WEBVTT 00:00:07.454 --> 00:00:09.370 CHRISTINE BREINER: Welcome back to recitation. 00:00:09.370 --> 00:00:12.580 In this video, I'd like us to consider the following problem. 00:00:12.580 --> 00:00:16.000 The first part is I'd like to know, for what values of b 00:00:16.000 --> 00:00:18.700 is this vector field F conservative? 00:00:18.700 --> 00:00:24.800 And F is defined as y*i plus the quantity x plus b*y*z j plus 00:00:24.800 --> 00:00:26.930 the quantity y squared plus 1, k. 00:00:26.930 --> 00:00:28.729 So as you can see, the only thing 00:00:28.729 --> 00:00:30.520 we're allowed to manipulate in this problem 00:00:30.520 --> 00:00:32.800 is b. b will be some real number. 00:00:32.800 --> 00:00:38.390 And I want to know, what real numbers can I put in for b so 00:00:38.390 --> 00:00:40.890 that this vector field is conservative. 00:00:40.890 --> 00:00:42.680 The second part of this problem is 00:00:42.680 --> 00:00:46.180 for each b-value you determined from one, 00:00:46.180 --> 00:00:47.680 find a potential function. 00:00:47.680 --> 00:00:52.200 So fix the b-value for one of the ones that is acceptable, 00:00:52.200 --> 00:00:55.990 based on number one, and then find the potential function. 00:00:55.990 --> 00:00:59.880 And then the third part says that you should explain 00:00:59.880 --> 00:01:02.750 why F dot dr is exact, and this is obviously 00:01:02.750 --> 00:01:04.990 for the b-values determined from one. 00:01:04.990 --> 00:01:09.000 So the second and third part are once you know the b-values. 00:01:09.000 --> 00:01:11.000 And you're only going to use those b-values that 00:01:11.000 --> 00:01:13.770 make F conservative, because that's the place where we can 00:01:13.770 --> 00:01:15.760 talk about finding a potential function, 00:01:15.760 --> 00:01:18.540 and that's where we can talk about F dot dr being exact, 00:01:18.540 --> 00:01:20.560 are exactly those values. 00:01:20.560 --> 00:01:21.250 OK. 00:01:21.250 --> 00:01:23.650 So why don't you pause the video, work on these three 00:01:23.650 --> 00:01:26.070 problems, and then when you're feeling good about them, 00:01:26.070 --> 00:01:28.153 bring the video back up, I'll show you what I did. 00:01:36.300 --> 00:01:37.560 OK, welcome back. 00:01:37.560 --> 00:01:40.711 Again, we're interested in doing three things with this vector 00:01:40.711 --> 00:01:41.210 field. 00:01:41.210 --> 00:01:43.290 The first thing we want to do is to find 00:01:43.290 --> 00:01:45.880 the values of b that make this vector field conservative. 00:01:45.880 --> 00:01:47.920 So I will start with that part. 00:01:47.920 --> 00:01:50.170 And as we know from the lecture, the thing 00:01:50.170 --> 00:01:53.750 I ultimately need to do is I need to find the curl of F. OK, 00:01:53.750 --> 00:01:55.880 so the curl of F is going to measure how far F 00:01:55.880 --> 00:01:57.190 is from being conservative. 00:01:57.190 --> 00:01:59.380 So if the curl of F is 0, I'm going 00:01:59.380 --> 00:02:01.190 to have F being conservative. 00:02:01.190 --> 00:02:04.560 So that's really what I'm interested in doing first. 