.080 ","End":"01:02.420","Text":"Here there are arrows this way,"},{"Start":"01:02.420 ","End":"01:04.460","Text":"and we\u0027ll see why in a moment."},{"Start":"01:04.460 ","End":"01:09.410","Text":"But a curve could go in any of 2 directions, a closed curve."},{"Start":"01:10.000 ","End":"01:16.760","Text":"Now there is a way of associating an orientation on S with an orientation on the curve"},{"Start":"01:16.760 ","End":"01:25.120","Text":"C. We imagine that where a normal vector walking along the curve."},{"Start":"01:25.120 ","End":"01:30.845","Text":"Let\u0027s say the stick figures pointing in the same direction as the normal vector."},{"Start":"01:30.845 ","End":"01:39.470","Text":"The idea is to walk along the curve in such a way that the surface is on your left."},{"Start":"01:39.470 ","End":"01:45.530","Text":"Then we say that this has a positive orientation with"},{"Start":"01:45.530 ","End":"01:51.560","Text":"respect to S or S induces an orientation on C, that\u0027s informal."},{"Start":"01:51.560 ","End":"01:53.345","Text":"I think that will do."},{"Start":"01:53.345 ","End":"02:02.235","Text":"But I would like to share with you something I found on a website called Math Insight,"},{"Start":"02:02.235 ","End":"02:10.760","Text":"and here we are on Math Insight and they\u0027ve created an interactive explanation."},{"Start":"02:10.760 ","End":"02:14.509","Text":"For one thing I can adjust the surface,"},{"Start":"02:14.509 ","End":"02:17.980","Text":"so let\u0027s just leave it like that and you can also warp it,"},{"Start":"02:17.980 ","End":"02:22.680","Text":"and just so in case you\u0027re thinking it\u0027s a circle, it\u0027s not."},{"Start":"02:22.730 ","End":"02:27.120","Text":"Also you can play with the normal vector."},{"Start":"02:27.120 ","End":"02:32.050","Text":"We can move it around."},{"Start":"02:32.470 ","End":"02:38.255","Text":"You could think that if you have a normal vector at any given place,"},{"Start":"02:38.255 ","End":"02:42.230","Text":"then it induces an orientation."},{"Start":"02:42.230 ","End":"02:45.550","Text":"This would be the direction you would turn in,"},{"Start":"02:45.550 ","End":"02:51.420","Text":"in order to get the effect of a right-hand screw."},{"Start":"02:54.890 ","End":"02:59.220","Text":"If you twist it, rotate it this way it will move in this direction."},{"Start":"02:59.220 ","End":"03:02.635","Text":"At each point we get these little orientations"},{"Start":"03:02.635 ","End":"03:06.800","Text":"and together they give us a direction along the curve."},{"Start":"03:06.800 ","End":"03:13.210","Text":"I guess this is similar to what I also said about if you\u0027re walking along the curve"},{"Start":"03:13.210 ","End":"03:19.405","Text":"in the direction with your head going towards the normal,"},{"Start":"03:19.405 ","End":"03:21.370","Text":"like you would be going this way,"},{"Start":"03:21.370 ","End":"03:24.110","Text":"and the surface is on your left,"},{"Start":"03:24.110 ","End":"03:27.305","Text":"then that would be called"},{"Start":"03:27.305 ","End":"03:33.125","Text":"the positive orientation of this curve with respect to this oriented surface."},{"Start":"03:33.125 ","End":"03:38.720","Text":"Just one more way of looking at positive orientation."},{"Start":"03:38.720 ","End":"03:41.630","Text":"We can\u0027t really talk about clockwise and"},{"Start":"03:41.630 ","End":"03:47.645","Text":"counterclockwise because it\u0027s in space and you could be looking at it from any direction,"},{"Start":"03:47.645 ","End":"03:52.070","Text":"but if you look at it from the direction of one of the normals,"},{"Start":"03:52.070 ","End":"03:55.570","Text":"sometimes you draw an eye, like this."},{"Start":"03:55.570 ","End":"03:57.720","Text":"I didn\u0027t do a very good job of it,"},{"Start":"03:57.720 ","End":"04:01.390","Text":"but this is an eye looking at it,"},{"Start":"04:01.390 ","End":"04:08.615","Text":"then the positive direction would be the counterclockwise direction."},{"Start":"04:08.615 ","End":"04:10.865","Text":"That\u0027s enough about orientation."},{"Start":"04:10.865 ","End":"04:13.330","Text":"Now let me show you the theorem."},{"Start":"04:13.330 ","End":"04:16.545","Text":"For this theorem, I need 3 ingredients."},{"Start":"04:16.545 ","End":"04:21.775","Text":"Ingredient number 1 would be a vector field,"},{"Start":"04:21.775 ","End":"04:26.730","Text":"which we have discussed quite at length."},{"Start":"04:27.940 ","End":"04:30.610","Text":"This would be in 3D."},{"Start":"04:30.610 ","End":"04:32.595","Text":"We\u0027re in 3D now."},{"Start":"04:32.595 ","End":"04:39.755","Text":"Number 2, would be a surface S and I\u0027ll give some further conditions in a moment."},{"Start":"04:39.755 ","End":"04:45.035","Text":"Number 3, it will be a curve C,"},{"Start":"04:45.035 ","End":"04:48.200","Text":"which is going to be the boundary"},{"Start":"04:48.200 ","End":"04:55.940","Text":"of S. Under certain conditions,"},{"Start":"04:55.940 ","End":"05:01.950","Text":"which I\u0027ll specify, what we can say is that the"},{"Start":"05:01.950 ","End":"05:04.500","Text":"line integral of"},{"Start":"05:04.500 ","End":"05:13.470","Text":"F.dr along C,"},{"Start":"05:13.470 ","End":"05:17.440","Text":"it\u0027s a closed curve, I guess so I\u0027ll put a circle here,"},{"Start":"05:18.020 ","End":"05:28.440","Text":"is equal to the double integral along the surface S of,"},{"Start":"05:28.440 ","End":"05:32.105","Text":"I\u0027ll write it as the word curl."},{"Start":"05:32.105 ","End":"05:38.060","Text":"Even though we sometimes write curl as del, cross."},{"Start":"05:38.060 ","End":"05:47.185","Text":"This is a vector and I want curlF.ds."},{"Start":"05:47.185 ","End":"05:50.675","Text":"This is what we learnt the previous lesson."},{"Start":"05:50.675 ","End":"05:54.755","Text":"Remember the curl of a vector field is also a vector field,"},{"Start":"05:54.755 ","End":"05:58.470","Text":"and this is what we learnt in the last lesson."},{"Start":"05:58.550 ","End":"06:01.095","Text":"Yeah, I mentioned some conditions."},{"Start":"06:01.095 ","End":"06:06.640","Text":"It\u0027s not just any old surface it has to be oriented."},{"Start":"06:08.150 ","End":"06:11.745","Text":"The curve is also oriented,"},{"Start":"06:11.745 ","End":"06:18.675","Text":"and it\u0027s oriented based on the orientation of S as I explained at length."},{"Start":"06:18.675 ","End":"06:25.320","Text":"S also we want to be smooth,"},{"Start":"06:25.320 ","End":"06:31.909","Text":"and it goes without saying that this being a boundary is also a closed curve."},{"Start":"06:31.909 ","End":"06:37.925","Text":"But the important thing is it has to have the positive orientation all right."},{"Start":"06:37.925 ","End":"06:41.240","Text":"They have to match with orientations."},{"Start":"06:41.240 ","End":"06:44.940","Text":"Let\u0027s highlight the theorem."},{"Start":"06:46.250 ","End":"06:50.610","Text":"Now let\u0027s do a couple of examples."}],"ID":8820},{"Watched":false,"Name":"Worked Example 1","Duration":"9m 35s","ChapterTopicVideoID":8753,"CourseChapterTopicPlaylistID":4973,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.414","Text":"Let\u0027s get to the first example."},{"Start":"00:02.414 ","End":"00:04.110","Text":"I\u0027m actually going to give 2 examples."},{"Start":"00:04.110 ","End":"00:07.590","Text":"In 1 case, we\u0027re going to use the equality 1 way,"},{"Start":"00:07.590 ","End":"00:10.680","Text":"and in the other case, vice versa."},{"Start":"00:10.680 ","End":"00:13.425","Text":"In other words, in 1 case I\u0027m going to give you"},{"Start":"00:13.425 ","End":"00:18.570","Text":"a line integral to evaluate by means of a surface integral"},{"Start":"00:18.570 ","End":"00:21.480","Text":"and in the other example, we\u0027ll do the opposite."},{"Start":"00:21.480 ","End":"00:26.835","Text":"In this case, I\u0027m going to ask you to compute a double integral"},{"Start":"00:26.835 ","End":"00:32.775","Text":"over a surface S of the curl of a vector function,"},{"Start":"00:32.775 ","End":"00:36.060","Text":"a vector field F.ds,"},{"Start":"00:36.060 ","End":"00:39.375","Text":"and I have to give you what is F and what is S,"},{"Start":"00:39.375 ","End":"00:41.865","Text":"and remember, S has to be oriented."},{"Start":"00:41.865 ","End":"00:51.680","Text":"I\u0027m going to take F of x, y, and z to equal z squared"},{"Start":"00:51.680 ","End":"00:57.080","Text":"in the first component minus 3xy in the second component,"},{"Start":"00:57.080 ","End":"01:00.410","Text":"and x cubed, y cubed in the third component,"},{"Start":"01:00.410 ","End":"01:03.400","Text":"and I\u0027m using the angular brackets notation."},{"Start":"01:03.400 ","End":"01:06.330","Text":"Now I\u0027m going to describe S."},{"Start":"01:06.330 ","End":"01:09.669","Text":"I\u0027ll bring in a picture."},{"Start":"01:09.669 ","End":"01:12.000","Text":"Here\u0027s a picture of S."},{"Start":"01:12.000 ","End":"01:15.380","Text":"It\u0027s an upside down circular paraboloid"},{"Start":"01:15.380 ","End":"01:20.150","Text":"and it\u0027s given by the equation where z is a function of x, y,"},{"Start":"01:20.150 ","End":"01:27.520","Text":"z equals 5 minus x squared minus y squared."},{"Start":"01:27.520 ","End":"01:32.390","Text":"The highest point it can be is 5 and it\u0027s centered on the z-axis."},{"Start":"01:32.390 ","End":"01:40.850","Text":"But I don\u0027t want the whole paraboloid just a bit above the plane where z equals 1."},{"Start":"01:40.850 ","End":"01:46.265","Text":"In other words, z is going to be bigger or equal to 1,"},{"Start":"01:46.265 ","End":"01:52.000","Text":"and the boundary C will be given by z equals 1."},{"Start":"01:52.000 ","End":"01:53.300","Text":"We\u0027ll do that in a moment."},{"Start":"01:53.300 ","End":"01:56.975","Text":"Let me just say that I have to also give an orientation."},{"Start":"01:56.975 ","End":"02:01.850","Text":"Now, as usual, when we have 1 variable as a function of the others,"},{"Start":"02:01.850 ","End":"02:04.970","Text":"the positive orientation is the one in this case"},{"Start":"02:04.970 ","End":"02:08.450","Text":"where z is increasing with z is positive."},{"Start":"02:08.450 ","End":"02:10.220","Text":"It\u0027s going to be upwards facing"},{"Start":"02:10.220 ","End":"02:12.350","Text":"or it\u0027s going to have at least an upward component."},{"Start":"02:12.350 ","End":"02:15.170","Text":"So the normal vector here or here,"},{"Start":"02:15.170 ","End":"02:18.925","Text":"it\u0027ll be exactly upwards and here it\u0027ll be partly,"},{"Start":"02:18.925 ","End":"02:20.960","Text":"at least it\u0027ll have an upward component."},{"Start":"02:20.960 ","End":"02:27.000","Text":"That will be the normal direction is leaning upwards."},{"Start":"02:27.490 ","End":"02:34.560","Text":"Let\u0027s compute this integral by means of the line integral over C."},{"Start":"02:34.560 ","End":"02:36.905","Text":"Because this is the normal,"},{"Start":"02:36.905 ","End":"02:38.870","Text":"the direction, like we discussed,"},{"Start":"02:38.870 ","End":"02:44.280","Text":"the orientation will actually be the one that\u0027s indicated by the arrows here."},{"Start":"02:45.020 ","End":"02:51.375","Text":"The equation of C is gotten by setting z equals 1,"},{"Start":"02:51.375 ","End":"02:53.995","Text":"and if we set z equals 1,"},{"Start":"02:53.995 ","End":"02:57.935","Text":"then we get 1 equals 5 minus x squared minus y squared,"},{"Start":"02:57.935 ","End":"02:59.600","Text":"and if we just play around with it,"},{"Start":"02:59.600 ","End":"03:05.285","Text":"we get that x squared plus y squared equals 4,"},{"Start":"03:05.285 ","End":"03:08.580","Text":"but still z equals 1."},{"Start":"03:08.630 ","End":"03:13.805","Text":"It\u0027s a circle of radius 2 because this is 2 squared."},{"Start":"03:13.805 ","End":"03:20.660","Text":"Radius 2 centered at the z-axis and hovering 1 unit above the x, y plane,"},{"Start":"03:20.660 ","End":"03:22.550","Text":"z is going to be equal to 1 there."},{"Start":"03:22.550 ","End":"03:27.920","Text":"We can parameterize it by using a parameter t,"},{"Start":"03:27.920 ","End":"03:29.420","Text":"which is the angle."},{"Start":"03:29.420 ","End":"03:32.080","Text":"We take this like the circle,"},{"Start":"03:32.080 ","End":"03:36.150","Text":"and the circle is r cosine t, r sine t."},{"Start":"03:36.150 ","End":"03:42.140","Text":"In this case, it\u0027s 2 cosine t, 2 sine t."},{"Start":"03:42.140 ","End":"03:44.210","Text":"But let\u0027s not forget we have the z component"},{"Start":"03:44.210 ","End":"03:46.835","Text":"and that\u0027s constantly equal to 1,"},{"Start":"03:46.835 ","End":"03:49.805","Text":"and t, we got the full circle."},{"Start":"03:49.805 ","End":"03:56.770","Text":"the only one that go around once is t goes from 0-2Pi in radians."},{"Start":"03:56.770 ","End":"03:59.894","Text":"Now we\u0027re going to apply Stokes\u0027 theorem."},{"Start":"03:59.894 ","End":"04:03.185","Text":"We\u0027re going to have to evaluate this integral."},{"Start":"04:03.185 ","End":"04:07.325","Text":"I\u0027ll copy it before I scroll down and it\u0027ll disappear."},{"Start":"04:07.325 ","End":"04:11.040","Text":"We need the integral over the curve,"},{"Start":"04:11.040 ","End":"04:18.440","Text":"I\u0027m just copying the left-hand side of the vector field dot product with dr,"},{"Start":"04:18.440 ","End":"04:20.975","Text":"and we\u0027ll see what this computes to."},{"Start":"04:20.975 ","End":"04:25.109","Text":"Now dr will equal,"},{"Start":"04:25.109 ","End":"04:27.555","Text":"in fact, I\u0027ll do that at the side."},{"Start":"04:27.555 ","End":"04:30.665","Text":"From this formula for r,"},{"Start":"04:30.665 ","End":"04:40.395","Text":"I can get that dr is just the derivative of r and then dt is derivative of this."},{"Start":"04:40.395 ","End":"04:51.980","Text":"It\u0027s minus 2 sine t, and then 2 cosine t and then 0."},{"Start":"04:51.980 ","End":"04:58.100","Text":"But all this dt, vector times a scalar for the vector,"},{"Start":"04:58.100 ","End":"05:01.220","Text":"and what we need now is the dot product."},{"Start":"05:01.220 ","End":"05:07.440","Text":"We have dr here and we have F here."},{"Start":"05:07.440 ","End":"05:11.665","Text":"Let me take this dot product with this."},{"Start":"05:11.665 ","End":"05:12.970","Text":"This is what we need,"},{"Start":"05:12.970 ","End":"05:22.230","Text":"except that we have to substitute for x, y, and z the components from r."},{"Start":"05:22.230 ","End":"05:24.734","Text":"This would be like x,"},{"Start":"05:24.734 ","End":"05:28.740","Text":"this would be y, and this would be z."},{"Start":"05:28.740 ","End":"05:34.610","Text":"Let me just write that, and say that what we want is the integral"},{"Start":"05:34.610 ","End":"05:39.315","Text":"where t goes from 0-2Pi."},{"Start":"05:39.315 ","End":"05:41.820","Text":"Now first the F bit."},{"Start":"05:41.820 ","End":"05:43.470","Text":"Along the curve, let\u0027s see,"},{"Start":"05:43.470 ","End":"05:48.105","Text":"z squared, z is 1, 1 squared is 1,"},{"Start":"05:48.105 ","End":"05:52.410","Text":"and then I need minus 3xy minus 3."},{"Start":"05:52.410 ","End":"05:56.075","Text":"This times this, 3 times 2 times 2 is 12,"},{"Start":"05:56.075 ","End":"06:00.740","Text":"so it\u0027s minus 12 cosine t sine t."},{"Start":"06:00.740 ","End":"06:04.370","Text":"Then in the last component,"},{"Start":"06:04.370 ","End":"06:06.935","Text":"I need x cubed, y cubed."},{"Start":"06:06.935 ","End":"06:13.625","Text":"Again from here, 2 cubed is 8 times another 8,"},{"Start":"06:13.625 ","End":"06:17.260","Text":"so that would be 64,"},{"Start":"06:17.260 ","End":"06:25.520","Text":"and then cosine cubed sine cubed, cosine cubed t, sine cubed t."},{"Start":"06:25.520 ","End":"06:31.040","Text":"All this dot product with what I got here,"},{"Start":"06:31.040 ","End":"06:40.850","Text":"which is minus 2 sine t, 2 cosine t, 0, dt."},{"Start":"06:40.850 ","End":"06:45.860","Text":"Let\u0027s gets some more space here."},{"Start":"06:45.860 ","End":"06:48.710","Text":"Now we\u0027ll do the dot product."},{"Start":"06:48.710 ","End":"06:54.705","Text":"What we get is the integral again from 0-2Pi."},{"Start":"06:54.705 ","End":"06:59.270","Text":"Now, the dot product is this times this plus this times this plus this times this,"},{"Start":"06:59.270 ","End":"07:01.955","Text":"and the last component is 0, we can see that."},{"Start":"07:01.955 ","End":"07:04.940","Text":"I\u0027ve got 1 times minus 2 sine t,"},{"Start":"07:04.940 ","End":"07:08.489","Text":"which is minus 2 sine t,"},{"Start":"07:08.489 ","End":"07:13.515","Text":"and then I\u0027ve got this with this."},{"Start":"07:13.515 ","End":"07:16.040","Text":"The 12 times 2 is 24,"},{"Start":"07:16.040 ","End":"07:20.240","Text":"so I\u0027ve got minus 24,"},{"Start":"07:20.240 ","End":"07:24.070","Text":"and then cosine squared sine,"},{"Start":"07:24.070 ","End":"07:30.420","Text":"cosine squared t, sine t, dt,"},{"Start":"07:30.420 ","End":"07:32.890","Text":"I need the brackets."},{"Start":"07:33.490 ","End":"07:39.080","Text":"I want to do a substitution because I see that I have a minus sine t,"},{"Start":"07:39.080 ","End":"07:41.210","Text":"which is the derivative of cosine t."},{"Start":"07:41.210 ","End":"07:45.890","Text":"Let me just rewrite this as the integral from 0-2Pi."},{"Start":"07:45.890 ","End":"07:50.045","Text":"I\u0027m going to take minus sine t outside the brackets."},{"Start":"07:50.045 ","End":"07:58.665","Text":"I\u0027ll write that this is 2 plus 24 cosine squared t,"},{"Start":"07:58.665 ","End":"08:02.400","Text":"all this times minus sine t, dt."},{"Start":"08:02.400 ","End":"08:09.650","Text":"The reason I need minus sine t is I want to substitute that,"},{"Start":"08:09.650 ","End":"08:11.480","Text":"say u is cosine t,"},{"Start":"08:11.480 ","End":"08:14.435","Text":"and then I\u0027ll already have the derivative of cosine here."},{"Start":"08:14.435 ","End":"08:15.710","Text":"In fact, let\u0027s do that."},{"Start":"08:15.710 ","End":"08:23.180","Text":"If I substitute, I can say, let u equal cosine t"},{"Start":"08:23.180 ","End":"08:28.580","Text":"and then du is minus sine t dt."},{"Start":"08:28.580 ","End":"08:31.565","Text":"If I substitute all that here,"},{"Start":"08:31.565 ","End":"08:37.230","Text":"then what this becomes is the integral,"},{"Start":"08:37.230 ","End":"08:40.850","Text":"and let\u0027s see, with the limits also have to be substituted."