00:02:04.560 --> 00:02:09.320 So I'm actually just going to rewrite what the curl of F 00:02:09.320 --> 00:02:10.570 actually is. 00:02:10.570 --> 00:02:14.210 So I'm going to let F-- I'm going to denote in our usual 00:02:14.210 --> 00:02:19.410 way by P, Q, R. And in this case, 00:02:19.410 --> 00:02:27.130 that's specifically y comma x plus b*y*z comma y squared plus 00:02:27.130 --> 00:02:29.600 1. 00:02:29.600 --> 00:02:33.670 OK, that's my P, Q, R. So P is y, Q is x plus b*y*z, 00:02:33.670 --> 00:02:36.160 and R is y squared plus 1. 00:02:36.160 --> 00:02:45.580 And now the curl of F-- which was found in the lecture, 00:02:45.580 --> 00:02:48.020 so I'm not going to show you again how to find it, 00:02:48.020 --> 00:02:50.410 I'm just going to write the formula for it-- is exactly 00:02:50.410 --> 00:02:52.880 the following vector. 00:02:52.880 --> 00:02:56.220 It's the derivative of the R-th component with respect to y, 00:02:56.220 --> 00:02:58.490 minus the Q-th component with respect to z. 00:02:58.490 --> 00:03:00.610 That's the i-value. 00:03:00.610 --> 00:03:08.650 The j-value is P sub z minus R sub x, j. 00:03:08.650 --> 00:03:15.990 The k-th value is Q sub x minus P sub y, k. 00:03:15.990 --> 00:03:17.930 OK, so there are three components here. 00:03:17.930 --> 00:03:21.410 And let's just start figuring out what these values are, 00:03:21.410 --> 00:03:24.857 and then we'll see what kind of restrictions we have on b. 00:03:24.857 --> 00:03:25.940 So let's start doing this. 00:03:25.940 --> 00:03:28.420 So again, this is P, this is Q, and this 00:03:28.420 --> 00:03:32.020 is R. So R sub y is the derivative of this component 00:03:32.020 --> 00:03:33.390 with respect to y. 00:03:33.390 --> 00:03:36.230 That's just 2y. 00:03:36.230 --> 00:03:39.320 Q sub z is the derivative of this with respect to z. 00:03:39.320 --> 00:03:41.030 Well, this is 0, and this is b*y. 00:03:44.560 --> 00:03:45.190 OK. 00:03:45.190 --> 00:03:47.410 Let's look at the rest first. 00:03:47.410 --> 00:03:51.535 P sub z: the derivative of this with respect to z is 0. 00:03:51.535 --> 00:03:56.782 The derivative of R with respect to x is 0. 00:03:56.782 --> 00:03:59.720 So that doesn't have any b's in it at all. 00:03:59.720 --> 00:04:01.580 And then Q sub x minus P sub y. 00:04:01.580 --> 00:04:02.440 Q is the middle one. 00:04:02.440 --> 00:04:03.970 Q sub x is 1. 00:04:07.240 --> 00:04:08.050 P is the first one. 00:04:08.050 --> 00:04:09.030 P sub y is 1. 00:04:12.010 --> 00:04:13.135 OK, so what do we get here? 00:04:13.135 --> 00:04:16.000 I should have written equals there, maybe. 00:04:16.000 --> 00:04:18.970 OK, so the j-th component is 0. 00:04:18.970 --> 00:04:20.910 And the k-th component is 0. 00:04:20.910 --> 00:04:28.140 So all I'm left with is 2y minus b*y, i. 00:04:28.140 --> 00:04:29.960 And if I want F to be conservative, 00:04:29.960 --> 00:04:31.880 this quantity has to be 0, so I see 00:04:31.880 --> 00:04:34.070 there's only one b-value that's going to work, 00:04:34.070 --> 00:04:36.420 and that is b is equal to 2. 00:04:40.190 --> 00:04:40.820 OK? 00:04:40.820 --> 00:04:43.740 So I know in part one, the answer to the question 00:04:43.740 --> 00:04:47.280 is just for b equals 2, is F conservative. 