},{"Start":"08:40.850 ","End":"08:47.360","Text":"What we get here, 2 plus 24, cosine t is u,"},{"Start":"08:47.360 ","End":"08:55.665","Text":"so it\u0027s u squared and minus sine t dt is du, and let\u0027s see."},{"Start":"08:55.665 ","End":"08:59.560","Text":"When t equals 0,"},{"Start":"08:59.560 ","End":"09:07.625","Text":"we get that u equals cosine 0 is 1,"},{"Start":"09:07.625 ","End":"09:10.550","Text":"and when t equals 2Pi,"},{"Start":"09:10.550 ","End":"09:14.450","Text":"then u equals cosine 2Pi,"},{"Start":"09:14.450 ","End":"09:18.335","Text":"which is the same as cosine 0, also equals 1."},{"Start":"09:18.335 ","End":"09:21.235","Text":"We\u0027ve got the integral from 1-1."},{"Start":"09:21.235 ","End":"09:23.330","Text":"We don\u0027t even have to evaluate it,"},{"Start":"09:23.330 ","End":"09:24.380","Text":"the integral of something to it."},{"Start":"09:24.380 ","End":"09:26.120","Text":"So I\u0027m just going to get 0."},{"Start":"09:26.120 ","End":"09:29.075","Text":"It\u0027s something minus the same thing is 0."},{"Start":"09:29.075 ","End":"09:31.420","Text":"So that\u0027s the answer."},{"Start":"09:31.420 ","End":"09:35.880","Text":"After the break, we\u0027ll do the other example in the other direction."}],"ID":8821},{"Watched":false,"Name":"Worked Example 2","Duration":"17m 39s","ChapterTopicVideoID":8754,"CourseChapterTopicPlaylistID":4973,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.200 ","End":"00:03.359","Text":"Let\u0027s move on to the second example."},{"Start":"00:03.359 ","End":"00:07.755","Text":"I\u0027ll scroll back up and I\u0027ll clear the board."},{"Start":"00:07.755 ","End":"00:12.150","Text":"In Example 2, which is like the reverse of Example 1,"},{"Start":"00:12.150 ","End":"00:16.155","Text":"in the sense of which way we\u0027re going to use the formula."},{"Start":"00:16.155 ","End":"00:18.750","Text":"I\u0027m going to give you a line integral and we\u0027re going"},{"Start":"00:18.750 ","End":"00:21.915","Text":"to evaluate it using a surface integral."},{"Start":"00:21.915 ","End":"00:24.690","Text":"Let\u0027s say what are F of x,"},{"Start":"00:24.690 ","End":"00:26.415","Text":"y, and z is,"},{"Start":"00:26.415 ","End":"00:30.060","Text":"and that will be, well,"},{"Start":"00:30.060 ","End":"00:33.300","Text":"let\u0027s use the i, j, k, notation this time."},{"Start":"00:33.300 ","End":"00:38.890","Text":"I\u0027ll let it be z squared in the i direction,"},{"Start":"00:38.890 ","End":"00:43.795","Text":"and y squared in the j direction,"},{"Start":"00:43.795 ","End":"00:48.405","Text":"and x in the k direction,"},{"Start":"00:48.405 ","End":"00:54.170","Text":"that\u0027s F. I also want to give you what is C, the curve."},{"Start":"00:54.170 ","End":"00:56.795","Text":"I\u0027ll just bring in a picture."},{"Start":"00:56.795 ","End":"01:02.090","Text":"Here\u0027s our picture which contains more than we need to know."},{"Start":"01:02.090 ","End":"01:07.245","Text":"The curve C connects the 3 points and let me write them."},{"Start":"01:07.245 ","End":"01:10.755","Text":"I have 0,0,1 over here,"},{"Start":"01:10.755 ","End":"01:16.820","Text":"I have 0,1,0 over here,"},{"Start":"01:16.820 ","End":"01:20.165","Text":"and I have 1, 0,"},{"Start":"01:20.165 ","End":"01:25.110","Text":"0 over here and I connect them with a triangle and we"},{"Start":"01:25.110 ","End":"01:30.780","Text":"go in the direction which is counterclockwise if you look from above."},{"Start":"01:30.780 ","End":"01:33.770","Text":"If I look from this direction,"},{"Start":"01:33.770 ","End":"01:37.180","Text":"then it\u0027s going to be counterclockwise."},{"Start":"01:37.180 ","End":"01:42.650","Text":"That describes C, and I\u0027ll come to this colored bit in the S in a moment."},{"Start":"01:42.650 ","End":"01:47.405","Text":"What we want is the integral over the curve"},{"Start":"01:47.405 ","End":"01:53.255","Text":"C in the direction stated of what\u0027s written here,"},{"Start":"01:53.255 ","End":"01:58.875","Text":"F.dr, the line integral."},{"Start":"01:58.875 ","End":"02:04.610","Text":"Now what I\u0027m going to say is when I use Stokes\u0027 theorem in this direction to express it"},{"Start":"02:04.610 ","End":"02:10.340","Text":"as the surface integral over some surface S sorry,"},{"Start":"02:10.340 ","End":"02:16.280","Text":"of the curl F, which we can write as also del cross"},{"Start":"02:16.280 ","End":"02:23.525","Text":"F. Just to get you practice with the alternative ways of writing, also ds."},{"Start":"02:23.525 ","End":"02:25.340","Text":"I don\u0027t have S yet,"},{"Start":"02:25.340 ","End":"02:27.380","Text":"but the most obvious thing,"},{"Start":"02:27.380 ","End":"02:32.660","Text":"or the simplest thing to do is to find a surface whose boundary is this C,"},{"Start":"02:32.660 ","End":"02:34.490","Text":"is just to take a plane."},{"Start":"02:34.490 ","End":"02:38.180","Text":"We\u0027re going to take S as"},{"Start":"02:38.180 ","End":"02:48.700","Text":"the plane through these points or through the curve C because it\u0027s planar."},{"Start":"02:48.700 ","End":"02:53.780","Text":"I need to give you also an orientation because it\u0027s not enough to give a surface,"},{"Start":"02:53.780 ","End":"02:56.250","Text":"we need to know oriented surface."},{"Start":"02:56.650 ","End":"03:00.575","Text":"At any point it\u0027s going to have the same normal because it\u0027s a plane."},{"Start":"03:00.575 ","End":"03:08.490","Text":"But we\u0027ll take the normal that faces upwards or at least that has an upward component,"},{"Start":"03:08.490 ","End":"03:11.190","Text":"so that will be our normal."},{"Start":"03:11.190 ","End":"03:14.870","Text":"What we expect is if we write it in vector form at the third component,"},{"Start":"03:14.870 ","End":"03:17.015","Text":"the z component will be positive."},{"Start":"03:17.015 ","End":"03:20.750","Text":"Then it will work out that the orientation"},{"Start":"03:20.750 ","End":"03:25.290","Text":"on the curve is the one that we get from the orientation on the surface."},{"Start":"03:28.070 ","End":"03:33.260","Text":"Let\u0027s next compute the cross product or the curl."},{"Start":"03:33.260 ","End":"03:36.360","Text":"I\u0027m going to use the determinant method for a change."},{"Start":"03:36.360 ","End":"03:39.470","Text":"If you haven\u0027t studied determinants really all that matters is"},{"Start":"03:39.470 ","End":"03:42.650","Text":"the result of this computation."},{"Start":"03:42.650 ","End":"03:44.220","Text":"There are other definitions."},{"Start":"03:44.220 ","End":"03:50.015","Text":"What we get is that the cross product of del,"},{"Start":"03:50.015 ","End":"03:54.845","Text":"cross with F is equal to the determinant."},{"Start":"03:54.845 ","End":"03:58.595","Text":"Here we put the 3 unit vectors, i,"},{"Start":"03:58.595 ","End":"04:03.570","Text":"j, and k. In the middle,"},{"Start":"04:03.570 ","End":"04:08.090","Text":"we put the first part of the cross,"},{"Start":"04:08.090 ","End":"04:12.845","Text":"which is this, which is just the operator d by dx,"},{"Start":"04:12.845 ","End":"04:16.800","Text":"d by dy, d by dz,"},{"Start":"04:16.800 ","End":"04:22.365","Text":"3 partial derivative operators and then we put the F, which is here,"},{"Start":"04:22.365 ","End":"04:27.695","Text":"so it\u0027s z squared, y squared,"},{"Start":"04:27.695 ","End":"04:34.400","Text":"and then x. I\u0027m going to use the method of co-factors."},{"Start":"04:34.400 ","End":"04:36.860","Text":"Again, if you haven\u0027t studied this,"},{"Start":"04:36.860 ","End":"04:41.070","Text":"really only the end result of this cross-product is important."},{"Start":"04:41.070 ","End":"04:43.014","Text":"In the i place,"},{"Start":"04:43.014 ","End":"04:47.180","Text":"I get this times this minus this times this,"},{"Start":"04:47.180 ","End":"04:50.560","Text":"derivative of this with respect to x is"},{"Start":"04:50.560 ","End":"04:56.225","Text":"nothing and derivative of y squared with respect to z is also nothing,"},{"Start":"04:56.225 ","End":"05:00.080","Text":"so we get 0 in the i position."},{"Start":"05:00.080 ","End":"05:02.845","Text":"Next, the j position,"},{"Start":"05:02.845 ","End":"05:06.060","Text":"it\u0027s actually with a minus,"},{"Start":"05:06.060 ","End":"05:09.585","Text":"we alternate, so there."},{"Start":"05:09.585 ","End":"05:17.060","Text":"Then we cross out the row and column with the j and then we\u0027re left with this,"},{"Start":"05:17.060 ","End":"05:19.700","Text":"with this minus this with this."},{"Start":"05:19.700 ","End":"05:22.850","Text":"It\u0027s d by dx of x is"},{"Start":"05:22.850 ","End":"05:31.565","Text":"1 minus d by dz of z squared is 2z,"},{"Start":"05:31.565 ","End":"05:36.605","Text":"all this will be in the j direction."},{"Start":"05:36.605 ","End":"05:39.890","Text":"Finally, for the k, we get this with this is"},{"Start":"05:39.890 ","End":"05:44.645","Text":"nothing because y squared with respect to x is 0 and similarly here."},{"Start":"05:44.645 ","End":"05:51.270","Text":"We\u0027re going to get 0k and if I just tidy this up a bit,"},{"Start":"05:51.270 ","End":"05:56.254","Text":"what it amounts to is 2z minus 1,"},{"Start":"05:56.254 ","End":"05:59.430","Text":"and it\u0027s all in the j direction."},{"Start":"05:59.430 ","End":"06:01.785","Text":"There\u0027s only a j component."},{"Start":"06:01.785 ","End":"06:04.415","Text":"That\u0027s the cross product."},{"Start":"06:04.415 ","End":"06:09.785","Text":"Now, we still have a surface integral."},{"Start":"06:09.785 ","End":"06:16.025","Text":"What we have is the surface integral over S,"},{"Start":"06:16.025 ","End":"06:17.660","Text":"which we haven\u0027t even computed yet,"},{"Start":"06:17.660 ","End":"06:23.870","Text":"we\u0027ve just described it in the picture, of this function."},{"Start":"06:23.870 ","End":"06:25.620","Text":"In fact, you know what,"},{"Start":"06:25.620 ","End":"06:29.735","Text":"I\u0027ll go back to the angular brackets."},{"Start":"06:29.735 ","End":"06:33.150","Text":"This I can write as 0,"},{"Start":"06:34.130 ","End":"06:43.230","Text":"2z i minus 1 comma 0 and then ds."},{"Start":"06:45.440 ","End":"06:51.575","Text":"The surface S could be described as the surface."},{"Start":"06:51.575 ","End":"06:52.790","Text":"If you just think about it,"},{"Start":"06:52.790 ","End":"06:57.180","Text":"it\u0027s just x plus y plus z equals 1."},{"Start":"06:57.520 ","End":"07:01.505","Text":"It\u0027s easy to check that it goes through these 3 points,"},{"Start":"07:01.505 ","End":"07:02.900","Text":"and each of these 3 points,"},{"Start":"07:02.900 ","End":"07:04.970","Text":"you have 1 coordinate which is 1,"},{"Start":"07:04.970 ","End":"07:08.240","Text":"and the other 2 are 0, so x plus y plus z is 1,"},{"Start":"07:08.240 ","End":"07:10.310","Text":"and here x plus y plus z is 1,"},{"Start":"07:10.310 ","End":"07:13.025","Text":"and here, so this is the plane."},{"Start":"07:13.025 ","End":"07:15.560","Text":"Otherwise you could take time to compute it,"},{"Start":"07:15.560 ","End":"07:18.740","Text":"but it\u0027 an easy case here."},{"Start":"07:18.740 ","End":"07:25.385","Text":"This would be like a level surface of f of x,"},{"Start":"07:25.385 ","End":"07:28.745","Text":"y, and z equals this,"},{"Start":"07:28.745 ","End":"07:31.745","Text":"which is level at 1."},{"Start":"07:31.745 ","End":"07:36.530","Text":"We can also write this as a function where z is g of x,"},{"Start":"07:36.530 ","End":"07:42.165","Text":"y, so there\u0027s a formula we can use."},{"Start":"07:42.165 ","End":"07:44.970","Text":"This is going to equal, let\u0027s see,"},{"Start":"07:44.970 ","End":"07:50.770","Text":"bring the x and the y over,1 minus x minus y."},{"Start":"07:54.530 ","End":"08:00.620","Text":"What we do, this is equal to a regular surface integral,"},{"Start":"08:00.620 ","End":"08:03.140","Text":"not a vector field, the scalar,"},{"Start":"08:03.140 ","End":"08:08.900","Text":"if we just take the double integral over S,"},{"Start":"08:08.900 ","End":"08:12.710","Text":"the scalar will be F.n."},{"Start":"08:12.710 ","End":"08:14.915","Text":"Other words, I take the function f,"},{"Start":"08:14.915 ","End":"08:21.660","Text":"which is this, and I dot-product it with n, a normal."},{"Start":"08:22.070 ","End":"08:27.185","Text":"I need the normal and one way of getting a normal,"},{"Start":"08:27.185 ","End":"08:28.880","Text":"I can get it from either of these."},{"Start":"08:28.880 ","End":"08:33.229","Text":"Or one way of getting a normal from a level curve is just by taking"},{"Start":"08:33.229 ","End":"08:40.820","Text":"the partial derivatives of this f. We can say that a normal,"},{"Start":"08:40.820 ","End":"08:42.950","Text":"not necessarily a unit normal,"},{"Start":"08:42.950 ","End":"08:47.355","Text":"would be f with respect to x,"},{"Start":"08:47.355 ","End":"08:49.650","Text":"f with respect to y,"},{"Start":"08:49.650 ","End":"08:52.160","Text":"f with respect to z."},{"Start":"08:52.160 ","End":"08:58.220","Text":"This would be a normal vector but if we want a unit normal,"},{"Start":"08:58.220 ","End":"09:07.320","Text":"then we have to divide by the magnitude of the same thing, fx, fy, fz."},{"Start":"09:08.570 ","End":"09:14.680","Text":"Now, these 3 partial derivatives are just 1,"},{"Start":"09:14.680 ","End":"09:18.290","Text":"each of them, the derivative of this with respect to x is 1 with respect to y,"},{"Start":"09:18.290 ","End":"09:20.870","Text":"it\u0027s 1, with respect to z, it\u0027s 1."},{"Start":"09:20.870 ","End":"09:25.910","Text":"So here we just get 1,1,1"},{"Start":"09:26.120 ","End":"09:35.050","Text":"over the magnitude of 1, 1, 1."},{"Start":"09:36.210 ","End":"09:39.474","Text":"This is what I plug into here,"},{"Start":"09:39.474 ","End":"09:43.060","Text":"because essentially what we want according to the definition,"},{"Start":"09:43.060 ","End":"09:51.025","Text":"is f.n, where n is a unit normal vector in the right direction."},{"Start":"09:51.025 ","End":"09:54.460","Text":"I want to point out that this isn\u0027t the correct direction,"},{"Start":"09:54.460 ","End":"09:56.800","Text":"because it has a positive z component,"},{"Start":"09:56.800 ","End":"10:00.880","Text":"so it is facing somewhat upwards,"},{"Start":"10:00.880 ","End":"10:02.155","Text":"which is what we said."},{"Start":"10:02.155 ","End":"10:07.764","Text":"Anyway, so it start with and all this is ds,"},{"Start":"10:07.764 ","End":"10:11.530","Text":"but this expression is now a scalar so we"},{"Start":"10:11.530 ","End":"10:15.910","Text":"have a regular surface integral not a vector field."},{"Start":"10:15.910 ","End":"10:20.575","Text":"We still want to reduce it 1 step further and get it to a double integral,"},{"Start":"10:20.575 ","End":"10:24.550","Text":"so we need to use yet another formula,"},{"Start":"10:24.550 ","End":"10:31.550","Text":"that this thing is equal to,"},{"Start":"10:31.560 ","End":"10:34.195","Text":"and I\u0027ll continue over here,"},{"Start":"10:34.195 ","End":"10:43.520","Text":"the double integral over D and I\u0027ll say what D is in a minute of this thing."},{"Start":"10:44.220 ","End":"10:47.440","Text":"Let me just copy it."},{"Start":"10:47.440 ","End":"10:54.550","Text":"If you review the chapter where we evaluate a ds in terms of da,"},{"Start":"10:54.550 ","End":"11:00.505","Text":"we said that ds is equal to the magnitude"},{"Start":"11:00.505 ","End":"11:09.400","Text":"of minus g_x, minus g_y, 1."},{"Start":"11:09.400 ","End":"11:14.020","Text":"This was in the case where we had z as a function of x and y,"},{"Start":"11:14.020 ","End":"11:15.490","Text":"which we do here."},{"Start":"11:15.490 ","End":"11:24.040","Text":"Now, minus g_x is just minus, minus 1,"},{"Start":"11:24.040 ","End":"11:27.205","Text":"which is 1 minus g_y,"},{"Start":"11:27.205 ","End":"11:30.100","Text":"if I\u0027m here is minus, minus 1,"},{"Start":"11:30.100 ","End":"11:32.995","Text":"which is 1, and 1 is just 1."},{"Start":"11:32.995 ","End":"11:35.305","Text":"Essentially what this means,"},{"Start":"11:35.305 ","End":"11:42.295","Text":"is that this magnitude here cancels out with this magnitude here."},{"Start":"11:42.295 ","End":"11:44.350","Text":"In each case it\u0027s 1, 1, 1."},{"Start":"11:44.350 ","End":"11:46.420","Text":"It actually comes out to the square root of 3,"},{"Start":"11:46.420 ","End":"11:48.100","Text":"but that doesn\u0027t matter."},{"Start":"11:48.100 ","End":"11:53.320","Text":"Then we need to put here a dA."},{"Start":"11:53.320 ","End":"11:56.950","Text":"The only thing we\u0027re missing is what is d?"},{"Start":"11:56.950 ","End":"12:00.325","Text":"Well, d is the part,"},{"Start":"12:00.325 ","End":"12:07.015","Text":"I guess I only see part of it is like this triangle here,"},{"Start":"12:07.015 ","End":"12:11.440","Text":"which I can highlight."},{"Start":"12:11.440 ","End":"12:16.900","Text":"It\u0027s best to put a fresh picture here and here it is."},{"Start":"12:16.900 ","End":"12:21.280","Text":"This is our D in the x y plane."},{"Start":"12:21.280 ","End":"12:23.245","Text":"If we let z equals 0,"},{"Start":"12:23.245 ","End":"12:28.090","Text":"that you see from here we get x plus y equals 1 and also from here,"},{"Start":"12:28.090 ","End":"12:30.190","Text":"if we let z equals 0,"},{"Start":"12:30.190 ","End":"12:32.260","Text":"we get 1 minus x minus y is 0."},{"Start":"12:32.260 ","End":"12:34.090","Text":"Again, x plus y is 1,"},{"Start":"12:34.090 ","End":"12:37.210","Text":"and so y is minus x plus 1."},{"Start":"12:37.210 ","End":"12:40.330","Text":"Now we have described the domain."},{"Start":"12:40.330 ","End":"12:45.940","Text":"Of course it is bounded by the y-axis on the left and the x-axis from below."},{"Start":"12:45.940 ","End":"12:52.645","Text":"Now we have to compute this integral and we can just simplify it with the dot-product,"},{"Start":"12:52.645 ","End":"12:55.690","Text":"because look, only the middle term is going to apply."},{"Start":"12:55.690 ","End":"13:01.000","Text":"It\u0027s going to be the double integral over D. Now the 0 and the 0,"},{"Start":"13:01.000 ","End":"13:02.320","Text":"as I say disappear."},{"Start":"13:02.320 ","End":"13:06.460","Text":"1 with 2 z minus 1 is just 2,"},{"Start":"13:06.460 ","End":"13:15.430","Text":"z minus 1 and all of this is dA, regular double integral."},{"Start":"13:15.430 ","End":"13:18.610","Text":"dA will either be dx dy or dy dx,"},{"Start":"13:18.610 ","End":"13:22.045","Text":"depending on which way we want to break it up into"},{"Start":"13:22.045 ","End":"13:26.530","Text":"vertical strips or horizontal strips doesn\u0027t really matter in this case."},{"Start":"13:26.530 ","End":"13:29.890","Text":"But let\u0027s say, in the case we have y in terms of x."},{"Start":"13:29.890 ","End":"13:32.380","Text":"That will take first of all,"},{"Start":"13:32.380 ","End":"13:37.720","Text":"the integral with respect to"},{"Start":"13:37.720 ","End":"13:45.050","Text":"y going from 0 to minus x plus 1."},{"Start":"13:46.110 ","End":"13:49.585","Text":"This is going to be dy."},{"Start":"13:49.585 ","End":"13:51.985","Text":"I\u0027ll fill in the thing in the minute,"},{"Start":"13:51.985 ","End":"13:57.580","Text":"and then we\u0027ll take the afterwards the integral dx and x goes from 0-1."},{"Start":"13:57.580 ","End":"14:02.620","Text":"As soon as we go from 0-1, after we do the middle bit will only have a function of x and"},{"Start":"14:02.620 ","End":"14:07.630","Text":"we\u0027ll integrate that from 0-1 and this is just this bit here."},{"Start":"14:07.630 ","End":"14:13.405","Text":"Now, the z I want here is the z from here,"},{"Start":"14:13.405 ","End":"14:15.985","Text":"expressed as a function of x and y."},{"Start":"14:15.985 ","End":"14:20.515","Text":"If I take twice this minus 1,"},{"Start":"14:20.515 ","End":"14:24.220","Text":"twice this will be 2 minus 2x minus 2y."},{"Start":"14:24.220 ","End":"14:31.390","Text":"If I subtract 1, it\u0027ll be 1 minus 2x minus 2y, dy, dx."},{"Start":"14:31.390 ","End":"14:34.255","Text":"That\u0027s straightforward."},{"Start":"14:34.255 ","End":"14:36.160","Text":"Let\u0027s do first of all,"},{"Start":"14:36.160 ","End":"14:41.350","Text":"the integral with respect to y. I\u0027ve got the integral from 0-1."},{"Start":"14:41.350 ","End":"14:46.105","Text":"Now, the integral with respect to y will be, for 1,"},{"Start":"14:46.105 ","End":"14:52.150","Text":"I\u0027ll get y minus 2x is also a constant,"},{"Start":"14:52.150 ","End":"14:55.120","Text":"so 2xy and 2y,"},{"Start":"14:55.120 ","End":"14:58.570","Text":"the integral of that will be y squared."},{"Start":"14:58.570 ","End":"15:08.950","Text":"All this has got to be taken where y goes from 0 to minus x plus 1 and then dx."},{"Start":"15:08.950 ","End":"15:11.185","Text":"I plug both in."},{"Start":"15:11.185 ","End":"15:14.020","Text":"Now, when y is 0,"},{"Start":"15:14.020 ","End":"15:15.580","Text":"everything is going to be 0."},{"Start":"15:15.580 ","End":"15:19.285","Text":"I just have to plug in minus x plus 1."},{"Start":"15:19.285 ","End":"15:21.970","Text":"I\u0027ve got the integral."},{"Start":"15:21.970 ","End":"15:24.970","Text":"If I let y be minus x plus 1,"},{"Start":"15:24.970 ","End":"15:33.790","Text":"I have minus x plus 1 minus 2x of minus x plus 1,"},{"Start":"15:33.790 ","End":"15:36.850","Text":"minus, minus x plus 1,"},{"Start":"15:36.850 ","End":"15:41.470","Text":"all squared this dx."},{"Start":"15:41.470 ","End":"15:43.750","Text":"Let\u0027s do a quick side computation."},{"Start":"15:43.750 ","End":"15:46.600","Text":"If I just expand this bit I\u0027ve got, let\u0027s see,"},{"Start":"15:46.600 ","End":"15:57.590","Text":"minus x plus 1 plus 2x squared minus 2x"},{"Start":"15:58.050 ","End":"16:04.809","Text":"minus and then this thing is going to be x squared minus"},{"Start":"16:04.809 ","End":"16:12.520","Text":"2x plus 1 using the special product binomial."},{"Start":"16:12.520 ","End":"16:15.670","Text":"Let\u0027s see what do we end up with?"},{"Start":"16:15.670 ","End":"16:20.665","Text":"The x squared will get 2x squared minus x squared is x squared."},{"Start":"16:20.665 ","End":"16:29.725","Text":"For x\u0027s I\u0027ll get x minus 2x plus 2x will just be"},{"Start":"16:29.725 ","End":"16:39.055","Text":"x and as for numbers I\u0027ll have a plus 1 here and a minus 1 here and that\u0027s nothing,"},{"Start":"16:39.055 ","End":"16:42.535","Text":"but I just noticed that this here was a minus,"},{"Start":"16:42.535 ","End":"16:45.205","Text":"so this here is a minus,"},{"Start":"16:45.205 ","End":"16:47.170","Text":"this also is a minus."},{"Start":"16:47.170 ","End":"16:51.294","Text":"Sorry about that. Now we can plug that back in here,"},{"Start":"16:51.294 ","End":"16:55.480","Text":"and we get integral from"},{"Start":"16:55.480 ","End":"17:02.230","Text":"0-1 of x squared minus x dx."},{"Start":"17:02.230 ","End":"17:05.140","Text":"This is now straight forward,"},{"Start":"17:05.140 ","End":"17:07.735","Text":"what have I got?"},{"Start":"17:07.735 ","End":"17:13.765","Text":"X squared, gives me x cubed over 3,"},{"Start":"17:13.765 ","End":"17:17.635","Text":"this gives me x squared over 2."},{"Start":"17:17.635 ","End":"17:20.800","Text":"All this we want between 0 and 1 when it\u0027s 0,"},{"Start":"17:20.800 ","End":"17:23.065","Text":"everything 0, when it\u0027s 1,"},{"Start":"17:23.065 ","End":"17:28.660","Text":"we get 1/3 minus 1/2,"},{"Start":"17:28.660 ","End":"17:33.550","Text":"which is minus 1/6 and"},{"Start":"17:33.550 ","End":"17:39.560","Text":"that is the answer to the second example and we\u0027re done with Stoke\u0027s theorem."}],"ID":8822},{"Watched":false,"Name":"Exercise 1 – Verified one direction","Duration":"10m 14s","ChapterTopicVideoID":8756,"CourseChapterTopicPlaylistID":4973,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.335","Text":"In this exercise, we have to verify Stokes\u0027 Theorem,"},{"Start":"00:04.335 ","End":"00:09.855","Text":"which is written briefly here for the following example,"},{"Start":"00:09.855 ","End":"00:13.410","Text":"where the vector field is given as follows,"},{"Start":"00:13.410 ","End":"00:17.520","Text":"and the surface S is part of the paraboloid,"},{"Start":"00:17.520 ","End":"00:19.410","Text":"and I\u0027ve sketched it here."},{"Start":"00:19.410 ","End":"00:21.450","Text":"This is the whole paraboloid,"},{"Start":"00:21.450 ","End":"00:26.445","Text":"but we just truncated above the x-y plane."},{"Start":"00:26.445 ","End":"00:29.505","Text":"Z is bigger or equal to 0."},{"Start":"00:29.505 ","End":"00:34.080","Text":"The curve C, according to Stokes Theorem,"},{"Start":"00:34.080 ","End":"00:37.815","Text":"is taken as the border of the surface"},{"Start":"00:37.815 ","End":"00:47.810","Text":"S. It\u0027s induced by the orientation of S. If we take the outward normal vector,"},{"Start":"00:47.810 ","End":"00:50.270","Text":"which is what we have to do here,"},{"Start":"00:50.270 ","End":"00:52.040","Text":"and we look from above,"},{"Start":"00:52.040 ","End":"00:55.375","Text":"then we take a positive direction,"},{"Start":"00:55.375 ","End":"01:05.465","Text":"it induces this direction on the curve C. Let\u0027s get started on this."},{"Start":"01:05.465 ","End":"01:09.620","Text":"I\u0027d like to expand this shorthand."},{"Start":"01:09.620 ","End":"01:14.145","Text":"If we call this part P, this part Q,"},{"Start":"01:14.145 ","End":"01:20.930","Text":"and this part R so that F is written in the standard form like this,"},{"Start":"01:20.930 ","End":"01:23.060","Text":"I like to use P, Q, and R;"},{"Start":"01:23.060 ","End":"01:25.700","Text":"some people use f, g, and h,"},{"Start":"01:25.700 ","End":"01:33.090","Text":"then the left-hand side can be expanded as this component-wise."},{"Start":"01:33.090 ","End":"01:35.480","Text":"I better put a circle here."},{"Start":"01:35.480 ","End":"01:41.480","Text":"I don\u0027t have to but it\u0027s usually what we do when it\u0027s a closed curve."},{"Start":"01:41.480 ","End":"01:45.680","Text":"I think we\u0027ll start with expanding the left-hand side."},{"Start":"01:45.680 ","End":"01:47.210","Text":"But while I\u0027m giving formulas,"},{"Start":"01:47.210 ","End":"01:50.400","Text":"let me remind you what the curl is."},{"Start":"01:50.450 ","End":"01:53.300","Text":"We have the formula for curl here but I\u0027m not"},{"Start":"01:53.300 ","End":"01:55.190","Text":"going to use it right away because as I said,"},{"Start":"01:55.190 ","End":"01:59.510","Text":"we\u0027re starting with left-hand side and the left-hand side will be the"},{"Start":"01:59.510 ","End":"02:05.160","Text":"integral over the closed curve C, just replacing P,"},{"Start":"02:05.160 ","End":"02:14.550","Text":"so it\u0027s 2z dx plus 3x dy"},{"Start":"02:14.550 ","End":"02:20.550","Text":"plus 5y dz."},{"Start":"02:20.550 ","End":"02:24.470","Text":"Next, we\u0027ll need a parametrization of C,"},{"Start":"02:24.470 ","End":"02:25.805","Text":"at least that\u0027s one way to go."},{"Start":"02:25.805 ","End":"02:28.175","Text":"Now what is this curve C?"},{"Start":"02:28.175 ","End":"02:33.810","Text":"It\u0027s actually where the paraboloid hits the x-y plane."},{"Start":"02:33.810 ","End":"02:39.615","Text":"It\u0027s the boundary, so it\u0027s where actually z equals 0."},{"Start":"02:39.615 ","End":"02:41.945","Text":"Where z equals 0,"},{"Start":"02:41.945 ","End":"02:51.110","Text":"then we\u0027ll get that for C. We get 4 minus x squared minus y squared equals 0."},{"Start":"02:51.110 ","End":"02:55.115","Text":"Now this 4 minus x squared minus y squared is 0 is"},{"Start":"02:55.115 ","End":"02:59.630","Text":"just the same as x squared plus y squared equals 4,"},{"Start":"02:59.630 ","End":"03:02.180","Text":"or write it as 2 squared."},{"Start":"03:02.180 ","End":"03:05.585","Text":"It\u0027s a circle of radius 2,"},{"Start":"03:05.585 ","End":"03:08.735","Text":"so for the parametrized form,"},{"Start":"03:08.735 ","End":"03:14.570","Text":"I can write C as x equals y equals"},{"Start":"03:14.570 ","End":"03:20.850","Text":"the standard parametrization to cosine of t,"},{"Start":"03:20.850 ","End":"03:28.760","Text":"Theta whatever, 2 sine t and our parameter t,"},{"Start":"03:28.760 ","End":"03:33.485","Text":"which is the angle is from 0 to 2 Pi."},{"Start":"03:33.485 ","End":"03:38.794","Text":"Well, not quite because we are in 3D and we forgot about z,"},{"Start":"03:38.794 ","End":"03:43.850","Text":"and with things in the x-y plane here, z equals 0."},{"Start":"03:43.850 ","End":"03:47.330","Text":"Let\u0027s see what we get here."},{"Start":"03:47.330 ","End":"03:53.780","Text":"The integral now becomes a simple integral based on the"},{"Start":"03:53.780 ","End":"04:00.105","Text":"parameter t. T goes from 0 to 2 Pi."},{"Start":"04:00.105 ","End":"04:02.475","Text":"We\u0027ll substitute each of these things,"},{"Start":"04:02.475 ","End":"04:06.495","Text":"and dx, dy, dz we\u0027ll just compute as needed."},{"Start":"04:06.495 ","End":"04:13.905","Text":"We get 2z, z is 0 times dx."},{"Start":"04:13.905 ","End":"04:16.320","Text":"I could have just not written it because z is 0,"},{"Start":"04:16.320 ","End":"04:18.410","Text":"but just to show I haven\u0027t forgotten it."},{"Start":"04:18.410 ","End":"04:26.045","Text":"Then 3x, x is 2 cosine t,"},{"Start":"04:26.045 ","End":"04:32.440","Text":"and dy is going to be just 2 cosine t dt."},{"Start":"04:33.970 ","End":"04:37.190","Text":"Maybe I\u0027ll just write them in here anyway."},{"Start":"04:37.190 ","End":"04:42.355","Text":"Dx is minus 2 sine t dt;"},{"Start":"04:42.355 ","End":"04:46.130","Text":"only I didn\u0027t need it because it\u0027s multiplied by 0."},{"Start":"04:46.130 ","End":"04:53.690","Text":"Dy is 2 cosine t dt and dz equals also 0,"},{"Start":"04:53.690 ","End":"04:56.120","Text":"or if you like 0 dt."},{"Start":"04:56.520 ","End":"05:00.575","Text":"Back here, we were up to the dy part,"},{"Start":"05:00.575 ","End":"05:01.965","Text":"which we said was"},{"Start":"05:01.965 ","End":"05:08.080","Text":"2 cosine t dt,"},{"Start":"05:08.080 ","End":"05:12.780","Text":"and then 5y."},{"Start":"05:12.780 ","End":"05:19.120","Text":"I\u0027m not going to expand this because dz is 0."},{"Start":"05:20.330 ","End":"05:25.410","Text":"The first and the last terms are both 0,"},{"Start":"05:25.410 ","End":"05:28.990","Text":"so we just have this middle term."},{"Start":"05:29.420 ","End":"05:37.460","Text":"Multiplying out, we\u0027ve got the integral from t equals 0 to 2 Pi."},{"Start":"05:37.460 ","End":"05:45.260","Text":"Now what we have is from here 3 times 2 times 2 is 12."},{"Start":"05:45.260 ","End":"05:49.610","Text":"But the 12 I\u0027m going to write as 6 times 2."},{"Start":"05:49.610 ","End":"05:50.840","Text":"You\u0027ll see in a moment why."},{"Start":"05:50.840 ","End":"05:55.505","Text":"It\u0027s convenient for me to take the 6 out but to leave 2 cosine"},{"Start":"05:55.505 ","End":"06:02.520","Text":"squared t dt here because there\u0027s a trigonometrical identity which uses the 2 here."},{"Start":"06:02.600 ","End":"06:10.350","Text":"In fact, this is equal to 1 plus cosine"},{"Start":"06:10.350 ","End":"06:21.000","Text":"2t dt from 0 to 2 Pi."},{"Start":"06:21.000 ","End":"06:23.710","Text":"I need some more space."},{"Start":"06:23.710 ","End":"06:26.524","Text":"This is a straightforward integral."},{"Start":"06:26.524 ","End":"06:29.195","Text":"What we get is 6."},{"Start":"06:29.195 ","End":"06:39.855","Text":"Now, 1, its integral is just t. I start off with sine of 2t,"},{"Start":"06:39.855 ","End":"06:43.170","Text":"but because there\u0027s a 2 here, it\u0027s not t,"},{"Start":"06:43.170 ","End":"06:45.225","Text":"I need to divide by 2,"},{"Start":"06:45.225 ","End":"06:48.100","Text":"so it\u0027s 1/2 sine 2t."},{"Start":"06:48.530 ","End":"06:55.410","Text":"All this is taken from 0 to 2 Pi."},{"Start":"06:55.410 ","End":"06:57.600","Text":"Let\u0027s see what we get."},{"Start":"06:57.600 ","End":"06:59.310","Text":"We get 6."},{"Start":"06:59.310 ","End":"07:04.800","Text":"Now, when t is 0,"},{"Start":"07:04.800 ","End":"07:12.660","Text":"I\u0027m going to get just 0 here and sine of twice 0 is 0."},{"Start":"07:12.660 ","End":"07:14.600","Text":"I don\u0027t know why I started with the second one,"},{"Start":"07:14.600 ","End":"07:18.720","Text":"but it\u0027s going to be minus 0."},{"Start":"07:19.840 ","End":"07:23.540","Text":"The first bit with 2 Pi,"},{"Start":"07:23.540 ","End":"07:30.440","Text":"I\u0027m going to get, when t is 2 Pi, it\u0027s 2 Pi."},{"Start":"07:30.440 ","End":"07:38.370","Text":"Now, a half of sine of 4 Pi is also 0."},{"Start":"07:40.220 ","End":"07:43.050","Text":"That\u0027s also 0, that part,"},{"Start":"07:43.050 ","End":"07:46.455","Text":"so I just get 2 Pi minus 0."},{"Start":"07:46.455 ","End":"07:51.970","Text":"In short, 12 Pi is the answer."},{"Start":"07:52.040 ","End":"07:58.455","Text":"That\u0027s just the part for the line integral."},{"Start":"07:58.455 ","End":"08:02.945","Text":"Now let\u0027s continue and do the other half of the exercise,"},{"Start":"08:02.945 ","End":"08:06.290","Text":"which is to figure out this."},{"Start":"08:06.290 ","End":"08:10.280","Text":"What I\u0027ll do is record the left-hand side and the 12 Pi."},{"Start":"08:10.280 ","End":"08:12.775","Text":"Now I\u0027m going to erase what I don\u0027t need."},{"Start":"08:12.775 ","End":"08:15.300","Text":"Let\u0027s start by computing the curl."},{"Start":"08:15.300 ","End":"08:17.170","Text":"Well, we have the formula here."},{"Start":"08:17.170 ","End":"08:20.240","Text":"We have P, Q, and R here. Let\u0027s see."},{"Start":"08:20.240 ","End":"08:24.050","Text":"R with respect to y is 5,"},{"Start":"08:24.050 ","End":"08:29.680","Text":"Q with respect to z is 0,"},{"Start":"08:29.740 ","End":"08:34.520","Text":"P with respect to z is 2,"},{"Start":"08:34.520 ","End":"08:38.915","Text":"R with respect to x is 0,"},{"Start":"08:38.915 ","End":"08:42.680","Text":"Q with respect to x is 3,"},{"Start":"08:42.680 ","End":"08:47.690","Text":"P with respect to y is 0."},{"Start":"08:47.690 ","End":"08:55.670","Text":"What we get is that the curl of F is"},{"Start":"08:55.670 ","End":"09:02.150","Text":"just 5i plus"},{"Start":"09:02.150 ","End":"09:09.170","Text":"2j plus 3k."},{"Start":"09:09.170 ","End":"09:11.840","Text":"Now it just so happens that we\u0027ve done this exercise"},{"Start":"09:11.840 ","End":"09:15.050","Text":"before in the chapter on surface integrals."},{"Start":"09:15.050 ","End":"09:17.300","Text":"At least it was phrased that this was"},{"Start":"09:17.300 ","End":"09:22.835","Text":"the original function F. It was given that this was F,"},{"Start":"09:22.835 ","End":"09:28.670","Text":"compute the double integral of F.n ds,"},{"Start":"09:28.670 ","End":"09:34.550","Text":"and S was the same surface."},{"Start":"09:34.550 ","End":"09:43.325","Text":"In that clip, we got the answer to be also 12 Pi,"},{"Start":"09:43.325 ","End":"09:45.170","Text":"and so we\u0027re done,"},{"Start":"09:45.170 ","End":"09:48.820","Text":"except that you might have missed that clip."},{"Start":"09:48.820 ","End":"09:51.740","Text":"Maybe you don\u0027t have it or you can\u0027t find it,"},{"Start":"09:51.740 ","End":"09:56.480","Text":"so what I\u0027m going to do is in the clip immediately following this one,"},{"Start":"09:56.480 ","End":"10:06.745","Text":"I\u0027ll repeat it and show you how the integral of this.n ds is equal to 12 Pi."},{"Start":"10:06.745 ","End":"10:11.280","Text":"I\u0027ll just note see next clip and it works out,"},{"Start":"10:11.280 ","End":"10:13.930","Text":"and so we\u0027re done."}],"ID":8823},{"Watched":false,"Name":"Exercise 1 – Verified second direction","Duration":"10m 56s","ChapterTopicVideoID":8755,"CourseChapterTopicPlaylistID":4973,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.700","Text":"In this exercise, we want to compute this type 2 surface integral,"},{"Start":"00:05.700 ","End":"00:10.260","Text":"where the vector field F is given as follows."},{"Start":"00:10.260 ","End":"00:14.370","Text":"It\u0027s actually a constant vector field."},{"Start":"00:14.370 ","End":"00:22.500","Text":"S is part of the paraboloid that\u0027s above the x-y plane."},{"Start":"00:22.500 ","End":"00:25.515","Text":"In other words, z is bigger or equal to 0."},{"Start":"00:25.515 ","End":"00:28.630","Text":"Sketch will really help here."},{"Start":"00:28.670 ","End":"00:34.785","Text":"Here\u0027s the Picture and what we\u0027re missing is,"},{"Start":"00:34.785 ","End":"00:39.910","Text":"I need a sketch of the projection onto the x-y plane."},{"Start":"00:39.910 ","End":"00:44.375","Text":"What I can do is to say because z is bigger or equal to 0,"},{"Start":"00:44.375 ","End":"00:49.250","Text":"that 4 minus x squared minus y squared bigger or equal to 0."