00:04:47.280 --> 00:04:48.960 That was, maybe, poorly phrased. 00:04:48.960 --> 00:04:51.580 F is conservative only when b is 2. 00:04:51.580 --> 00:04:52.080 OK. 00:04:52.080 --> 00:04:55.810 And that's because the curl of F is 0 only when b is 2. 00:04:55.810 --> 00:04:56.310 All right. 00:04:56.310 --> 00:04:58.300 So now we can move on to the second part. 00:04:58.300 --> 00:05:01.230 And the second part is for this particular value of b, 00:05:01.230 --> 00:05:02.980 find a potential function. 00:05:02.980 --> 00:05:04.610 And our strategy for that is going 00:05:04.610 --> 00:05:06.720 to be one of the methods from lecture. 00:05:06.720 --> 00:05:08.761 And it's going to be the method from lecture that 00:05:08.761 --> 00:05:11.820 in three dimensions is much easier than the other. 00:05:11.820 --> 00:05:15.140 So the one method in lecture that's easy in three dimensions 00:05:15.140 --> 00:05:17.220 is where you start at the origin, 00:05:17.220 --> 00:05:20.680 and you integrate F dot dr along a curve that's 00:05:20.680 --> 00:05:21.950 made up of line segments. 00:05:21.950 --> 00:05:25.130 So this strategy I've done before in two dimensions, 00:05:25.130 --> 00:05:27.170 in one of the problems in recitation. 00:05:27.170 --> 00:05:29.044 Now, we'll see it in three dimensions. 00:05:29.044 --> 00:05:30.460 So what we're going to do is we're 00:05:30.460 --> 00:05:34.930 going to integrate along a certain curve F dot dr. 00:05:34.930 --> 00:05:38.630 And this curve is going to go from the origin to 00:05:38.630 --> 00:05:41.500 (x_1, y_1, z_1). 00:05:41.500 --> 00:05:49.950 And that will give us f of (x_1, y_1, z_1). 00:05:49.950 --> 00:05:50.450 OK. 00:05:50.450 --> 00:05:52.600 So this is a sort of general strategy, 00:05:52.600 --> 00:05:54.880 and now we'll talk about it specifically. 00:05:54.880 --> 00:05:58.090 This will actually be f of (x_1, y_1, z_1) plus a constant, 00:05:58.090 --> 00:06:00.470 but we'll deal with that part right at the end. 00:06:00.470 --> 00:06:03.400 OK, so C in this case is going to be made up of three curves. 00:06:03.400 --> 00:06:06.970 And I'm going to draw them, in a picture, 00:06:06.970 --> 00:06:09.720 and then we're going to describe them. 00:06:09.720 --> 00:06:12.380 So I'm going to start at (0, 0, 0). 00:06:12.380 --> 00:06:20.310 My first curve will go out to x_1 comma 0, 0, 00:06:20.310 --> 00:06:22.280 and that's going to be the curve C_1. 00:06:22.280 --> 00:06:23.060 Oops. 00:06:23.060 --> 00:06:24.560 I want that to go the other way. 00:06:24.560 --> 00:06:25.590 That way. 00:06:25.590 --> 00:06:29.300 OK, C_1 is going to go from the origin to x_1 comma 0, 0. 00:06:29.300 --> 00:06:31.860 So the y- and z-values are going to be 00:06:31.860 --> 00:06:35.016 0 and 0 all the way along, and the x-value is going to change. 00:06:35.016 --> 00:06:36.640 My next one-- I'm going to make it long 00:06:36.640 --> 00:06:39.770 so I can have enough room to write-- that's 00:06:39.770 --> 00:06:41.900 going to be my C_2. 00:06:41.900 --> 00:06:49.180 And that's going to be x_1 comma y_1 comma 0. 