},{"Start":"00:49.250 ","End":"00:55.475","Text":"That will give me that x squared plus y squared is less than or equal to 4,"},{"Start":"00:55.475 ","End":"00:57.590","Text":"which is 2 squared,"},{"Start":"00:57.590 ","End":"01:02.180","Text":"so this will give this region,"},{"Start":"01:02.180 ","End":"01:06.860","Text":"I\u0027ll call it R. Here it would be like if I trace this"},{"Start":"01:06.860 ","End":"01:12.335","Text":"out here and then there\u0027s an invisible hidden part here."},{"Start":"01:12.335 ","End":"01:14.465","Text":"Anyway, you get the idea,"},{"Start":"01:14.465 ","End":"01:16.725","Text":"project this down, I get this."},{"Start":"01:16.725 ","End":"01:19.935","Text":"On here I have this as the function, vector 1."},{"Start":"01:19.935 ","End":"01:21.290","Text":"I give this a letter,"},{"Start":"01:21.290 ","End":"01:28.050","Text":"I\u0027ll call this g of x and y so z is g of x, y."},{"Start":"01:28.050 ","End":"01:32.830","Text":"We have formulas for the case where z is a function of x and y."},{"Start":"01:32.830 ","End":"01:35.570","Text":"Well, actually there\u0027s 2 formulas."},{"Start":"01:35.570 ","End":"01:40.245","Text":"It depends on whether, let\u0027s take a point on this paraboloid,"},{"Start":"01:40.245 ","End":"01:43.980","Text":"the normal has an upward component."},{"Start":"01:43.980 ","End":"01:45.875","Text":"I\u0027m not saying it\u0027s vertical,"},{"Start":"01:45.875 ","End":"01:49.505","Text":"but it has a positive z component,"},{"Start":"01:49.505 ","End":"01:51.485","Text":"and when this is the case,"},{"Start":"01:51.485 ","End":"01:55.865","Text":"then the formula that applies is from the theorem,"},{"Start":"01:55.865 ","End":"01:58.280","Text":"I\u0027ll just label this S, is the surface,"},{"Start":"01:58.280 ","End":"02:04.520","Text":"is that the integral over S"},{"Start":"02:04.520 ","End":"02:12.930","Text":"of F.n ds is equal,"},{"Start":"02:14.980 ","End":"02:17.090","Text":"not the surface integral,"},{"Start":"02:17.090 ","End":"02:20.840","Text":"the regular double integral over the region R"},{"Start":"02:20.840 ","End":"02:27.965","Text":"of F. and I\u0027m going to use angular brackets notation,"},{"Start":"02:27.965 ","End":"02:35.725","Text":"for example, F and angular brackets notation would be the vector 5, 2, 3."},{"Start":"02:35.725 ","End":"02:38.400","Text":"Some people use round brackets."},{"Start":"02:38.530 ","End":"02:49.375","Text":"Here we have the formula is minus g_x, minus g_y, 1."},{"Start":"02:49.375 ","End":"02:51.320","Text":"If it was downward facing,"},{"Start":"02:51.320 ","End":"02:53.540","Text":"if it had a downward component on the normal,"},{"Start":"02:53.540 ","End":"02:57.290","Text":"we would just reverse all the signs here, plus, plus, minus."},{"Start":"02:57.290 ","End":"02:58.490","Text":"But we don\u0027t need that,"},{"Start":"02:58.490 ","End":"02:59.840","Text":"we just need this."},{"Start":"02:59.840 ","End":"03:02.330","Text":"Let\u0027s see what we get in our case."},{"Start":"03:02.330 ","End":"03:08.285","Text":"We get the double integral over S of F.n"},{"Start":"03:08.285 ","End":"03:16.530","Text":"ds equals the double integral over R of,"},{"Start":"03:16.530 ","End":"03:24.940","Text":"f is the vector 5, 2,"},{"Start":"03:24.940 ","End":"03:29.270","Text":"3 dot, now let\u0027s see,"},{"Start":"03:29.270 ","End":"03:34.855","Text":"minus g with respect to x is minus 2x."},{"Start":"03:34.855 ","End":"03:37.965","Text":"But I needed minus,"},{"Start":"03:37.965 ","End":"03:41.150","Text":"so I\u0027m going to actually erase this minus."},{"Start":"03:41.150 ","End":"03:45.050","Text":"I\u0027ll just write it as plus to show that it\u0027s minus minus."},{"Start":"03:45.050 ","End":"03:48.110","Text":"Then I need minus g_y,"},{"Start":"03:48.110 ","End":"03:57.720","Text":"which is plus 2y, and then 1."},{"Start":"03:59.000 ","End":"04:02.510","Text":"All that is, where is it?"},{"Start":"04:02.510 ","End":"04:03.650","Text":"Oh yeah, I forgot to write."},{"Start":"04:03.650 ","End":"04:09.755","Text":"That\u0027s dA. It\u0027s a bit crowded here."},{"Start":"04:09.755 ","End":"04:13.260","Text":"This is equal to,"},{"Start":"04:13.390 ","End":"04:18.525","Text":"let\u0027s see, I just do the dot product."},{"Start":"04:18.525 ","End":"04:24.735","Text":"We have 5 times 2x is 10x,"},{"Start":"04:24.735 ","End":"04:31.320","Text":"and then 2 times 2y is 4y,"},{"Start":"04:31.320 ","End":"04:35.960","Text":"and then 3 times 1 is 3."},{"Start":"04:35.960 ","End":"04:38.020","Text":"We want this integral,"},{"Start":"04:38.020 ","End":"04:47.560","Text":"dA over R. Now I want to convert this to polar coordinates."},{"Start":"04:47.560 ","End":"04:51.630","Text":"I\u0027ll remind you of the equations for polar conversion,"},{"Start":"04:51.630 ","End":"04:53.120","Text":"I\u0027ll write them over here."},{"Start":"04:53.120 ","End":"04:56.885","Text":"We have that x equals r cosine Theta."},{"Start":"04:56.885 ","End":"04:59.195","Text":"We go from x, y to r and Theta."},{"Start":"04:59.195 ","End":"05:03.465","Text":"We put y equals r sine Theta."},{"Start":"05:03.465 ","End":"05:09.765","Text":"Instead of dA, we put rdr dTheta."},{"Start":"05:09.765 ","End":"05:17.820","Text":"There\u0027s another useful formula that x squared plus y squared equals r squared."},{"Start":"05:17.820 ","End":"05:20.530","Text":"I don\u0027t know if we\u0027ll use that one here."},{"Start":"05:20.540 ","End":"05:26.115","Text":"Anyway, back here and we\u0027ll get,"},{"Start":"05:26.115 ","End":"05:29.470","Text":"and I will change color for polar."},{"Start":"05:29.470 ","End":"05:32.030","Text":"I also want to change the region."},{"Start":"05:32.030 ","End":"05:36.725","Text":"We\u0027ve seen this kind of region before when we have a disc,"},{"Start":"05:36.725 ","End":"05:38.660","Text":"I\u0027ll just quickly remind you."},{"Start":"05:38.660 ","End":"05:41.030","Text":"Theta goes all the way around."},{"Start":"05:41.030 ","End":"05:45.115","Text":"It goes from 0 all the way around to 2 Pi."},{"Start":"05:45.115 ","End":"05:46.850","Text":"From here, it goes all the way around,"},{"Start":"05:46.850 ","End":"05:49.204","Text":"comes back here, and r,"},{"Start":"05:49.204 ","End":"05:54.665","Text":"the radius goes from being 0 here to being 2 here."},{"Start":"05:54.665 ","End":"06:01.110","Text":"So we\u0027ve got the integral from 0-2 Pi for Theta."},{"Start":"06:01.390 ","End":"06:06.340","Text":"For r we go from 0-2."},{"Start":"06:06.340 ","End":"06:10.445","Text":"Then we have these which we substitute as 10."},{"Start":"06:10.445 ","End":"06:15.530","Text":"Now x is r cosine Theta and here we have"},{"Start":"06:15.530 ","End":"06:25.750","Text":"4r sine Theta plus 3r dr dTheta."},{"Start":"06:25.750 ","End":"06:27.740","Text":"Notice the r here,"},{"Start":"06:27.740 ","End":"06:29.225","Text":"one tends to forget it."},{"Start":"06:29.225 ","End":"06:31.889","Text":"Let\u0027s see what we get."},{"Start":"06:32.590 ","End":"06:39.240","Text":"Just multiply this by r. We go from 0-2 Pi with Theta,"},{"Start":"06:39.240 ","End":"06:42.810","Text":"and then we have, multiplying everything by r,"},{"Start":"06:42.810 ","End":"06:47.925","Text":"10r squared cosine Theta plus 4r"},{"Start":"06:47.925 ","End":"06:55.470","Text":"squared sine Theta and then plus 3r,"},{"Start":"06:55.470 ","End":"07:00.180","Text":"all this, dr dTheta,"},{"Start":"07:00.180 ","End":"07:01.970","Text":"still not the integral."},{"Start":"07:01.970 ","End":"07:04.380","Text":"Now we\u0027ll do the integral."},{"Start":"07:05.440 ","End":"07:09.500","Text":"I forgot to write r goes from 0-2."},{"Start":"07:09.500 ","End":"07:13.895","Text":"Then this equals the integral from 0-2 Pi."},{"Start":"07:13.895 ","End":"07:15.800","Text":"The integral of this,"},{"Start":"07:15.800 ","End":"07:17.570","Text":"I raise the power by 1,"},{"Start":"07:17.570 ","End":"07:19.505","Text":"cosine Theta is a constant of course,"},{"Start":"07:19.505 ","End":"07:29.269","Text":"so I get 10 over 3r cubed cosine Theta,"},{"Start":"07:29.269 ","End":"07:36.029","Text":"and here 4/3r cubed sine Theta,"},{"Start":"07:36.029 ","End":"07:40.570","Text":"and here 3/2r squared."},{"Start":"07:40.570 ","End":"07:42.580","Text":"That\u0027s really the integral, dr,"},{"Start":"07:42.580 ","End":"07:47.569","Text":"but I do have to substitute that it goes from 0-2,"},{"Start":"07:48.590 ","End":"07:53.560","Text":"and then after that I still have to do an integral dTheta we\u0027ll get rid of r this way."},{"Start":"07:53.560 ","End":"07:55.300","Text":"This is r from 0-2."},{"Start":"07:55.300 ","End":"07:59.830","Text":"Now if I plug in r equals 0,"},{"Start":"07:59.830 ","End":"08:04.130","Text":"everything\u0027s going to be 0, so I just need to plug in r equals 2."},{"Start":"08:04.130 ","End":"08:08.085","Text":"Anyway, integral from 0-2 Pi."},{"Start":"08:08.085 ","End":"08:11.520","Text":"When r is 2, 2 cubed is 8,"},{"Start":"08:11.520 ","End":"08:17.100","Text":"so it\u0027s 80/3 cosine Theta."},{"Start":"08:17.100 ","End":"08:21.505","Text":"Then also here, r cubed is 8."},{"Start":"08:21.505 ","End":"08:31.780","Text":"8 times 4 is 32/3 sine Theta."},{"Start":"08:31.780 ","End":"08:35.845","Text":"Here, when r is 2,"},{"Start":"08:35.845 ","End":"08:41.170","Text":"2 squared is 4, 4 times 3/2 is 6."},{"Start":"08:41.170 ","End":"08:44.365","Text":"So I\u0027ve got just 6,"},{"Start":"08:44.365 ","End":"08:48.085","Text":"and this is dTheta."},{"Start":"08:48.085 ","End":"08:52.950","Text":"Continuing. Now we can do the integral."},{"Start":"08:52.950 ","End":"08:55.870","Text":"The integral of cosine is"},{"Start":"09:04.670 ","End":"09:10.720","Text":"sine Theta without the minus."},{"Start":"09:10.990 ","End":"09:14.705","Text":"The integral of sine is minus cosine."},{"Start":"09:14.705 ","End":"09:18.720","Text":"So minus 32/3 cosine Theta,"},{"Start":"09:18.720 ","End":"09:21.689","Text":"and the integral of 6 is just 6 Theta,"},{"Start":"09:21.689 ","End":"09:26.700","Text":"all this has to be taken from 0-2 Pi."},{"Start":"09:26.700 ","End":"09:29.055","Text":"Let\u0027s see what we get."},{"Start":"09:29.055 ","End":"09:32.549","Text":"If we put Theta equals 2 Pi,"},{"Start":"09:32.549 ","End":"09:37.770","Text":"sine of 2 Pi is 0 and cosine of 2 Pi is the same"},{"Start":"09:37.770 ","End":"09:44.355","Text":"as cosine 0 is 1,"},{"Start":"09:44.355 ","End":"09:49.980","Text":"so this is minus 32/3,"},{"Start":"09:49.980 ","End":"09:53.895","Text":"and 6 Theta is just 12 Pi."},{"Start":"09:53.895 ","End":"09:56.260","Text":"That\u0027s the 2 Pi part."},{"Start":"09:56.260 ","End":"09:59.644","Text":"Now the 0 part, which I subtract,"},{"Start":"09:59.644 ","End":"10:02.074","Text":"sine of 0 is 0,"},{"Start":"10:02.074 ","End":"10:04.955","Text":"cosine of 0 is 1."},{"Start":"10:04.955 ","End":"10:17.960","Text":"So it\u0027s minus 32/3,"},{"Start":"10:17.960 ","End":"10:20.000","Text":"and here it should be a minus of course,"},{"Start":"10:20.000 ","End":"10:23.720","Text":"because cosine Theta, cosine 2 Pi is also 1,"},{"Start":"10:23.720 ","End":"10:25.130","Text":"but the minus was there,"},{"Start":"10:25.130 ","End":"10:29.250","Text":"just cosine 2 Pi and cosine 0 are the same thing,"},{"Start":"10:29.570 ","End":"10:35.120","Text":"minus, and then here I have also 0 and Theta 0."},{"Start":"10:35.120 ","End":"10:37.595","Text":"So this minus this,"},{"Start":"10:37.595 ","End":"10:40.505","Text":"so what does this give me?"},{"Start":"10:40.505 ","End":"10:44.630","Text":"Minus 32/3 cancels with plus 32/3."},{"Start":"10:44.630 ","End":"10:48.515","Text":"If you like, it\u0027s this thing minus the same thing anyway, this cancels,"},{"Start":"10:48.515 ","End":"10:52.630","Text":"and all we\u0027re left with is 12 Pi,"},{"Start":"10:52.630 ","End":"10:54.760","Text":"which I\u0027ll highlight."},{"Start":"10:54.760 ","End":"10:57.300","Text":"We are done."}],"ID":8824},{"Watched":false,"Name":"Exercise 2 – Verified one direction","Duration":"13m 21s","ChapterTopicVideoID":8758,"CourseChapterTopicPlaylistID":4973,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.390","Text":"In this exercise, we have to verify Stokes\u0027 theorem,"},{"Start":"00:03.390 ","End":"00:07.500","Text":"which in condensed form can be written like this."},{"Start":"00:07.500 ","End":"00:12.435","Text":"We have the following case where the vector field F is given as follows,"},{"Start":"00:12.435 ","End":"00:14.895","Text":"and the surface S is"},{"Start":"00:14.895 ","End":"00:23.760","Text":"the upper hemisphere that lies above the xy plane and has radius 4,"},{"Start":"00:23.760 ","End":"00:25.590","Text":"and that\u0027s sketched here."},{"Start":"00:25.590 ","End":"00:29.040","Text":"Sphere, meaning just the shell, the surface."},{"Start":"00:29.230 ","End":"00:35.005","Text":"The boundary would be this curve here, C,"},{"Start":"00:35.005 ","End":"00:42.600","Text":"which goes counterclockwise according to the outward normal."},{"Start":"00:43.880 ","End":"00:46.790","Text":"If we look at it from above,"},{"Start":"00:46.790 ","End":"00:49.010","Text":"then counterclockwise is this."},{"Start":"00:49.010 ","End":"00:55.230","Text":"In short, this induces the counterclockwise orientation here."},{"Start":"00:55.640 ","End":"01:00.459","Text":"I think we\u0027ll start with the left-hand side."},{"Start":"01:00.530 ","End":"01:04.265","Text":"There\u0027s a formula that the"},{"Start":"01:04.265 ","End":"01:13.440","Text":"integral of F.DR in general,"},{"Start":"01:13.440 ","End":"01:16.560","Text":"is equal to this."},{"Start":"01:16.560 ","End":"01:19.550","Text":"I have to tell you what what PQ and R mean."},{"Start":"01:19.550 ","End":"01:23.000","Text":"These are just the components of F. So in our case,"},{"Start":"01:23.000 ","End":"01:26.090","Text":"the first bit here is P, this is Q,"},{"Start":"01:26.090 ","End":"01:29.960","Text":"and this is R. Some people like to use little f, g,"},{"Start":"01:29.960 ","End":"01:31.940","Text":"and h. I prefer to use capital P,"},{"Start":"01:31.940 ","End":"01:35.330","Text":"Q and R. It doesn\u0027t matter though."},{"Start":"01:35.330 ","End":"01:42.665","Text":"We put a circle here because it\u0027s a closed curve,"},{"Start":"01:42.665 ","End":"01:50.084","Text":"and what we get is the integral over the curve."},{"Start":"01:50.084 ","End":"01:51.485","Text":"I\u0027ll just write it out,"},{"Start":"01:51.485 ","End":"01:57.490","Text":"x squared plus y minus 4 dx,"},{"Start":"01:57.490 ","End":"02:00.230","Text":"and then we have q, d,"},{"Start":"02:00.230 ","End":"02:02.015","Text":"y, but there\u0027s a minus,"},{"Start":"02:02.015 ","End":"02:06.110","Text":"minus 3 xy, dy,"},{"Start":"02:06.110 ","End":"02:15.060","Text":"and then plus 2 xz plus z squared dz."},{"Start":"02:16.520 ","End":"02:24.020","Text":"The way we do this is to parameterize the curve C. Well,"},{"Start":"02:24.020 ","End":"02:28.550","Text":"we know how to parameterize a circle based on the radius 4."},{"Start":"02:28.550 ","End":"02:30.410","Text":"When it\u0027s centered at the origin,"},{"Start":"02:30.410 ","End":"02:36.180","Text":"we just have that x is the radius which is 4 cosine of,"},{"Start":"02:36.180 ","End":"02:38.675","Text":"we\u0027ll take the angle as the parameter,"},{"Start":"02:38.675 ","End":"02:43.325","Text":"call it t. T would be actually the angle"},{"Start":"02:43.325 ","End":"02:48.635","Text":"from the x-axis going counterclockwise, but it doesn\u0027t matter."},{"Start":"02:48.635 ","End":"02:54.395","Text":"We just know it as the formula x equals radius cosine t,"},{"Start":"02:54.395 ","End":"03:04.215","Text":"y equals the radius sine t. T goes between 0 and 2 Pi,"},{"Start":"03:04.215 ","End":"03:05.910","Text":"but that\u0027s in 2-dimensions."},{"Start":"03:05.910 ","End":"03:08.770","Text":"In 3-dimensions, we also have a z,"},{"Start":"03:08.770 ","End":"03:10.750","Text":"but this circle is in the x-y plane,"},{"Start":"03:10.750 ","End":"03:13.190","Text":"so z equals 0."},{"Start":"03:13.860 ","End":"03:20.920","Text":"I guess we\u0027ll also need the dx, dy, and dz."},{"Start":"03:21.650 ","End":"03:32.520","Text":"Yeah, so dx would be minus 4 sine t. Dy is 4, oh yeah,"},{"Start":"03:32.520 ","End":"03:36.015","Text":"there\u0027s a dt, of course 4 cosine t,"},{"Start":"03:36.015 ","End":"03:42.135","Text":"dt and dz is 0 or 0dt whatever,"},{"Start":"03:42.135 ","End":"03:45.620","Text":"and now we can go and substitute here."},{"Start":"03:45.620 ","End":"03:48.850","Text":"What we get is the integral."},{"Start":"03:48.850 ","End":"03:55.270","Text":"The parameter t goes from 0-2 Pi."},{"Start":"03:55.270 ","End":"03:56.380","Text":"We just substitute everything."},{"Start":"03:56.380 ","End":"04:06.540","Text":"X squared is 16 cosine squared t and y"},{"Start":"04:06.540 ","End":"04:11.110","Text":"is 4 sine t minus"},{"Start":"04:11.110 ","End":"04:17.450","Text":"4 times dx is minus 4 sine tdt."},{"Start":"04:17.450 ","End":"04:26.190","Text":"Just write that, minus 4 sine t. We\u0027ll put 1 big dt at the end."},{"Start":"04:26.190 ","End":"04:30.465","Text":"So I\u0027ll put a square brackets in here,"},{"Start":"04:30.465 ","End":"04:33.110","Text":"and then afterwards, we\u0027ll close it with a dt."},{"Start":"04:33.110 ","End":"04:35.675","Text":"Now minus 3xy, dy."},{"Start":"04:35.675 ","End":"04:43.145","Text":"So we have minus 3x is 4."},{"Start":"04:43.145 ","End":"04:48.029","Text":"Cosine ty is 4 sine t,"},{"Start":"04:48.029 ","End":"04:52.755","Text":"and dy is another 4 cosine tdt."},{"Start":"04:52.755 ","End":"05:00.480","Text":"I thought I was running out of space,"},{"Start":"05:00.480 ","End":"05:04.260","Text":"but look dz is 0,"},{"Start":"05:04.260 ","End":"05:09.435","Text":"and so because dz is 0,"},{"Start":"05:09.435 ","End":"05:12.640","Text":"we can just end it here."},{"Start":"05:13.520 ","End":"05:16.650","Text":"Yeah, the dt belongs to both."},{"Start":"05:16.650 ","End":"05:20.230","Text":"Okay, so we\u0027re all set here."},{"Start":"05:20.740 ","End":"05:24.350","Text":"Now let\u0027s expand the brackets and see what we get."},{"Start":"05:24.350 ","End":"05:30.655","Text":"We get the integral from 0-2 Pi."},{"Start":"05:30.655 ","End":"05:34.310","Text":"Here we have 3 terms that\u0027s due each 2."},{"Start":"05:34.310 ","End":"05:40.975","Text":"Minus 4 times 16 is minus 64,"},{"Start":"05:40.975 ","End":"05:45.645","Text":"cosine squared t sine t,"},{"Start":"05:45.645 ","End":"05:52.415","Text":"and then we have minus 16 sine squared t,"},{"Start":"05:52.415 ","End":"06:01.690","Text":"and then plus 16 sine t. Here,"},{"Start":"06:01.690 ","End":"06:03.150","Text":"3 times 4 times,"},{"Start":"06:03.150 ","End":"06:07.280","Text":"4 times 4 comes out to be a 192."