00:06:49.180 --> 00:06:51.830 So in the end, what I've done is I've taken my x-value, 00:06:51.830 --> 00:06:53.580 I've kept it fixed all the way along here, 00:06:53.580 --> 00:06:56.210 but I'm varying the y-value out to y_1. 00:06:56.210 --> 00:06:59.730 And then the last one is going to go straight up. 00:06:59.730 --> 00:07:01.920 Right there. 00:07:01.920 --> 00:07:05.500 And so it's going to be with the x-value and y-value fixed. 00:07:05.500 --> 00:07:10.020 And at the end, I will be at x_1, y_1, z_1. 00:07:10.020 --> 00:07:11.110 And this is C3. 00:07:11.110 --> 00:07:13.790 So those are my three curves-- And this one, 00:07:13.790 --> 00:07:15.590 I'm going to move in this direction. 00:07:15.590 --> 00:07:17.000 Those are my three curves. 00:07:17.000 --> 00:07:19.450 And I want to point out that in order 00:07:19.450 --> 00:07:22.190 to understand how to simplify this problem, 00:07:22.190 --> 00:07:25.030 I'm going to have to remind myself what F dot dr is. 00:07:25.030 --> 00:07:25.530 OK. 00:07:25.530 --> 00:07:37.310 So F dot dr is P*dx plus Q*dy plus R*dz. 00:07:37.310 --> 00:07:38.120 Right? 00:07:38.120 --> 00:07:39.550 That's what F dot dr is. 00:07:39.550 --> 00:07:41.100 And so what I'm interested in, I'm 00:07:41.100 --> 00:07:44.630 going to integrate each of these things along C_1, C_2, C_3. 00:07:44.630 --> 00:07:48.700 But let's notice what happens along certain numbers 00:07:48.700 --> 00:07:51.500 of these curves. 00:07:51.500 --> 00:07:52.660 If we come back over here. 00:07:52.660 --> 00:07:56.220 On C_1, y is fixed and z as fixed. 00:07:56.220 --> 00:07:58.630 So dy and dz are both 0. 00:07:58.630 --> 00:08:02.490 So on C1, I only have to integrate P. OK, 00:08:02.490 --> 00:08:04.240 so I'm going to keep track of that. 00:08:04.240 --> 00:08:06.770 On C_1-- which, C_1 is parametrized 00:08:06.770 --> 00:08:10.420 by x, 0 to x-- I only need to worry 00:08:10.420 --> 00:08:14.380 about the P. P of x, 0, 0, dx. 00:08:14.380 --> 00:08:17.200 This is my C_1 component, and there's nothing else there, 00:08:17.200 --> 00:08:19.680 because these two are both 0. 00:08:19.680 --> 00:08:20.180 Right? 00:08:20.180 --> 00:08:24.470 Now let's consider what happens on C_2. 00:08:24.470 --> 00:08:25.930 If I look here. 00:08:25.930 --> 00:08:29.400 On C_2, the x-value is fixed and the z-value is fixed. 00:08:29.400 --> 00:08:32.340 x is fixed at x_1 and z is fixed at 0. 00:08:32.340 --> 00:08:36.590 And so dx and dz are both 0, because x and z are not 00:08:36.590 --> 00:08:37.190 changing. 00:08:37.190 --> 00:08:39.560 So there's only a dy component. 00:08:39.560 --> 00:08:45.120 So on C_2, which is parametrized in y-- from 0 to y_1-- 00:08:45.120 --> 00:08:52.300 I'm only interested in Q at x_1 comma y comma 0, dy. 00:08:52.300 --> 00:08:57.730 Again, this component is 0 on C_2 because dx is 0. 00:08:57.730 --> 00:09:00.770 And this component is 0 on C_2 because dz is 0. 00:09:00.770 --> 00:09:03.200 And this component-- I'm evaluating it-- 00:09:03.200 --> 00:09:06.440 x is fixed at x_1, z is fixed at 0, 00:09:06.440 --> 00:09:09.070 and the y is varying from 0 to y1. 00:09:09.070 --> 00:09:10.932 And then there's one more component, 00:09:10.932 --> 00:09:12.640 and I'm going to write it below, and then 00:09:12.640 --> 00:09:14.250 we'll do the rest over here. 00:09:14.250 --> 00:09:16.850 And the third component is the C_3 component. 