},{"Start":"06:07.280 ","End":"06:12.170","Text":"So we have minus 192 cosine"},{"Start":"06:12.170 ","End":"06:18.830","Text":"squared t sine t,"},{"Start":"06:18.830 ","End":"06:23.900","Text":"and all this dt."},{"Start":"06:23.900 ","End":"06:28.650","Text":"The only thing we can collect together would be,"},{"Start":"06:28.760 ","End":"06:32.915","Text":"it\u0027s kind of ugly, but let\u0027s continue."},{"Start":"06:32.915 ","End":"06:37.414","Text":"I\u0027m actually going to break it up into separate integrals."},{"Start":"06:37.414 ","End":"06:43.460","Text":"The first part, I\u0027ll take the cosine squared t sine t. I\u0027ve got 2 of them."},{"Start":"06:43.460 ","End":"06:48.560","Text":"I\u0027ve got a minus 64 here and a minus 192 here."},{"Start":"06:48.560 ","End":"06:57.630","Text":"So I\u0027ve got minus 256 cosine squared t,"},{"Start":"06:57.630 ","End":"07:03.230","Text":"sine t, dt, and then I\u0027ll take"},{"Start":"07:03.230 ","End":"07:08.345","Text":"the other integral minus"},{"Start":"07:08.345 ","End":"07:13.975","Text":"in fact why don\u0027t I take this out in front."},{"Start":"07:13.975 ","End":"07:17.430","Text":"I just copy the integral there,"},{"Start":"07:17.430 ","End":"07:19.965","Text":"okay, that\u0027s putting this in front,"},{"Start":"07:19.965 ","End":"07:28.065","Text":"and then 16 integral from 0-2 Pi sine square tdt,"},{"Start":"07:28.065 ","End":"07:31.085","Text":"and then from here,"},{"Start":"07:31.085 ","End":"07:38.670","Text":"16 times the integral from 0 to 2 Pi of sine tdt."},{"Start":"07:42.160 ","End":"07:45.410","Text":"Let me do some of these integrals at the side."},{"Start":"07:45.410 ","End":"07:47.960","Text":"Now, this 1 is,"},{"Start":"07:47.960 ","End":"07:53.780","Text":"the integral of this would be minus 1/3,"},{"Start":"07:53.780 ","End":"07:58.070","Text":"cosine sine cubed t,"},{"Start":"07:58.070 ","End":"08:02.660","Text":"the indefinite integral because you could think of it as a substitution."},{"Start":"08:02.660 ","End":"08:05.390","Text":"If I substitute for cosine t,"},{"Start":"08:05.390 ","End":"08:08.390","Text":"let\u0027s say u is cosine of t,"},{"Start":"08:08.390 ","End":"08:13.095","Text":"then du would be minus sine t,"},{"Start":"08:13.095 ","End":"08:15.575","Text":"and when you take something squared,"},{"Start":"08:15.575 ","End":"08:17.885","Text":"it\u0027s 1/3 of that thing cubed."},{"Start":"08:17.885 ","End":"08:22.685","Text":"In short, you can differentiate this and see that you get cosine"},{"Start":"08:22.685 ","End":"08:28.790","Text":"squared t. This I take from 0-2 Pi,"},{"Start":"08:28.790 ","End":"08:32.450","Text":"and then we get, well,"},{"Start":"08:32.450 ","End":"08:38.719","Text":"it\u0027s going to be 0 because the cosine of 2 Pi and the cosine of 0 are the same,"},{"Start":"08:38.719 ","End":"08:41.510","Text":"so we\u0027re going to be subtracting the same thing,"},{"Start":"08:41.510 ","End":"08:45.440","Text":"whatever it is, so that would be 0."},{"Start":"08:45.440 ","End":"08:48.200","Text":"That first bit is 0,"},{"Start":"08:48.200 ","End":"08:49.685","Text":"then the second bit,"},{"Start":"08:49.685 ","End":"08:54.110","Text":"the sine squared t. Well,"},{"Start":"08:54.110 ","End":"08:55.310","Text":"I\u0027m sure we\u0027ve done this before."},{"Start":"08:55.310 ","End":"08:57.110","Text":"We use trigonometric identities,"},{"Start":"08:57.110 ","End":"09:06.115","Text":"I\u0027ll just give you the answer that it\u0027s 1.5t minus 1/4 sine of 2t."},{"Start":"09:06.115 ","End":"09:11.665","Text":"I have to take this from 0-2 Pi."},{"Start":"09:11.665 ","End":"09:15.260","Text":"Now the sine part doesn\u0027t give me anything because sine of"},{"Start":"09:15.260 ","End":"09:20.390","Text":"0 and sine of 4 Pi are the same, that will cancel out."},{"Start":"09:20.390 ","End":"09:22.795","Text":"So I just need to substitute here."},{"Start":"09:22.795 ","End":"09:31.255","Text":"I get 0.5 of 2 Pi is Pi minus 0 and just end up with Pi."},{"Start":"09:31.255 ","End":"09:36.260","Text":"The last 1, the integral of sine t is"},{"Start":"09:36.260 ","End":"09:44.300","Text":"minus cosine t from 0-2 Pi."},{"Start":"09:44.300 ","End":"09:49.444","Text":"Again, cosine of 0 and cosine of 2 Pi are the same,"},{"Start":"09:49.444 ","End":"09:51.440","Text":"so I\u0027ll be subtracting the same thing."},{"Start":"09:51.440 ","End":"09:53.300","Text":"So I\u0027ll be left with 0."},{"Start":"09:53.300 ","End":"10:00.020","Text":"All I\u0027m left with is that this bit here is Pi,"},{"Start":"10:00.020 ","End":"10:01.670","Text":"this bit is 0."},{"Start":"10:01.670 ","End":"10:11.770","Text":"So I\u0027ve got 0 minus 16 Pi plus 0 and so I end up with minus 16 Pi."},{"Start":"10:11.770 ","End":"10:16.595","Text":"But remember that\u0027s just the left-hand side."},{"Start":"10:16.595 ","End":"10:23.770","Text":"We have also the right-hand side to check."},{"Start":"10:23.770 ","End":"10:26.685","Text":"Let me record here the result."},{"Start":"10:26.685 ","End":"10:30.105","Text":"Minus 16 Pi for the left-hand side,"},{"Start":"10:30.105 ","End":"10:32.510","Text":"and before the right-hand side,"},{"Start":"10:32.510 ","End":"10:36.210","Text":"I\u0027ll just clear the board. That\u0027s better."},{"Start":"10:36.210 ","End":"10:42.760","Text":"I want the formula for the curl."},{"Start":"10:43.160 ","End":"10:45.690","Text":"Here\u0027s the formula."},{"Start":"10:45.690 ","End":"10:52.390","Text":"In our case, r with respect to y is 0,"},{"Start":"10:52.390 ","End":"10:57.040","Text":"q with respect to z is 0,"},{"Start":"10:57.040 ","End":"11:01.795","Text":"P with respect to z is 0,"},{"Start":"11:01.795 ","End":"11:08.470","Text":"R with respect to x is 2z."},{"Start":"11:11.000 ","End":"11:18.330","Text":"Q with respect to x would be minus 3y,"},{"Start":"11:18.330 ","End":"11:25.305","Text":"p with respect to y would be 1,"},{"Start":"11:25.305 ","End":"11:28.175","Text":"and I haven\u0027t written the minus in."},{"Start":"11:28.175 ","End":"11:34.345","Text":"But what we get is that the curl"},{"Start":"11:34.345 ","End":"11:42.285","Text":"of F is just nothing I think I\u0027ll keep the 0 in 0 i."},{"Start":"11:42.285 ","End":"11:45.975","Text":"Then we have minus 2z,"},{"Start":"11:45.975 ","End":"11:51.645","Text":"j and here we have minus 3y minus 1."},{"Start":"11:51.645 ","End":"11:55.169","Text":"So I\u0027ll write it in brackets,"},{"Start":"11:55.169 ","End":"11:58.830","Text":"so it\u0027s minus 3y minus"},{"Start":"11:58.830 ","End":"12:04.850","Text":"1k and we need"},{"Start":"12:04.850 ","End":"12:11.179","Text":"the double integral of this over S. Now,"},{"Start":"12:11.179 ","End":"12:16.865","Text":"we did this in a previous exercise in the section on surface integrals."},{"Start":"12:16.865 ","End":"12:22.340","Text":"The original F was given as this not the curl of something,"},{"Start":"12:22.340 ","End":"12:26.270","Text":"but we were given that F is equal to this."},{"Start":"12:26.270 ","End":"12:32.030","Text":"All the rest of the setup was the same with the upper 0.5 Sphere radius 4 and so on."},{"Start":"12:32.030 ","End":"12:38.975","Text":"We had to compute the double integral over S. There it was written as F not as curl F,"},{"Start":"12:38.975 ","End":"12:43.760","Text":"but it was the same thing here, DS."},{"Start":"12:43.760 ","End":"12:49.055","Text":"The answer to that exercise came out to be minus 16 Pi,"},{"Start":"12:49.055 ","End":"12:54.620","Text":"just the same, and so indeed they are equal."},{"Start":"12:54.620 ","End":"13:02.620","Text":"But in case you don\u0027t have that exercise or you can\u0027t find it with the surface integrals,"},{"Start":"13:02.620 ","End":"13:08.725","Text":"then I will put it immediately in the following clip."},{"Start":"13:08.725 ","End":"13:13.034","Text":"I\u0027ll just write see next clip."},{"Start":"13:13.034 ","End":"13:18.469","Text":"There I\u0027ll do the calculation for this surface integral."},{"Start":"13:18.469 ","End":"13:21.750","Text":"Meanwhile here we\u0027re done."}],"ID":8825},{"Watched":false,"Name":"Exercise 2 – Verified second direction","Duration":"13m ","ChapterTopicVideoID":8757,"CourseChapterTopicPlaylistID":4973,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.770","Text":"In this exercise, we have to compute this type 2 surface integral F.ndS."},{"Start":"00:07.770 ","End":"00:12.360","Text":"We\u0027re given the vector field F. We\u0027re given the i,"},{"Start":"00:12.360 ","End":"00:13.709","Text":"j and k components."},{"Start":"00:13.709 ","End":"00:18.930","Text":"Though I might decide to use the angular brackets notation,"},{"Start":"00:18.930 ","End":"00:28.500","Text":"in which case it\u0027s 0 minus 2z and minus 3y minus 1."},{"Start":"00:28.500 ","End":"00:30.990","Text":"Use both notations, brackets or i, j,"},{"Start":"00:30.990 ","End":"00:39.285","Text":"k. S is the hemisphere that\u0027s above the xy plane, half a sphere."},{"Start":"00:39.285 ","End":"00:44.450","Text":"One of the reasons we have a hemisphere or that it\u0027s useful to"},{"Start":"00:44.450 ","End":"00:50.165","Text":"have a hemisphere is that we can then write z as a function of x and y."},{"Start":"00:50.165 ","End":"00:52.850","Text":"In fact, for radius 4,"},{"Start":"00:52.850 ","End":"00:56.840","Text":"we could write the equation that z equals"},{"Start":"00:56.840 ","End":"01:07.040","Text":"the square root of 4 squared minus x squared minus y squared."},{"Start":"01:07.040 ","End":"01:11.330","Text":"This comes from the equation of the sphere where x"},{"Start":"01:11.330 ","End":"01:15.185","Text":"squared plus y squared plus z squared equals r squared."},{"Start":"01:15.185 ","End":"01:17.590","Text":"But in our case,"},{"Start":"01:17.590 ","End":"01:22.915","Text":"z is bigger or equal to 0 because we\u0027re above the xy plane."},{"Start":"01:22.915 ","End":"01:28.730","Text":"We take the positive square root after we transfer the x and y to the other side."},{"Start":"01:28.730 ","End":"01:31.520","Text":"This will be our g of x,"},{"Start":"01:31.520 ","End":"01:33.455","Text":"y, which we\u0027ll use,"},{"Start":"01:33.455 ","End":"01:36.950","Text":"and there are formulas that tell us how to"},{"Start":"01:36.950 ","End":"01:41.700","Text":"convert from the surface integral to a regular double integral."},{"Start":"01:41.830 ","End":"01:44.810","Text":"There are actually 2 formulas,"},{"Start":"01:44.810 ","End":"01:46.700","Text":"and they depend on whether"},{"Start":"01:46.700 ","End":"01:52.820","Text":"the normal vector has an upward component or a downward component."},{"Start":"01:52.820 ","End":"01:56.855","Text":"In this case, you can see that everywhere on the upper hemisphere, the normal,"},{"Start":"01:56.855 ","End":"01:58.250","Text":"I\u0027m not saying it\u0027s vertical,"},{"Start":"01:58.250 ","End":"02:03.230","Text":"but it has a vertical component that is in the direction of the positive z-axis."},{"Start":"02:03.230 ","End":"02:06.545","Text":"I\u0027ll write down the formula for this case."},{"Start":"02:06.545 ","End":"02:08.450","Text":"It turns out to be similar."},{"Start":"02:08.450 ","End":"02:11.600","Text":"If I just minus when the normal"},{"Start":"02:11.600 ","End":"02:16.099","Text":"is going downwards as it would say in the lower hemisphere."},{"Start":"02:16.099 ","End":"02:21.890","Text":"Anyway, the formula tells us that the double integral over a surface"},{"Start":"02:21.890 ","End":"02:31.865","Text":"of F.ndS is the double integral over R,"},{"Start":"02:31.865 ","End":"02:39.480","Text":"where R is the projection of this surface down onto the xy plane of"},{"Start":"02:40.460 ","End":"02:47.190","Text":"F dot and this"},{"Start":"02:47.190 ","End":"02:52.520","Text":"will be minus g with respect to x. g is this function here,"},{"Start":"02:52.520 ","End":"03:00.150","Text":"partial derivative with respect to x minus g with respect to y, and 1 dA."},{"Start":"03:00.150 ","End":"03:03.709","Text":"Like I said, if the normal was facing downwards,"},{"Start":"03:03.709 ","End":"03:06.080","Text":"we\u0027d reverse all the signs here."},{"Start":"03:06.080 ","End":"03:10.290","Text":"But I\u0027m not going to write the other formula because you don\u0027t need it."},{"Start":"03:10.450 ","End":"03:14.880","Text":"Let\u0027s do this dot product."},{"Start":"03:15.010 ","End":"03:17.450","Text":"Now, let\u0027s go over here,"},{"Start":"03:17.450 ","End":"03:20.180","Text":"what we need is the double integral over R,"},{"Start":"03:20.180 ","End":"03:26.760","Text":"where R is this in the xy plane disk of radius 4."},{"Start":"03:26.780 ","End":"03:34.025","Text":"We need F. I\u0027ll do it in the vector notation."},{"Start":"03:34.025 ","End":"03:38.765","Text":"I mean in the brackets notation 0"},{"Start":"03:38.765 ","End":"03:45.495","Text":"minus 2z minus 3y minus 1 dot."},{"Start":"03:45.495 ","End":"03:50.160","Text":"Now, what is gx?"},{"Start":"03:50.160 ","End":"03:56.985","Text":"gx will be, not so simple but not so difficult,"},{"Start":"03:56.985 ","End":"03:58.890","Text":"have a square root,"},{"Start":"03:58.890 ","End":"04:07.200","Text":"so 1 over twice the square root of 16 minus x squared minus y squared."},{"Start":"04:07.200 ","End":"04:10.325","Text":"On the numerator, the derivative,"},{"Start":"04:10.325 ","End":"04:13.220","Text":"the inner derivative, and it\u0027s with respect to x."},{"Start":"04:13.220 ","End":"04:16.195","Text":"It\u0027s minus 2x."},{"Start":"04:16.195 ","End":"04:19.455","Text":"But the 2 cancels."},{"Start":"04:19.455 ","End":"04:22.790","Text":"Then same thing just with y on the top,"},{"Start":"04:22.790 ","End":"04:25.100","Text":"it\u0027s clear that will get the same thing."},{"Start":"04:25.100 ","End":"04:30.455","Text":"Square root of 16 minus x squared minus y squared."},{"Start":"04:30.455 ","End":"04:35.130","Text":"Lastly, just 1 dA."},{"Start":"04:37.790 ","End":"04:44.510","Text":"This equals double integral over R. Now,"},{"Start":"04:44.510 ","End":"04:46.850","Text":"if we multiply out the first one,"},{"Start":"04:46.850 ","End":"04:49.790","Text":"0 times this is nothing."},{"Start":"04:49.790 ","End":"04:54.530","Text":"Then I have minus 2z times minus y over this."},{"Start":"04:54.530 ","End":"04:59.465","Text":"It\u0027s just 2zy over"},{"Start":"04:59.465 ","End":"05:05.930","Text":"the square root of 16 minus x squared minus y squared."},{"Start":"05:05.930 ","End":"05:11.815","Text":"Then the last component is just minus"},{"Start":"05:11.815 ","End":"05:21.170","Text":"3y minus 1 and all this is dA."},{"Start":"05:21.170 ","End":"05:26.165","Text":"However, we don\u0027t want z here."},{"Start":"05:26.165 ","End":"05:30.410","Text":"We\u0027re replacing z by g of x,"},{"Start":"05:30.410 ","End":"05:32.195","Text":"y, which is this."},{"Start":"05:32.195 ","End":"05:40.350","Text":"Actually, z equals the denominator here."},{"Start":"05:40.350 ","End":"05:46.415","Text":"What I\u0027m saying is that I can cancel because z is equal to this."},{"Start":"05:46.415 ","End":"05:49.140","Text":"This will cancel out."},{"Start":"05:50.150 ","End":"05:56.960","Text":"We\u0027ll be left with just the double integral over R,"},{"Start":"05:56.960 ","End":"06:06.120","Text":"2y minus 3y is minus y minus 1 dA."},{"Start":"06:07.840 ","End":"06:12.245","Text":"What I wrote here is g with respect to x."},{"Start":"06:12.245 ","End":"06:16.535","Text":"I almost forgot that I need minus gx from the formula."},{"Start":"06:16.535 ","End":"06:24.950","Text":"The minus will knock out this minus and 2 will cancel with 2."},{"Start":"06:24.950 ","End":"06:32.890","Text":"Next, well, it\u0027ll be the same thing just with y instead of x,"},{"Start":"06:32.890 ","End":"06:35.630","Text":"because the inner derivative would be minus 2y,"},{"Start":"06:35.630 ","End":"06:38.545","Text":"the 2 would cancel again the minus with the minus."},{"Start":"06:38.545 ","End":"06:49.220","Text":"I just get y over square root of 16 minus x squared minus y squared."},{"Start":"06:49.820 ","End":"06:56.230","Text":"The last component is just 1, and that\u0027s dA."},{"Start":"06:58.640 ","End":"07:04.340","Text":"Actually, what I have to do here is replace z."},{"Start":"07:05.160 ","End":"07:08.785","Text":"We only want the x and y over this region,"},{"Start":"07:08.785 ","End":"07:12.835","Text":"or replace z by what it\u0027s equal to which is"},{"Start":"07:12.835 ","End":"07:20.125","Text":"the square root of 16 minus x squared minus y squared."},{"Start":"07:20.125 ","End":"07:23.095","Text":"Let\u0027s do the multiplication."},{"Start":"07:23.095 ","End":"07:27.365","Text":"The dot product we get,"},{"Start":"07:27.365 ","End":"07:31.515","Text":"the first component, 0 times anything is 0."},{"Start":"07:31.515 ","End":"07:34.230","Text":"That\u0027s just 0 plus."},{"Start":"07:34.230 ","End":"07:36.100","Text":"Now, let\u0027s see what else do I get?"},{"Start":"07:36.100 ","End":"07:43.030","Text":"I get minus 2 square root times y over square root."},{"Start":"07:43.030 ","End":"07:48.250","Text":"It\u0027s just minus 2y because"},{"Start":"07:48.250 ","End":"07:50.920","Text":"the square root in the numerator and the denominator will"},{"Start":"07:50.920 ","End":"07:54.250","Text":"cancel each other out and I just get the minus 2 with the y."},{"Start":"07:54.250 ","End":"07:56.290","Text":"Then the last component,"},{"Start":"07:56.290 ","End":"08:00.920","Text":"I get minus 3y minus 1,"},{"Start":"08:02.180 ","End":"08:06.485","Text":"and all this dA,"},{"Start":"08:06.485 ","End":"08:09.250","Text":"which is just the integral."},{"Start":"08:09.250 ","End":"08:17.360","Text":"Let\u0027s just simplify this as we have minus 5y minus 1 dA."},{"Start":"08:17.360 ","End":"08:20.770","Text":"Now, how am I going to do this integral?"},{"Start":"08:20.770 ","End":"08:24.220","Text":"I say it\u0027s best we do it in polar coordinates"},{"Start":"08:24.220 ","End":"08:29.060","Text":"because the region is a circle centered at the origin or a disk."},{"Start":"08:32.450 ","End":"08:39.489","Text":"We\u0027ll do it in polar and I\u0027ll remind you what the equations of a polar."},{"Start":"08:39.489 ","End":"08:44.860","Text":"We have x equals r cosine Theta,"},{"Start":"08:44.860 ","End":"08:50.260","Text":"y equals r sine Theta,"},{"Start":"08:50.260 ","End":"08:57.765","Text":"and dA is equal to r dr dTheta."},{"Start":"08:57.765 ","End":"09:00.585","Text":"There\u0027s a 4th formula for x squared plus y squared,"},{"Start":"09:00.585 ","End":"09:02.870","Text":"but we won\u0027t need it here."},{"Start":"09:02.870 ","End":"09:11.500","Text":"The other thing we have to do is convert or describe the disk of radius 4 in polar terms."},{"Start":"09:11.500 ","End":"09:14.400","Text":"We\u0027ve done this so many times before."},{"Start":"09:14.400 ","End":"09:16.400","Text":"Let me just write it."},{"Start":"09:16.400 ","End":"09:18.565","Text":"We go the full circle."},{"Start":"09:18.565 ","End":"09:22.985","Text":"Theta goes from 0 to 2Pi or 360."},{"Start":"09:22.985 ","End":"09:25.925","Text":"R goes from the center to the circumference."},{"Start":"09:25.925 ","End":"09:28.755","Text":"It\u0027s 0 to 4."},{"Start":"09:28.755 ","End":"09:32.125","Text":"R goes from 0 to 4."},{"Start":"09:32.125 ","End":"09:35.780","Text":"Then I replace this,"},{"Start":"09:35.780 ","End":"09:38.330","Text":"I get minus 5,"},{"Start":"09:38.330 ","End":"09:45.120","Text":"y is r sine Theta and minus 1,"},{"Start":"09:45.120 ","End":"09:51.130","Text":"and dA is r dr dTheta."},{"Start":"09:52.210 ","End":"09:59.780","Text":"Continuing, got the integral from 0 to 2Pi."},{"Start":"09:59.780 ","End":"10:02.645","Text":"If I multiply out the r,"},{"Start":"10:02.645 ","End":"10:10.320","Text":"I\u0027ve got the integral from 0 to 4 minus"},{"Start":"10:10.320 ","End":"10:21.010","Text":"5r squared sine Theta minus r dr dTheta."},{"Start":"10:21.770 ","End":"10:26.150","Text":"Then I get the integral from 0 to 2Pi."},{"Start":"10:26.150 ","End":"10:30.310","Text":"This integral with respect to r,"},{"Start":"10:30.310 ","End":"10:33.310","Text":"I raise the power by 1 and divide by it."},{"Start":"10:33.310 ","End":"10:40.115","Text":"I\u0027ve got minus 5/3r cubed"},{"Start":"10:40.115 ","End":"10:43.605","Text":"and sine Theta is a constant just stays."},{"Start":"10:43.605 ","End":"10:47.640","Text":"Here I have minus r squared over 2."},{"Start":"10:47.640 ","End":"10:56.135","Text":"I have to take this from 0 to 4 and all this dTheta."},{"Start":"10:56.135 ","End":"11:01.210","Text":"We get, if we plug in r equals 0,"},{"Start":"11:01.210 ","End":"11:02.775","Text":"we just get 0."},{"Start":"11:02.775 ","End":"11:05.040","Text":"That doesn\u0027t give us anything."},{"Start":"11:05.040 ","End":"11:08.190","Text":"If I plug in r equals 4,"},{"Start":"11:08.190 ","End":"11:10.725","Text":"let\u0027s see what we\u0027ll get."},{"Start":"11:10.725 ","End":"11:14.325","Text":"At 4, we get"},{"Start":"11:14.325 ","End":"11:22.110","Text":"4 cubed is 64."},{"Start":"11:22.110 ","End":"11:26.670","Text":"64 times 5 over 3."},{"Start":"11:26.670 ","End":"11:37.590","Text":"That\u0027s 320 over 3 sine Theta."},{"Start":"11:37.590 ","End":"11:41.510","Text":"When here r is 4,"},{"Start":"11:41.510 ","End":"11:43.730","Text":"I\u0027ve got minus 16 over 2,"},{"Start":"11:43.730 ","End":"11:52.290","Text":"which is minus 8, this integral dTheta."},{"Start":"11:52.540 ","End":"11:56.945","Text":"This equals continuing."},{"Start":"11:56.945 ","End":"12:02.910","Text":"The integral of sine is minus cosine."},{"Start":"12:02.910 ","End":"12:10.270","Text":"I\u0027ve got 320 over 3 cosine Theta."},{"Start":"12:11.300 ","End":"12:16.780","Text":"From here, I\u0027ve got minus 8 Theta."},{"Start":"12:17.600 ","End":"12:23.410","Text":"This is from 0 to 2Pi."},{"Start":"12:23.410 ","End":"12:29.300","Text":"Now, cosine of 0 and cosine of 2Pi are the same."},{"Start":"12:29.300 ","End":"12:31.380","Text":"They both happened to equal to 1,"},{"Start":"12:31.380 ","End":"12:32.990","Text":"but it doesn\u0027t matter they are the same,"},{"Start":"12:32.990 ","End":"12:35.420","Text":"so they will cancel each other out."},{"Start":"12:35.420 ","End":"12:39.300","Text":"Only the minus 8 Theta will matter."},{"Start":"12:39.340 ","End":"12:45.720","Text":"What I will get will be 8 Theta when Theta is"},{"Start":"12:45.720 ","End":"12:51.900","Text":"2Pi is minus 16Pi and when Theta is 0, it\u0027s nothing."},{"Start":"12:51.900 ","End":"12:55.480","Text":"I\u0027m just left with minus 16Pi."},{"Start":"12:55.550 ","End":"13:01.900","Text":"That is the final answer and we are done."}],"ID":8826},{"Watched":false,"Name":"Exercise 3","Duration":"13m 15s","ChapterTopicVideoID":8759,"CourseChapterTopicPlaylistID":4973,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.439","Text":"In this exercise, after compute an integral over a closed curve,"},{"Start":"00:04.439 ","End":"00:06.780","Text":"line integral as follows,"},{"Start":"00:06.780 ","End":"00:10.320","Text":"where the curve is rectangular shaped"},{"Start":"00:10.320 ","End":"00:16.155","Text":"and here\u0027s the points, 0, 0, 0 and so on to here, to here, to here."},{"Start":"00:16.155 ","End":"00:17.820","Text":"I phrased this badly,"},{"Start":"00:17.820 ","End":"00:20.730","Text":"I should have said, and back to the start again."},{"Start":"00:20.730 ","End":"00:22.710","Text":"It\u0027s a closed curve,"},{"Start":"00:22.710 ","End":"00:24.450","Text":"could have phrased it better."},{"Start":"00:24.450 ","End":"00:29.340","Text":"Now, the general idea is to use Stokes\u0027 theorem which converts an integral"},{"Start":"00:29.340 ","End":"00:35.759","Text":"over a curve to a line integral to a surface integral"},{"Start":"00:35.759 ","End":"00:40.470","Text":"and the surface is going to be a rectangle in this case."},{"Start":"00:40.470 ","End":"00:43.190","Text":"It turns out these 4 points,"},{"Start":"00:43.190 ","End":"00:45.200","Text":"the corners of a rectangle,"},{"Start":"00:45.200 ","End":"00:49.790","Text":"and we\u0027ll have a single surface integral over this."},{"Start":"00:49.790 ","End":"00:52.340","Text":"What I want do is some notation."},{"Start":"00:52.340 ","End":"00:55.820","Text":"I\u0027d like to call this function the x squared or call it P."},{"Start":"00:55.820 ","End":"01:01.310","Text":"This function I\u0027ll call Q and this function I\u0027ll call R."},{"Start":"01:01.310 ","End":"01:11.120","Text":"Then if I define a vector field F to be with components P, Q, and R,"},{"Start":"01:11.120 ","End":"01:19.470","Text":"then this is just the line integral of P dx plus Q dy plus R dz."},{"Start":"01:19.470 ","End":"01:24.120","Text":"I should write a little circle here."},{"Start":"01:24.120 ","End":"01:27.750","Text":"This is just in vector form,"},{"Start":"01:27.750 ","End":"01:36.665","Text":"F. dr over c and by Stokes\u0027 theorem,"},{"Start":"01:36.665 ","End":"01:40.730","Text":"this is equal to the surface integral over S."},{"Start":"01:40.730 ","End":"01:42.785","Text":"Now I\u0027ve drawn S in here,"},{"Start":"01:42.785 ","End":"01:46.025","Text":"I\u0027ll get back to it in a minute concerning orientation."},{"Start":"01:46.025 ","End":"01:59.230","Text":"The double integral over S of the curl of F.ndS."},{"Start":"01:59.230 ","End":"02:05.090","Text":"About the orientation, n is a unit normal vector,"},{"Start":"02:05.090 ","End":"02:11.150","Text":"but the orientation has to match the orientation on C."},{"Start":"02:11.150 ","End":"02:14.390","Text":"Now, C is going clockwise,"},{"Start":"02:14.390 ","End":"02:16.175","Text":"which is a negative direction,"},{"Start":"02:16.175 ","End":"02:24.235","Text":"which means that if I take a point here that the unit normal vector will be facing down."},{"Start":"02:24.235 ","End":"02:28.185","Text":"It has a negative z component."},{"Start":"02:28.185 ","End":"02:34.450","Text":"Now I\u0027d like to write the equation of this surface as z as a function of x, y."},{"Start":"02:34.450 ","End":"02:41.380","Text":"In other words, I want S to be some function g of x and y."},{"Start":"02:41.380 ","End":"02:43.570","Text":"I\u0027m going to give you the answer,"},{"Start":"02:43.570 ","End":"02:46.585","Text":"and I\u0027ll postpone that for later."},{"Start":"02:46.585 ","End":"02:51.460","Text":"Turns out that this g of x, y is just y."},{"Start":"02:51.460 ","End":"02:56.380","Text":"In other words, this surface could be written as z equals y,"},{"Start":"02:56.380 ","End":"02:58.775","Text":"x doesn\u0027t appear in fact."},{"Start":"02:58.775 ","End":"03:01.135","Text":"Now, I owe you this one,"},{"Start":"03:01.135 ","End":"03:03.655","Text":"and I\u0027ll show you this later on in the clip."},{"Start":"03:03.655 ","End":"03:05.080","Text":"Just don\u0027t want to interrupt the flow."},{"Start":"03:05.080 ","End":"03:08.995","Text":"This involves linear algebra and not much calculus,"},{"Start":"03:08.995 ","End":"03:11.330","Text":"so let\u0027s save it for later."},{"Start":"03:11.330 ","End":"03:18.395","Text":"Now, we\u0027ve studied a theorem or formula that converts the integral over a surface."},{"Start":"03:18.395 ","End":"03:21.920","Text":"When we have the surface as z as a function of x and y,"},{"Start":"03:21.920 ","End":"03:29.900","Text":"we can convert it to a regular double integral over the projected region."},{"Start":"03:29.900 ","End":"03:34.780","Text":"In the x y plane, I\u0027ll call it R and I\u0027ll highlight it."},{"Start":"03:34.780 ","End":"03:43.050","Text":"What we get is that this equals the double integral over the projected region R"},{"Start":"03:43.050 ","End":"03:48.800","Text":"that\u0027s below S in the xy plane of essentially the same thing"},{"Start":"03:48.800 ","End":"03:51.500","Text":"which I\u0027m copying curl F."},{"Start":"03:51.500 ","End":"03:55.760","Text":"Only thing is that when we have this expression in x, y, and z,"},{"Start":"03:55.760 ","End":"04:01.295","Text":"we just have to remember to replace a z by g of x, y in general."},{"Start":"04:01.295 ","End":"04:11.344","Text":"Then this part, we replace by g with respect to x,"},{"Start":"04:11.344 ","End":"04:15.050","Text":"g with respect to y minus 1."},{"Start":"04:15.050 ","End":"04:16.520","Text":"There\u0027s 2 separate formulas."},{"Start":"04:16.520 ","End":"04:20.420","Text":"One for when the normal is facing up and then it\u0027s minus, minus, plus."},{"Start":"04:20.420 ","End":"04:22.655","Text":"When the normal is downward facing,"},{"Start":"04:22.655 ","End":"04:24.305","Text":"this is the way it is."},{"Start":"04:24.305 ","End":"04:27.415","Text":"This is just a regular dA,"},{"Start":"04:27.415 ","End":"04:32.820","Text":"regular double integral over region in the plane."},{"Start":"04:32.820 ","End":"04:36.750","Text":"Now, we want to do some computations."},{"Start":"04:36.750 ","End":"04:41.920","Text":"I brought in a formula for the curl because"},{"Start":"04:41.920 ","End":"04:44.100","Text":"it\u0027s a hard one to remember."},{"Start":"04:44.100 ","End":"04:46.875","Text":"Let\u0027s see if we can compute it in R case."},{"Start":"04:46.875 ","End":"04:48.970","Text":"We have P, Q and R."},{"Start":"04:48.970 ","End":"04:53.700","Text":"R with respect to y, 2xy."},{"Start":"04:53.700 ","End":"04:59.890","Text":"Q with respect to z is 0."},{"Start":"04:59.890 ","End":"05:04.105","Text":"P with respect to z is 0."},{"Start":"05:04.105 ","End":"05:09.760","Text":"R with respect to x is y squared."},{"Start":"05:09.760 ","End":"05:15.400","Text":"Q with respect to x is 4y cubed."},{"Start":"05:15.400 ","End":"05:20.065","Text":"P with respect to y is 0."},{"Start":"05:20.065 ","End":"05:25.880","Text":"What we get if I do it in vector notation is that"},{"Start":"05:25.880 ","End":"05:34.970","Text":"this is equal to the double integral over this region R of the curl of F."},{"Start":"05:34.970 ","End":"05:41.188","Text":"I\u0027ll write it in bracket notation also."},{"Start":"05:41.188 ","End":"05:48.800","Text":"2xy minus 0, and that\u0027s in the j part, y squared,"},{"Start":"05:48.800 ","End":"05:53.090","Text":"but it\u0027s minus y squared because I didn\u0027t take the minus in here,"},{"Start":"05:53.090 ","End":"05:56.735","Text":"these are all minus, minus, minus."},{"Start":"05:56.735 ","End":"06:03.120","Text":"The next one x-component is 4y cubed dot."},{"Start":"06:03.120 ","End":"06:07.005","Text":"Now I need this g_x, g_y minus 1."},{"Start":"06:07.005 ","End":"06:10.935","Text":"Here\u0027s g, g of x, y is just y."},{"Start":"06:10.935 ","End":"06:16.215","Text":"With respect to x it\u0027s 0, with respect to y,"},{"Start":"06:16.215 ","End":"06:22.540","Text":"it\u0027s just 1 and then I\u0027m copying the minus 1 dA."},{"Start":"06:24.140 ","End":"06:26.690","Text":"I do the dot product."},{"Start":"06:26.690 ","End":"06:32.670","Text":"This time 0 is nothing minus y squared."},{"Start":"06:32.780 ","End":"06:37.430","Text":"Basically, what I get is the double integral over R."},{"Start":"06:37.430 ","End":"06:38.480","Text":"I\u0027ll just write it,"},{"Start":"06:38.480 ","End":"06:41.270","Text":"this one with this 1 is minus y squared."},{"Start":"06:41.270 ","End":"06:48.150","Text":"This with this is minus 4y cubed dA."},{"Start":"06:49.540 ","End":"06:55.415","Text":"I\u0027m going to sketch this region R again in the xy plane might make it a bit clearer,"},{"Start":"06:55.415 ","End":"06:59.615","Text":"x goes from 0 to 1, let\u0027s say this is 1,"},{"Start":"06:59.615 ","End":"07:02.665","Text":"y goes from 0 to 3."},{"Start":"07:02.665 ","End":"07:06.920","Text":"Basically, get a rectangle like this."},{"Start":"07:06.920 ","End":"07:10.880","Text":"This is 1, this is 3, this is R."},{"Start":"07:10.880 ","End":"07:15.935","Text":"Then we can write this as an iterated integral."},{"Start":"07:15.935 ","End":"07:20.645","Text":"Let\u0027s do the x first from 0 to 1."},{"Start":"07:20.645 ","End":"07:22.730","Text":"Then for each x,"},{"Start":"07:22.730 ","End":"07:28.775","Text":"y goes from 0 to 3 of this thing,"},{"Start":"07:28.775 ","End":"07:32.300","Text":"minus y squared minus 4y cubed."},{"Start":"07:32.300 ","End":"07:37.110","Text":"It\u0027s iterated now, so I can write it as dy, dx"},{"Start":"07:37.110 ","End":"07:41.250","Text":"and we evaluate it from the inside out."},{"Start":"07:41.330 ","End":"07:46.790","Text":"This is the first one that we do is the dy integral."},{"Start":"07:46.790 ","End":"07:50.720","Text":"I\u0027ll do this one with the side exercise."},{"Start":"07:50.720 ","End":"07:53.630","Text":"Let\u0027s see, minus y squared,"},{"Start":"07:53.630 ","End":"07:59.985","Text":"so that gives me minus y cubed over 3."},{"Start":"07:59.985 ","End":"08:08.830","Text":"Then I get just minus y^4th because the 4 cancels."},{"Start":"08:08.830 ","End":"08:14.455","Text":"This has to be taken from 0 to 3."},{"Start":"08:14.455 ","End":"08:16.820","Text":"When I put in 0, I don\u0027t get anything,"},{"Start":"08:16.820 ","End":"08:19.180","Text":"so I just have to plug in the 3."},{"Start":"08:19.180 ","End":"08:20.420","Text":"I\u0027ve got what?"},{"Start":"08:20.420 ","End":"08:23.975","Text":"Minus 3 cubed over 3 is minus 3 squared."},{"Start":"08:23.975 ","End":"08:26.540","Text":"That\u0027s minus 9."},{"Start":"08:26.540 ","End":"08:40.910","Text":"Then I have minus y^4th, 3^4th is 81."},{"Start":"08:40.910 ","End":"08:43.535","Text":"This is the 3, the 0 doesn\u0027t give anything."},{"Start":"08:43.535 ","End":"08:48.820","Text":"Altogether we just get minus 90."},{"Start":"08:48.820 ","End":"08:54.765","Text":"The highlighted bit is minus 90 and that\u0027s a constant and I can bring that in front."},{"Start":"08:54.765 ","End":"09:02.080","Text":"What I\u0027m left with is minus 90 times the integral of x from 0 to 1."},{"Start":"09:02.080 ","End":"09:05.210","Text":"There\u0027s nothing left here it\u0027s just 1dx."},{"Start":"09:05.210 ","End":"09:09.985","Text":"This integral is just 1 minus 0 is 1."},{"Start":"09:09.985 ","End":"09:13.720","Text":"The answer will be minus 90."},{"Start":"09:13.720 ","End":"09:14.721","Text":"That\u0027s the answer."},{"Start":"09:16.720 ","End":"09:25.755","Text":"This is a lot easier than it would have been to take 4 separate line integrals."},{"Start":"09:25.755 ","End":"09:27.560","Text":"You have to parameterize each one."},{"Start":"09:27.560 ","End":"09:32.490","Text":"I get an integral of t. It would be work."},{"Start":"09:32.620 ","End":"09:40.055","Text":"But I\u0027m not quite done because I still owe you to show that"},{"Start":"09:40.055 ","End":"09:48.705","Text":"the surface S is given by the plane equation z equals y."},{"Start":"09:48.705 ","End":"09:51.040","Text":"Let me now do that."},{"Start":"09:51.040 ","End":"09:53.230","Text":"I\u0027ll write the 4 points."},{"Start":"09:53.230 ","End":"09:55.545","Text":"Let\u0027s see."},{"Start":"09:55.545 ","End":"10:02.310","Text":"There was 0, 0, 0."},{"Start":"10:02.310 ","End":"10:05.625","Text":"There was 0, 3, 3."},{"Start":"10:05.625 ","End":"10:16.725","Text":"There was 1, 3, 3, and the 4th point was 1, 0, 0."},{"Start":"10:16.725 ","End":"10:19.760","Text":"I want to get an equation of a plane."},{"Start":"10:19.760 ","End":"10:24.900","Text":"The general equation of a plane would be,"},{"Start":"10:25.330 ","End":"10:29.075","Text":"actually, I won\u0027t take the general equation of a plane."},{"Start":"10:29.075 ","End":"10:33.350","Text":"I know that this plane is of the form z as a function of x and y,"},{"Start":"10:33.350 ","End":"10:38.570","Text":"I can take it as z equals ax plus by plus c."},{"Start":"10:38.570 ","End":"10:40.490","Text":"I know I can isolate z,"},{"Start":"10:40.490 ","End":"10:44.250","Text":"the plane is not parallel to the z-axis."},{"Start":"10:44.320 ","End":"10:47.615","Text":"Now I can substitute points."},{"Start":"10:47.615 ","End":"10:52.460","Text":"For example, if I substitute 0, 0, 0,"},{"Start":"10:52.460 ","End":"11:03.590","Text":"then I will get that 0 equals a times 0 plus b times 0 plus C."},{"Start":"11:03.590 ","End":"11:10.690","Text":"That gives me that C equals 0."},{"Start":"11:11.200 ","End":"11:13.500","Text":"I\u0027ve done this point."},{"Start":"11:13.500 ","End":"11:23.925","Text":"Now, let\u0027s try this point, 1, 0, 0 will give us that z is 0,"},{"Start":"11:23.925 ","End":"11:37.120","Text":"x is 1 plus b times y, b times 0 plus c, but c is 0."},{"Start":"11:37.190 ","End":"11:45.915","Text":"From here, we get 0 equals a plus 0 plus 0,"},{"Start":"11:45.915 ","End":"11:49.470","Text":"a equals 0 also."},{"Start":"11:49.470 ","End":"11:52.500","Text":"Now, let\u0027s try the next point."},{"Start":"11:52.500 ","End":"11:55.290","Text":"Let\u0027s try putting in 0, 3, 3,"},{"Start":"11:55.290 ","End":"12:04.595","Text":"so we get that z is 3 and then x is 0."