00:09:16.850 --> 00:09:20.200 Now, not surprisingly-- if I come back over here-- 00:09:20.200 --> 00:09:23.290 because x and y are fixed all along the C3 component, 00:09:23.290 --> 00:09:25.130 the only thing that's changing is z. 00:09:25.130 --> 00:09:29.340 So dx and dy are 0, so I'm only worried about the dz part. 00:09:29.340 --> 00:09:29.840 OK. 00:09:29.840 --> 00:09:32.852 So again, as happened before, I only 00:09:32.852 --> 00:09:34.810 had P in the first one and Q in the second one, 00:09:34.810 --> 00:09:37.000 and now I have R, only, in the third one. 00:09:37.000 --> 00:09:39.840 And it's parametrized in z, from 0 to z_1. 00:09:39.840 --> 00:09:41.730 That's what z varies over. 00:09:41.730 --> 00:09:48.830 And it's going to be R at x_1 comma y_1 comma z dz. 00:09:48.830 --> 00:09:52.390 Because the x's are fixed at x_1, the y is fixed at y_1, 00:09:52.390 --> 00:09:54.415 but z is varying from 0 to z_1. 00:09:54.415 --> 00:09:55.020 All right. 00:09:55.020 --> 00:09:57.050 So I have these three parts, and now 00:09:57.050 --> 00:10:00.830 I just have to fill them in with the vector field that I have. 00:10:00.830 --> 00:10:03.390 I want to find what P is at (x, 0, 0), 00:10:03.390 --> 00:10:07.870 what Q is at (x_1, y_1, 0), and what R is at (x_1, y_1, z). 00:10:07.870 --> 00:10:09.080 And then integrate. 00:10:09.080 --> 00:10:11.360 So I have two steps left. 00:10:11.360 --> 00:10:13.770 One is plugging in and one is evaluating. 00:10:13.770 --> 00:10:17.940 So let me remind us what P, Q, and R actually are, and then 00:10:17.940 --> 00:10:20.950 we'll see what we get. 00:10:20.950 --> 00:10:22.260 Let me write it again. 00:10:22.260 --> 00:10:23.630 Maybe here. 00:10:23.630 --> 00:10:31.870 [P, Q, R] was equal to y comma x plus 2y*z-- 00:10:31.870 --> 00:10:34.860 I'll put it here so you don't have to look and I don't have 00:10:34.860 --> 00:10:38.410 to look-- and then y squared plus 1. 00:10:38.410 --> 00:10:39.270 OK. 00:10:39.270 --> 00:10:44.380 So P at x comma 0 comma 0. 00:10:44.380 --> 00:10:48.450 Well, if I plug in 0 for y, P is 0. 00:10:48.450 --> 00:10:53.160 So P at (x, 0, 0) is equal to 0. 00:10:53.160 --> 00:10:55.250 So I get nothing to integrate in the first part. 00:10:55.250 --> 00:10:56.340 That's nice. 00:10:56.340 --> 00:10:57.040 OK. 00:10:57.040 --> 00:11:02.320 Now, what is Q at x_1 comma y comma 0? 00:11:02.320 --> 00:11:04.600 Well, that would be an x_1 here. 00:11:04.600 --> 00:11:06.550 0 for y makes this term go away. 00:11:06.550 --> 00:11:09.130 So it's just equal to x_1. 00:11:09.130 --> 00:11:09.930 Right? 00:11:09.930 --> 00:11:18.660 And then R at x_1 comma y_1 comma z is y_1 squared plus 1. 00:11:21.480 --> 00:11:23.087 So now I'm going to substitute these 00:11:23.087 --> 00:11:24.170 into what I'm integrating. 00:11:24.170 --> 00:11:27.192 So in the first one, there's nothing there. 