},{"Start":"12:04.595 ","End":"12:07.790","Text":"I\u0027m going for this one, x is 0."},{"Start":"12:07.790 ","End":"12:14.000","Text":"I get, well, in any event a is 0, 0 times 0,"},{"Start":"12:14.000 ","End":"12:22.770","Text":"b is, we don\u0027t know, but y is 3."},{"Start":"12:22.770 ","End":"12:28.110","Text":"Then we get c is 0."},{"Start":"12:28.110 ","End":"12:32.105","Text":"This gives us a 3 equals b times 3,"},{"Start":"12:32.105 ","End":"12:35.730","Text":"which gives us that b equals 1."},{"Start":"12:35.960 ","End":"12:44.135","Text":"That means that the equation if b is 1 and a is 0 and c is 0,"},{"Start":"12:44.135 ","End":"12:49.010","Text":"it gives us just z equals y."},{"Start":"12:49.010 ","End":"12:50.360","Text":"I can still see it on the screen,"},{"Start":"12:50.360 ","End":"12:52.430","Text":"which is what we said before."},{"Start":"12:52.430 ","End":"12:56.165","Text":"To be technically correct,"},{"Start":"12:56.165 ","End":"13:02.770","Text":"we have to also check the fourth point that it fits onto this z equals y."},{"Start":"13:02.770 ","End":"13:04.550","Text":"If I check this one,"},{"Start":"13:04.550 ","End":"13:08.450","Text":"z does equal to y, y is 3, z is 3."},{"Start":"13:08.450 ","End":"13:10.100","Text":"The fourth point also,"},{"Start":"13:10.100 ","End":"13:11.975","Text":"so we\u0027re really covered."},{"Start":"13:11.975 ","End":"13:13.640","Text":"That was what I owed you."},{"Start":"13:13.640 ","End":"13:16.380","Text":"Now we are ready."}],"ID":8827},{"Watched":false,"Name":"Exercise 4","Duration":"10m 34s","ChapterTopicVideoID":8760,"CourseChapterTopicPlaylistID":4973,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.330","Text":"In this exercise, we have to compute the line integral over the closed curve C,"},{"Start":"00:06.330 ","End":"00:11.130","Text":"which we described in a moment of vector field F."},{"Start":"00:11.130 ","End":"00:15.600","Text":"This is type 2 line integral."},{"Start":"00:15.600 ","End":"00:18.780","Text":"We\u0027re given the formula for F,"},{"Start":"00:18.780 ","End":"00:22.890","Text":"and C is the perimeter of the triangle, it\u0027s in the picture."},{"Start":"00:22.890 ","End":"00:28.515","Text":"It has the vertices 1, 0, 0, that would be where x is 1."},{"Start":"00:28.515 ","End":"00:31.590","Text":"That second one is where just y is 1,"},{"Start":"00:31.590 ","End":"00:34.440","Text":"the third one is where z is 1."},{"Start":"00:34.440 ","End":"00:40.670","Text":"We have 3 sides of a triangle closed curve"},{"Start":"00:40.670 ","End":"00:45.455","Text":"and it\u0027s in the anticlockwise direction when we\u0027re looking from above."},{"Start":"00:45.455 ","End":"00:47.940","Text":"These are the arrows."},{"Start":"00:48.440 ","End":"00:51.885","Text":"We\u0027re in the chapter on Stokes\u0027 theorem."},{"Start":"00:51.885 ","End":"00:54.470","Text":"Instead of doing this line integral,"},{"Start":"00:54.470 ","End":"00:57.110","Text":"which would be 3 separate computations"},{"Start":"00:57.110 ","End":"01:00.950","Text":"and each of them requiring work to parametrize and so on,"},{"Start":"01:00.950 ","End":"01:05.910","Text":"what we\u0027ll do is we\u0027ll find a surface that this is the boundary of"},{"Start":"01:05.910 ","End":"01:10.250","Text":"and the obvious thing is the plane that goes through these points."},{"Start":"01:10.250 ","End":"01:13.790","Text":"Let me call this surface S."},{"Start":"01:13.790 ","End":"01:17.960","Text":"What we\u0027ll do is we\u0027ll call in Stokes\u0027 theorem"},{"Start":"01:17.960 ","End":"01:23.630","Text":"and instead of the integral over C of F.dr,"},{"Start":"01:26.330 ","End":"01:40.220","Text":"we will get the surface integral over S of the curl of"},{"Start":"01:40.220 ","End":"01:47.600","Text":"the vector field dot normal unit vector dS."},{"Start":"01:47.600 ","End":"01:50.675","Text":"We have to just figure out which direction the normal is in."},{"Start":"01:50.675 ","End":"01:56.270","Text":"If this is going counterclockwise and I take a point on this plane,"},{"Start":"01:56.270 ","End":"01:58.580","Text":"it\u0027s just hard to describe them."},{"Start":"01:58.580 ","End":"02:00.755","Text":"Imagine I\u0027m on this triangle,"},{"Start":"02:00.755 ","End":"02:05.205","Text":"it goes somewhat upwards and outwards."},{"Start":"02:05.205 ","End":"02:08.955","Text":"In any event, it got an upward component."},{"Start":"02:08.955 ","End":"02:11.430","Text":"I\u0027ll label it n."},{"Start":"02:11.430 ","End":"02:14.945","Text":"Actually it\u0027s going to be a constant because this is a plane."},{"Start":"02:14.945 ","End":"02:21.185","Text":"Now this computation would be easier if I could get S as a function,"},{"Start":"02:21.185 ","End":"02:24.215","Text":"z of x and y,"},{"Start":"02:24.215 ","End":"02:26.865","Text":"z equals g of x, y."},{"Start":"02:26.865 ","End":"02:29.845","Text":"If I can get it in this form,"},{"Start":"02:29.845 ","End":"02:39.105","Text":"then I can write this surface integral as a regular double integral over R,"},{"Start":"02:39.105 ","End":"02:48.410","Text":"where R is the part in the x, y plane of the same thing,"},{"Start":"02:48.410 ","End":"02:52.280","Text":"which would be curl of F."},{"Start":"02:52.280 ","End":"02:54.695","Text":"But instead of dot n,"},{"Start":"02:54.695 ","End":"03:01.005","Text":"we would have dot and in angular bracket form,"},{"Start":"03:01.005 ","End":"03:10.285","Text":"we\u0027ll get here the vector minus g_x, minus g_y, 1, dA."},{"Start":"03:10.285 ","End":"03:12.530","Text":"That\u0027s the formula we\u0027ve used before."},{"Start":"03:12.530 ","End":"03:13.640","Text":"There\u0027s 2 versions."},{"Start":"03:13.640 ","End":"03:15.230","Text":"One where n goes upwards,"},{"Start":"03:15.230 ","End":"03:18.320","Text":"which is this and there\u0027s this slightly different version"},{"Start":"03:18.320 ","End":"03:22.235","Text":"for when n has a downward component."},{"Start":"03:22.235 ","End":"03:24.125","Text":"But we\u0027re going to use this."},{"Start":"03:24.125 ","End":"03:26.420","Text":"Let me give you the answer now"},{"Start":"03:26.420 ","End":"03:29.930","Text":"and I\u0027ll show you why later that this function,"},{"Start":"03:29.930 ","End":"03:35.490","Text":"g of x, y is 1 minus x minus y."},{"Start":"03:35.660 ","End":"03:42.800","Text":"If you happen to spot that all these 3 have components that add up to 1,"},{"Start":"03:42.800 ","End":"03:44.520","Text":"you could have said right away, oh, yeah,"},{"Start":"03:44.520 ","End":"03:48.800","Text":"x plus y plus z is 1 and z is 1 minus x minus y,"},{"Start":"03:48.800 ","End":"03:52.010","Text":"but I\u0027ll show you more formally at the end."},{"Start":"03:52.010 ","End":"03:54.080","Text":"I\u0027ll just write the word later,"},{"Start":"03:54.080 ","End":"03:56.695","Text":"which means I owe you this one."},{"Start":"03:56.695 ","End":"04:02.520","Text":"Now I\u0027ve broaden the definition of curl because it\u0027s hard to remember,"},{"Start":"04:02.520 ","End":"04:09.710","Text":"presuming that this part is P, this is Q, and this is R,"},{"Start":"04:09.710 ","End":"04:14.765","Text":"then the curl of the vector field F is given as follows."},{"Start":"04:14.765 ","End":"04:16.655","Text":"Let\u0027s compute it."},{"Start":"04:16.655 ","End":"04:20.815","Text":"R with respect to y is 0,"},{"Start":"04:20.815 ","End":"04:25.185","Text":"Q with respect to z is 2z,"},{"Start":"04:25.185 ","End":"04:29.400","Text":"P with respect to z is nothing,"},{"Start":"04:29.400 ","End":"04:33.495","Text":"R with respect to x is 2x,"},{"Start":"04:33.495 ","End":"04:38.500","Text":"Q with respect to z is 2z,"},{"Start":"04:38.590 ","End":"04:44.870","Text":"and P with respect to y is 2y"},{"Start":"04:44.870 ","End":"04:50.895","Text":"because I have to put minuses when I\u0027m computing this."},{"Start":"04:50.895 ","End":"04:55.010","Text":"I can now get compute this integral."},{"Start":"04:55.010 ","End":"05:03.945","Text":"This is equal to the double integral over r of,"},{"Start":"05:03.945 ","End":"05:08.110","Text":"I\u0027ll write it in angular bracket notation."},{"Start":"05:08.110 ","End":"05:11.925","Text":"Here I\u0027ve got minus 2z,"},{"Start":"05:11.925 ","End":"05:15.870","Text":"here I have minus 2x."},{"Start":"05:17.180 ","End":"05:19.605","Text":"This is not 2z."},{"Start":"05:19.605 ","End":"05:21.990","Text":"I thought this was a z,"},{"Start":"05:21.990 ","End":"05:26.295","Text":"this is an x, q with respect to x is nothing."},{"Start":"05:26.295 ","End":"05:31.380","Text":"This is minus 2y dot."},{"Start":"05:31.380 ","End":"05:35.390","Text":"Let\u0027s see."},{"Start":"05:35.390 ","End":"05:45.395","Text":"G with respect to x is just minus 1 and g with respect to y is also minus 1."},{"Start":"05:45.395 ","End":"05:53.828","Text":"This vector because minus, minus 1 is 1, 1, 1, dA"},{"Start":"05:53.828 ","End":"06:00.705","Text":"and this is equal to the double integral of,"},{"Start":"06:00.705 ","End":"06:05.080","Text":"this times this plus this times this, we just get,"},{"Start":"06:05.390 ","End":"06:08.250","Text":"and I can even change the order,"},{"Start":"06:08.250 ","End":"06:21.405","Text":"minus 2x, minus 2y, minus 2z, dA."},{"Start":"06:21.405 ","End":"06:25.070","Text":"Not quite, when we apply this formula,"},{"Start":"06:25.070 ","End":"06:31.100","Text":"we also have to change z to what it is in terms of x and y."},{"Start":"06:31.100 ","End":"06:38.740","Text":"Here I have a z, but z is 1 minus x minus y."},{"Start":"06:38.990 ","End":"06:42.670","Text":"Instead of the whole minus 2z,"},{"Start":"06:42.670 ","End":"06:51.235","Text":"what I have to put in is minus 2, plus 2x, plus 2y."},{"Start":"06:51.235 ","End":"06:54.990","Text":"Basically, all this just boils down to 2 because"},{"Start":"06:54.990 ","End":"07:01.170","Text":"the 2x and the minus 2x cancel 2y with minus 2."},{"Start":"07:01.170 ","End":"07:07.265","Text":"We can take the minus 2 out the brackets and get this as minus 2,"},{"Start":"07:07.265 ","End":"07:11.365","Text":"double integral over R of just 1dA."},{"Start":"07:11.365 ","End":"07:22.520","Text":"Now, this integral of 1, should ring a bell because there\u0027s a formula that"},{"Start":"07:22.520 ","End":"07:28.250","Text":"the double integral of 1 over a region in the x, y plane is just the area."},{"Start":"07:28.250 ","End":"07:35.120","Text":"So we\u0027ve got minus 2 times the area of R."},{"Start":"07:35.120 ","End":"07:38.060","Text":"Now what is this R?"},{"Start":"07:38.060 ","End":"07:40.330","Text":"We know that this is,"},{"Start":"07:40.330 ","End":"07:46.530","Text":"I\u0027m just marking the coordinates, 1."},{"Start":"07:46.530 ","End":"07:49.270","Text":"We have a triangle,"},{"Start":"07:52.610 ","End":"07:55.580","Text":"you can look at it as the base is 1,"},{"Start":"07:55.580 ","End":"08:01.380","Text":"the height is 1, and using the half base times height formula,"},{"Start":"08:01.750 ","End":"08:07.735","Text":"this area is 1/2."},{"Start":"08:07.735 ","End":"08:10.350","Text":"Base is 1 if we take the x as the base,"},{"Start":"08:10.350 ","End":"08:11.970","Text":"it doesn\u0027t matter which one is the base,"},{"Start":"08:11.970 ","End":"08:17.890","Text":"times 1, times 1 and this equals a 1/2."},{"Start":"08:17.960 ","End":"08:22.230","Text":"Altogether we have minus 2 times a 1/2,"},{"Start":"08:22.230 ","End":"08:25.330","Text":"which is minus 1."},{"Start":"08:25.820 ","End":"08:31.310","Text":"That\u0027s the answer, but we\u0027re not quite done because I still owe you"},{"Start":"08:31.310 ","End":"08:38.005","Text":"to show you how I got the equation of the plane through these 3 points."},{"Start":"08:38.005 ","End":"08:40.560","Text":"I\u0027m going to just write the 3 points."},{"Start":"08:40.560 ","End":"08:43.735","Text":"First of all we had the points,"},{"Start":"08:43.735 ","End":"08:45.650","Text":"I can still see them. It doesn\u0027t matter."},{"Start":"08:45.650 ","End":"08:58.565","Text":"I\u0027ll write them again, 1, 0, 0, I have 0, 1, 0, and 0, 0, 1."},{"Start":"08:58.565 ","End":"09:01.100","Text":"Now, the equation of a plane,"},{"Start":"09:01.100 ","End":"09:04.609","Text":"if I can extract z in terms of x and y,"},{"Start":"09:04.609 ","End":"09:07.280","Text":"then it\u0027s not a totally general equation of a plane,"},{"Start":"09:07.280 ","End":"09:10.190","Text":"but it\u0027s a plane where z is a function of x, y"},{"Start":"09:10.190 ","End":"09:15.830","Text":"would be something like ax plus by plus c."},{"Start":"09:15.830 ","End":"09:18.340","Text":"Now it goes through these 3 points,"},{"Start":"09:18.340 ","End":"09:23.525","Text":"so I can substitute them and get 3 equations in 3 unknowns."},{"Start":"09:23.525 ","End":"09:26.130","Text":"Let me do that."},{"Start":"09:26.870 ","End":"09:29.085","Text":"For the first point,"},{"Start":"09:29.085 ","End":"09:34.290","Text":"I\u0027ll get 0 equals a plus c."},{"Start":"09:34.290 ","End":"09:49.520","Text":"In the next one, z is 0 and this time x is 0, y is 1, so I get b plus c."},{"Start":"09:49.520 ","End":"09:58.260","Text":"In the last one, z is 1 and this part is 0, gives me 1 equals c."},{"Start":"09:58.260 ","End":"10:03.420","Text":"I can plug in c equals 1 here and here"},{"Start":"10:03.420 ","End":"10:08.495","Text":"and this is going to give me that a is minus 1,"},{"Start":"10:08.495 ","End":"10:11.570","Text":"and here I\u0027m going to get that b is minus 1,"},{"Start":"10:11.570 ","End":"10:21.650","Text":"and therefore, z is minus 1x, minus 1y, plus 1,"},{"Start":"10:21.650 ","End":"10:24.380","Text":"which is the same as what we got here,"},{"Start":"10:24.380 ","End":"10:25.520","Text":"just in a different order,"},{"Start":"10:25.520 ","End":"10:27.335","Text":"1 minus x minus y."},{"Start":"10:27.335 ","End":"10:28.370","Text":"Same thing."},{"Start":"10:28.370 ","End":"10:35.590","Text":"We verified that and now we really are done."}],"ID":8828},{"Watched":false,"Name":"Exercise 5","Duration":"8m 41s","ChapterTopicVideoID":8761,"CourseChapterTopicPlaylistID":4973,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.080 ","End":"00:03.480","Text":"In this exercise, amongst other things,"},{"Start":"00:03.480 ","End":"00:05.865","Text":"we\u0027re practicing different notations."},{"Start":"00:05.865 ","End":"00:10.845","Text":"This is yet another way of writing the curl of"},{"Start":"00:10.845 ","End":"00:13.320","Text":"the vector field F. It\u0027s"},{"Start":"00:13.320 ","End":"00:18.485","Text":"the grad symbol cross and there were reasons for this as I explained at the time."},{"Start":"00:18.485 ","End":"00:20.285","Text":"This is just the curl."},{"Start":"00:20.285 ","End":"00:28.550","Text":"Now, here\u0027s the vector field given as follows and S is described in words,"},{"Start":"00:28.550 ","End":"00:30.830","Text":"but picture explains better."},{"Start":"00:30.830 ","End":"00:33.350","Text":"What we have is part of the sphere."},{"Start":"00:33.350 ","End":"00:37.850","Text":"This is a sphere of radius 2 because 4 is 2 squared. Here\u0027s the sphere."},{"Start":"00:37.850 ","End":"00:41.120","Text":"The cylinder x-squared plus y-squared equals 1 is"},{"Start":"00:41.120 ","End":"00:45.080","Text":"just like the circle x squared plus y squared equals 1 in the plane,"},{"Start":"00:45.080 ","End":"00:47.945","Text":"but extended upward and downward indefinitely."},{"Start":"00:47.945 ","End":"00:51.425","Text":"But we only want above the xy plane,"},{"Start":"00:51.425 ","End":"00:55.475","Text":"so everything that\u0027s not above is omitted."},{"Start":"00:55.475 ","End":"00:57.830","Text":"If it didn\u0027t say above the xy plane,"},{"Start":"00:57.830 ","End":"01:00.530","Text":"there are 2 parts of the sphere inside the cylinder,"},{"Start":"01:00.530 ","End":"01:04.550","Text":"there would be this part and there would be a corresponding 1 below."},{"Start":"01:04.550 ","End":"01:07.790","Text":"But we don\u0027t want that 1 because we only want above."},{"Start":"01:07.790 ","End":"01:14.120","Text":"This yellow highlight bit is what we want."},{"Start":"01:14.120 ","End":"01:16.370","Text":"Now unless stated otherwise,"},{"Start":"01:16.370 ","End":"01:21.545","Text":"the normal would be outward facing on the sphere."},{"Start":"01:21.545 ","End":"01:25.715","Text":"If it\u0027s outward facing, then it induces."},{"Start":"01:25.715 ","End":"01:29.824","Text":"If you just take the counterclockwise direction everywhere,"},{"Start":"01:29.824 ","End":"01:35.150","Text":"we get this direction on the line here."},{"Start":"01:35.150 ","End":"01:36.890","Text":"Yeah, I\u0027m getting ahead of myself."},{"Start":"01:36.890 ","End":"01:40.310","Text":"Obviously, we\u0027re in the chapter on Stokes\u0027 theorem so"},{"Start":"01:40.310 ","End":"01:43.820","Text":"we\u0027re going to do the surface integral with the help of a line integral over"},{"Start":"01:43.820 ","End":"01:49.215","Text":"a closed curve and this will be the curve C. This part I\u0027ll call"},{"Start":"01:49.215 ","End":"01:54.900","Text":"S the cap and we\u0027re going to use Stokes\u0027 theorem,"},{"Start":"01:54.900 ","End":"01:56.535","Text":"which I\u0027ll remind you."},{"Start":"01:56.535 ","End":"02:01.700","Text":"The short form says that I\u0027ll reverse"},{"Start":"02:01.700 ","End":"02:03.500","Text":"the equals because we\u0027re starting with"},{"Start":"02:03.500 ","End":"02:06.770","Text":"a surface integral and working towards a line integral,"},{"Start":"02:06.770 ","End":"02:12.215","Text":"the integral on S and instead of curl I\u0027ll stick to the way it was written here,"},{"Start":"02:12.215 ","End":"02:16.150","Text":"of grad cross, F.n,"},{"Start":"02:17.210 ","End":"02:19.880","Text":"I didn\u0027t label this n,"},{"Start":"02:19.880 ","End":"02:28.240","Text":"ds is equal to the line integral over the curve C,"},{"Start":"02:28.240 ","End":"02:31.330","Text":"which is the boundary of S oriented"},{"Start":"02:31.330 ","End":"02:39.890","Text":"appropriately of f dot with dr."},{"Start":"02:40.610 ","End":"02:44.050","Text":"I\u0027ll write this out in full."},{"Start":"02:45.290 ","End":"02:48.180","Text":"This is equal to this,"},{"Start":"02:48.180 ","End":"02:50.850","Text":"well I\u0027ll have to tell you what P, Q and R are."},{"Start":"02:50.850 ","End":"02:55.449","Text":"This is P, this part is Q,"},{"Start":"02:55.449 ","End":"02:58.435","Text":"and this part is R. The components of the vector field"},{"Start":"02:58.435 ","End":"03:06.935","Text":"F. What I need next is a parametrization of C. First of all, let\u0027s see."},{"Start":"03:06.935 ","End":"03:11.060","Text":"If I intersect the sphere with the cylinder."},{"Start":"03:11.060 ","End":"03:13.595","Text":"If I take these 2 equations,"},{"Start":"03:13.595 ","End":"03:15.140","Text":"I could say, okay,"},{"Start":"03:15.140 ","End":"03:22.205","Text":"x squared plus y squared is 1 and put it in here and get 1 plus z squared is 4."},{"Start":"03:22.205 ","End":"03:25.490","Text":"That would give me that z squared is 3."},{"Start":"03:25.490 ","End":"03:30.125","Text":"So z is a constant square root of 3,"},{"Start":"03:30.125 ","End":"03:31.850","Text":"not the negative part,"},{"Start":"03:31.850 ","End":"03:35.315","Text":"because we already said that we\u0027re above the xy plane."},{"Start":"03:35.315 ","End":"03:37.490","Text":"Other than z equals 3,"},{"Start":"03:37.490 ","End":"03:41.000","Text":"I still have that x squared plus y squared equals 1."},{"Start":"03:41.000 ","End":"03:48.390","Text":"It\u0027s like a whole circle like this 1 only raised root 3 units above the xy plane."},{"Start":"03:48.390 ","End":"03:51.065","Text":"If I want to parametrize this,"},{"Start":"03:51.065 ","End":"03:57.540","Text":"I could get just parametrize the circle and let z equals root 3."},{"Start":"03:57.540 ","End":"04:04.370","Text":"The usual parametrization for the circle is in general,"},{"Start":"04:04.370 ","End":"04:05.510","Text":"when the radius is R,"},{"Start":"04:05.510 ","End":"04:08.375","Text":"it\u0027s R cosine t. But here the radius is 1."},{"Start":"04:08.375 ","End":"04:10.490","Text":"So it\u0027s cosine t,"},{"Start":"04:10.490 ","End":"04:13.755","Text":"y equals 1, sine t,"},{"Start":"04:13.755 ","End":"04:15.360","Text":"and z is a constant."},{"Start":"04:15.360 ","End":"04:17.825","Text":"Now I have a parametrization of the curve."},{"Start":"04:17.825 ","End":"04:24.005","Text":"I need to tell you that t goes from 0-2Pi, it\u0027s a full circle."},{"Start":"04:24.005 ","End":"04:27.440","Text":"Actually, t could be the Theta from polar coordinates if you like."},{"Start":"04:27.440 ","End":"04:29.150","Text":"It\u0027s the angle from"},{"Start":"04:29.150 ","End":"04:35.670","Text":"the x angle going counterclockwise that you could use as t. It doesn\u0027t matter,"},{"Start":"04:35.670 ","End":"04:37.815","Text":"this is a Parametrization."},{"Start":"04:37.815 ","End":"04:46.380","Text":"Now I have to plug that in here and then I get an integral from t from 0-2Pi."},{"Start":"04:46.380 ","End":"04:49.230","Text":"I\u0027m also going to need dx, dy, and dz."},{"Start":"04:49.230 ","End":"04:55.105","Text":"I\u0027ll just write those dx, dy, and dz."},{"Start":"04:55.105 ","End":"04:59.590","Text":"Dx would be minus sine t dt,"},{"Start":"04:59.590 ","End":"05:08.345","Text":"dy would be cosine t dt and dz is 0,"},{"Start":"05:08.345 ","End":"05:10.895","Text":"dt or just plain 0."},{"Start":"05:10.895 ","End":"05:17.210","Text":"Now in here what I get is the integral,"},{"Start":"05:17.210 ","End":"05:23.490","Text":"but over the parameter t from 0-2Pi,"},{"Start":"05:23.490 ","End":"05:32.470","Text":"P is yz, but yz I substitute from here."},{"Start":"05:33.530 ","End":"05:36.390","Text":"Here I am P, yz"},{"Start":"05:36.390 ","End":"05:46.250","Text":"sine t times root 3,"},{"Start":"05:46.250 ","End":"05:48.950","Text":"which is z, times dx,"},{"Start":"05:48.950 ","End":"05:53.900","Text":"which is minus sine t dt."},{"Start":"05:53.900 ","End":"05:59.615","Text":"Next bit, Q dy, Q is xz,"},{"Start":"05:59.615 ","End":"06:06.420","Text":"which is cosine t and z"},{"Start":"06:06.420 ","End":"06:14.070","Text":"is root 3 and dy is cosine t dt."},{"Start":"06:14.070 ","End":"06:17.325","Text":"The last bit, I need x,"},{"Start":"06:17.325 ","End":"06:22.760","Text":"y, which is R time dz."},{"Start":"06:22.760 ","End":"06:28.295","Text":"So x, y is x is cosine t,"},{"Start":"06:28.295 ","End":"06:32.225","Text":"y is sine t,"},{"Start":"06:32.225 ","End":"06:35.780","Text":"and dz is 0."},{"Start":"06:35.780 ","End":"06:38.975","Text":"I didn\u0027t even have to write these 0 dt."},{"Start":"06:38.975 ","End":"06:41.435","Text":"But this part can be omitted."},{"Start":"06:41.435 ","End":"06:45.755","Text":"I want to take the dt outside the brackets."},{"Start":"06:45.755 ","End":"06:51.045","Text":"I get the integral dt from 0-2Pi."},{"Start":"06:51.045 ","End":"06:57.800","Text":"Here I get minus root 3 sine squared"},{"Start":"06:57.800 ","End":"07:03.450","Text":"t. The dt at the end plus root"},{"Start":"07:03.450 ","End":"07:10.800","Text":"3 cosine squared t and all this dt."},{"Start":"07:10.800 ","End":"07:18.785","Text":"But there\u0027s a formula from trigonometry that cosine squared"},{"Start":"07:18.785 ","End":"07:27.105","Text":"t minus sine squared t is equal to cosine 2t."},{"Start":"07:27.105 ","End":"07:30.975","Text":"Usually is written in terms of our Alpha or Theta or X or something,"},{"Start":"07:30.975 ","End":"07:34.560","Text":"but it could be t. Over here,"},{"Start":"07:34.560 ","End":"07:37.680","Text":"we can take the root 3 in front of the"},{"Start":"07:37.680 ","End":"07:43.025","Text":"integral and then we would get cosine squared minus sine squared,"},{"Start":"07:43.025 ","End":"07:48.740","Text":"which we said is cosine 2t, dt."},{"Start":"07:48.740 ","End":"07:51.800","Text":"That\u0027s a straightforward integral."},{"Start":"07:51.800 ","End":"08:00.930","Text":"This equals the integral of cosine 2t is sine 2t but over 2 so we have root 3."},{"Start":"08:00.930 ","End":"08:05.265","Text":"Then I\u0027ll write the over 2 sine 2t."},{"Start":"08:05.265 ","End":"08:07.230","Text":"The over 2 is because it\u0027s not t,"},{"Start":"08:07.230 ","End":"08:10.635","Text":"its 2t which we divide by the inner derivative."},{"Start":"08:10.635 ","End":"08:17.215","Text":"This has to be taken from 0-2Pi."},{"Start":"08:17.215 ","End":"08:24.515","Text":"Now, sine of 0 is 0 and sine of 4Pi is the same it\u0027s also 0."},{"Start":"08:24.515 ","End":"08:29.935","Text":"This thing is just all equal to 0."},{"Start":"08:29.935 ","End":"08:35.600","Text":"The answer to this original integral with the help of Stokes\u0027 theorem"},{"Start":"08:35.600 ","End":"08:42.030","Text":"via the line integral is 0 and that\u0027s the answer and we\u0027re done."}],"ID":8829},{"Watched":false,"Name":"Exercise 6","Duration":"10m 31s","ChapterTopicVideoID":8762,"CourseChapterTopicPlaylistID":4973,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.410","Text":"In this exercise, we\u0027re given a surface integral to compute,"},{"Start":"00:04.410 ","End":"00:12.330","Text":"and I want to remind you that this notation is equivalent to saying the curl of F."},{"Start":"00:12.330 ","End":"00:16.200","Text":"Boldface is equivalent to an arrow,"},{"Start":"00:16.200 ","End":"00:17.550","Text":"shows what\u0027s a vector"},{"Start":"00:17.550 ","End":"00:23.690","Text":"and we\u0027re given the vector field described as follows."},{"Start":"00:23.690 ","End":"00:26.690","Text":"S is given to be part of the cone,"},{"Start":"00:26.690 ","End":"00:31.650","Text":"z equals this expression that\u0027s above the xy plane,"},{"Start":"00:31.650 ","End":"00:33.675","Text":"or what is this look like?"},{"Start":"00:33.675 ","End":"00:39.830","Text":"Here\u0027s a Picture, I won\u0027t go into all the details of why but briefly,"},{"Start":"00:39.830 ","End":"00:44.659","Text":"normal equation of a cone is z squared equals x squared plus y squared"},{"Start":"00:44.659 ","End":"00:48.515","Text":"and that gives us a full cone but centered at the origin."},{"Start":"00:48.515 ","End":"00:53.780","Text":"If I take the square root of both sides and just take the minus part,"},{"Start":"00:53.780 ","End":"00:58.460","Text":"I get the part of the cone and that\u0027s just below the xy plane."},{"Start":"00:58.460 ","End":"00:59.990","Text":"But If I add a 2 now,"},{"Start":"00:59.990 ","End":"01:02.170","Text":"it brings it 2 units upwards,"},{"Start":"01:02.170 ","End":"01:04.515","Text":"so anyway this is a cone,"},{"Start":"01:04.515 ","End":"01:11.880","Text":"well this is 2 and this has to be the z-axis."},{"Start":"01:11.880 ","End":"01:16.930","Text":"I guess that makes this the y-axis and this the x-axis."},{"Start":"01:16.930 ","End":"01:21.609","Text":"Let me just label the cone S,"},{"Start":"01:21.609 ","End":"01:24.170","Text":"it doesn\u0027t include the base,"},{"Start":"01:24.170 ","End":"01:29.685","Text":"it\u0027s just around and I\u0027m going to use Stokes\u0027 theorem."},{"Start":"01:29.685 ","End":"01:33.710","Text":"This integral would normally be hard to compute the surface integral,"},{"Start":"01:33.710 ","End":"01:38.075","Text":"this function is a bit of a mess and so is this."},{"Start":"01:38.075 ","End":"01:47.240","Text":"We\u0027ll use Stokes\u0027 theorem to do it as a line integral over the curve C,"},{"Start":"01:47.240 ","End":"01:55.920","Text":"now normally this normal here is outward so this would be the normal."},{"Start":"01:55.920 ","End":"02:03.380","Text":"This induces, if you look at my little counterclockwise circles everywhere,"},{"Start":"02:03.380 ","End":"02:08.580","Text":"they induce an orientation on the curve here."},{"Start":"02:08.950 ","End":"02:13.445","Text":"Well, it\u0027s going to be counterclockwise in the positive direction,"},{"Start":"02:13.445 ","End":"02:15.305","Text":"now what is this curve?"},{"Start":"02:15.305 ","End":"02:18.870","Text":"If I let z equals 0 here,"},{"Start":"02:19.000 ","End":"02:25.970","Text":"well on this curve z equals 0 but then I also get that this thing is 0"},{"Start":"02:25.970 ","End":"02:32.375","Text":"so if I just bring the square root to the other side and square it,"},{"Start":"02:32.375 ","End":"02:39.565","Text":"I get x squared plus y squared equals 4."},{"Start":"02:39.565 ","End":"02:42.720","Text":"Because it\u0027s 2 squared, which is 4,"},{"Start":"02:42.720 ","End":"02:46.095","Text":"that shows us that this is in the xy plane,"},{"Start":"02:46.095 ","End":"02:50.820","Text":"the circle of radius 2 so everything\u0027s 2."},{"Start":"02:50.820 ","End":"02:55.090","Text":"Now I\u0027ll call in Stokes\u0027 theorem and Stokes\u0027 theorem says that"},{"Start":"02:55.090 ","End":"03:03.115","Text":"this double integral is equal to the integral over C,"},{"Start":"03:03.115 ","End":"03:12.490","Text":"the closed curve C of F.dr."},{"Start":"03:12.490 ","End":"03:15.110","Text":"Now if we expand this,"},{"Start":"03:15.110 ","End":"03:21.270","Text":"this is equal to this but I have to tell you a P, Q and R mean."},{"Start":"03:21.270 ","End":"03:24.660","Text":"The P is just the i component,"},{"Start":"03:24.660 ","End":"03:26.650","Text":"Q is the j component,"},{"Start":"03:26.650 ","End":"03:31.150","Text":"and R is the k component of the vector field F,"},{"Start":"03:31.150 ","End":"03:34.340","Text":"3 functions of x, y, z."},{"Start":"03:34.880 ","End":"03:42.130","Text":"What we want to do now is write this as an integral of the variable t,"},{"Start":"03:42.130 ","End":"03:44.750","Text":"which is a parametrization of this curve."},{"Start":"03:44.750 ","End":"03:49.000","Text":"The almost obvious way to parametrize it is not the only way,"},{"Start":"03:49.000 ","End":"03:53.060","Text":"is to say, z equals 0 that plots the easy part."},{"Start":"03:53.060 ","End":"03:54.695","Text":"What about x and y?"},{"Start":"03:54.695 ","End":"03:57.560","Text":"Well, they\u0027re on a circle of radius 2,"},{"Start":"03:57.560 ","End":"04:01.765","Text":"so the usual thing is to let t be like Theta,"},{"Start":"04:01.765 ","End":"04:07.440","Text":"to be the angle from the x-axis going counterclockwise."},{"Start":"04:07.440 ","End":"04:12.200","Text":"Then the parametrization becomes x equals the radius,"},{"Start":"04:12.200 ","End":"04:18.215","Text":"which is 2 times cosine t,"},{"Start":"04:18.215 ","End":"04:20.940","Text":"though it would work that way also."},{"Start":"04:21.040 ","End":"04:25.760","Text":"Y equals 2 sine t,"},{"Start":"04:25.760 ","End":"04:29.630","Text":"the radius times sine t and the z as we said is 0,"},{"Start":"04:29.630 ","End":"04:34.890","Text":"I have to say where t goes from,"},{"Start":"04:34.890 ","End":"04:39.435","Text":"t goes from 0 to 2 Pi, 1 complete circle."},{"Start":"04:39.435 ","End":"04:43.800","Text":"I see that we will also need dx, dy, and dz,"},{"Start":"04:43.800 ","End":"04:49.575","Text":"so might as well say what those are, dx, dy, dz in terms of dt,"},{"Start":"04:49.575 ","End":"04:54.855","Text":"this will be minus 2 sine t dt."},{"Start":"04:54.855 ","End":"04:59.220","Text":"This will be 2 cosine t dt,"},{"Start":"04:59.220 ","End":"05:05.460","Text":"and this will be just 0 dt or plane 0."},{"Start":"05:05.460 ","End":"05:11.209","Text":"That should be everything and now if we expand this integral,"},{"Start":"05:11.209 ","End":"05:16.955","Text":"we will get the integral of the parameter t from 0 to 2 Pi."},{"Start":"05:16.955 ","End":"05:19.250","Text":"Now I need P dx,"},{"Start":"05:19.250 ","End":"05:22.115","Text":"P is x minus z,"},{"Start":"05:22.115 ","End":"05:25.744","Text":"I need to look up x minus z,"},{"Start":"05:25.744 ","End":"05:30.240","Text":"so x is 2 cosine t,"},{"Start":"05:30.770 ","End":"05:36.135","Text":"z is 0 so this is just 2 cosine t,"},{"Start":"05:36.135 ","End":"05:39.670","Text":"I\u0027m reading off here P dx."},{"Start":"05:39.670 ","End":"05:41.765","Text":"Then I need dx,"},{"Start":"05:41.765 ","End":"05:48.920","Text":"which is minus 2 sine t, dt,"},{"Start":"05:48.920 ","End":"05:51.680","Text":"that\u0027s just the first part."},{"Start":"05:51.680 ","End":"05:54.320","Text":"Now, the second part Q,"},{"Start":"05:54.320 ","End":"05:56.405","Text":"Q is x cubed,"},{"Start":"05:56.405 ","End":"06:07.185","Text":"which is x cubed would be 8 cosine cubed t plus yz."},{"Start":"06:07.185 ","End":"06:10.455","Text":"That\u0027s 0 because z is 0,"},{"Start":"06:10.455 ","End":"06:24.585","Text":"and then dy is 2 cosine t, dt times 2 cosine t, dt,"},{"Start":"06:24.585 ","End":"06:32.550","Text":"and the last component is the R dz."},{"Start":"06:32.550 ","End":"06:38.000","Text":"Well, I don\u0027t need to compute R because I see that dz is 0,"},{"Start":"06:38.000 ","End":"06:39.920","Text":"so I\u0027ll just write plus 0,"},{"Start":"06:39.920 ","End":"06:41.990","Text":"we don\u0027t need anymore."},{"Start":"06:41.990 ","End":"06:48.445","Text":"That gives us the integral from 0 to 2 Pi,"},{"Start":"06:48.445 ","End":"06:52.305","Text":"I\u0027ll take it 1 time dt so what are we left with?"},{"Start":"06:52.305 ","End":"07:01.695","Text":"Minus 4 cosine t sine t and then from the second bit,"},{"Start":"07:01.695 ","End":"07:09.300","Text":"I get 8 times 2 is 16,"},{"Start":"07:09.300 ","End":"07:22.960","Text":"so plus 16 cosine to the 4rth t, and all this is dt,"},{"Start":"07:22.960 ","End":"07:28.765","Text":"I\u0027d like to remind you of a trigonometrical identity that 2 cosine t,"},{"Start":"07:28.765 ","End":"07:33.310","Text":"sine t is sine 2t."},{"Start":"07:35.540 ","End":"07:42.500","Text":"This I can write as I can take minus 2"},{"Start":"07:42.500 ","End":"07:44.120","Text":"and just be left with the 2"},{"Start":"07:44.120 ","End":"07:47.480","Text":"and then use the formula so minus 2,"},{"Start":"07:47.480 ","End":"07:53.460","Text":"the integral of sine 2t."},{"Start":"07:56.330 ","End":"07:58.695","Text":"I\u0027ll write it separately,"},{"Start":"07:58.695 ","End":"08:00.720","Text":"I\u0027ll separate them into 2 integrals,"},{"Start":"08:00.720 ","End":"08:04.170","Text":"0 to 2 Pi sine 2t dt,"},{"Start":"08:04.170 ","End":"08:07.480","Text":"I could have kept them separate."},{"Start":"08:08.570 ","End":"08:22.020","Text":"Plus 16 integral from 0 to 2 Pi of cosine to the 4t, dt."},{"Start":"08:22.020 ","End":"08:24.340","Text":"Now the first one is easy and the second one"},{"Start":"08:24.340 ","End":"08:28.045","Text":"I\u0027m going to use Table of Integrals for not waste a lot of time."},{"Start":"08:28.045 ","End":"08:39.460","Text":"The first one, the integral is minus a 1/2 cosine 2t,"},{"Start":"08:39.460 ","End":"08:41.150","Text":"the minus a 1/2 for the minus 2,"},{"Start":"08:41.150 ","End":"08:49.965","Text":"it will just give me cosine of 2t and you can check by differentiating."},{"Start":"08:49.965 ","End":"08:55.540","Text":"The derivative of cosine is minus sine and also the 2t gives me an extra 2,"},{"Start":"08:55.540 ","End":"09:01.990","Text":"so this works out so this is cosine 2t from 0 to 2 Pi."},{"Start":"09:01.990 ","End":"09:06.760","Text":"Plus, now I have 16,"},{"Start":"09:06.760 ","End":"09:10.670","Text":"I just wrote the results from the table of integrals"},{"Start":"09:10.670 ","End":"09:16.145","Text":"and I have to substitute 0 and 2 Pi."},{"Start":"09:16.145 ","End":"09:20.830","Text":"Now I claim that all these trigonometric parts come out to be 0"},{"Start":"09:20.830 ","End":"09:29.490","Text":"because if I substitute for cosine or for sine 0 or 2 Pi or 4 Pi,"},{"Start":"09:29.490 ","End":"09:33.470","Text":"it\u0027s going to be the same thing it\u0027s the same angle basically."},{"Start":"09:33.470 ","End":"09:39.700","Text":"Same here, 0 or 2 Pi, or 4 Pi,"},{"Start":"09:39.980 ","End":"09:42.390","Text":"it\u0027s all going to be the same"},{"Start":"09:42.390 ","End":"09:49.990","Text":"so all I\u0027m left with really that\u0027s of importance is this bit here."},{"Start":"09:50.150 ","End":"09:53.710","Text":"Now, this is equal to,"},{"Start":"09:53.900 ","End":"10:00.800","Text":"I can multiply the 16 by the 3/8 and get 8 goes into 16 twice so 2 times 3 is 6."},{"Start":"10:00.800 ","End":"10:11.944","Text":"This is just equal to 6t from 0 to 2 Pi,"},{"Start":"10:11.944 ","End":"10:13.890","Text":"at 0 I get nothing,"},{"Start":"10:13.890 ","End":"10:21.195","Text":"at 2 Pi I get 6 times 2 Pi is 12 Pi so the answer is just 12 Pi."},{"Start":"10:21.195 ","End":"10:27.035","Text":"This wasn\u0027t really very hard with the help of Stokes\u0027 theorem,"},{"Start":"10:27.035 ","End":"10:32.940","Text":"let me just highlight that final answer and declare that we are done."}],"ID":8830}],"Thumbnail":null,"ID":4973}]

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