00:11:27.192 --> 00:11:28.820 Let me just write it right here. 00:11:28.820 --> 00:11:30.330 The Q is going to be the integral 00:11:30.330 --> 00:11:34.150 from 0 to y_1 of x_1 dy. 00:11:34.150 --> 00:11:39.160 And the R part is going to be the integral from 0 to z_1 00:11:39.160 --> 00:11:43.090 of y_1 squared plus 1 dz. 00:11:43.090 --> 00:11:46.060 OK, so the P part was disappeared. 00:11:46.060 --> 00:11:47.780 This is the Q part evaluated where 00:11:47.780 --> 00:11:50.580 I needed it to be evaluated. 00:11:50.580 --> 00:11:52.310 It's just x_1 dy. 00:11:52.310 --> 00:11:55.200 And the R part evaluated at (x_1, y_1, z) 00:11:55.200 --> 00:11:57.130 is just y_1 squared plus 1. 00:11:57.130 --> 00:11:59.500 And so I integrate that in z. 00:11:59.500 --> 00:12:03.910 So if I integrate this in y, all I get is x_1*y evaluated 0 00:12:03.910 --> 00:12:04.720 and y1. 00:12:04.720 --> 00:12:08.070 So here I just get an x_1*y_1. 00:12:08.070 --> 00:12:09.510 Right? 00:12:09.510 --> 00:12:12.800 And then here, if I integrate this in z, I just get a z-- 00:12:12.800 --> 00:12:14.600 and so I evaluate that at z_1 and 0-- 00:12:14.600 --> 00:12:18.890 I just get z_1 times y1 squared plus 1. 00:12:18.890 --> 00:12:19.400 OK. 00:12:19.400 --> 00:12:22.330 So this is actually my potential function. 00:12:22.330 --> 00:12:23.932 And so let me write it formally. 00:12:23.932 --> 00:12:25.890 I should actually say, this is my final answer. 00:12:25.890 --> 00:12:26.760 Right? 00:12:26.760 --> 00:12:27.920 I was integrating. 00:12:27.920 --> 00:12:29.730 This is actually what I get. 00:12:29.730 --> 00:12:31.180 And so what I was trying to find, 00:12:31.180 --> 00:12:33.429 if you remember-- I'm going to come back here and just 00:12:33.429 --> 00:12:34.190 mention it again. 00:12:34.190 --> 00:12:36.740 What I was doing was I was integrating along a curve F dot 00:12:36.740 --> 00:12:39.210 dr, to give me f of (x_1, y_1, z_1). 00:12:39.210 --> 00:12:40.010 Right? 00:12:40.010 --> 00:12:41.530 So now I've found it. 00:12:41.530 --> 00:12:43.650 The only thing I said is we also have to allow 00:12:43.650 --> 00:12:44.941 for there to be a constant. 00:12:44.941 --> 00:12:45.440 OK. 00:12:45.440 --> 00:12:50.000 So the potential function is actually exactly this function 00:12:50.000 --> 00:12:52.990 plus a constant. 00:12:52.990 --> 00:12:53.950 OK. 00:12:53.950 --> 00:12:56.280 So this is f of (x_1, y_1, z_1). 00:12:56.280 --> 00:12:57.905 And since I don't have much room above, 00:12:57.905 --> 00:12:59.373 I'll just write it below. 00:13:03.450 --> 00:13:05.330 This is f of x_1, y_1, z_1. 00:13:05.330 --> 00:13:10.810 So that's my potential function for this vector field, capital 00:13:10.810 --> 00:13:12.920 F, when it is conservative. 00:13:12.920 --> 00:13:15.290 So when b is equal to 2. 00:13:15.290 --> 00:13:17.470 OK, and there was one last part to this question. 00:13:17.470 --> 00:13:17.970 Right? 00:13:17.970 --> 00:13:19.920 So if we come all the way back over, 00:13:19.920 --> 00:13:22.790 we're reminded of one last part. 00:13:22.790 --> 00:13:26.160 It was explain why F dot dr is exact for the b values 00:13:26.160 --> 00:13:27.790 determined from number one. 00:13:27.790 --> 00:13:32.410 And the reason is exactly because of the following thing. 00:13:32.410 --> 00:13:36.360 F is conservative based on the fact that b is 2. 00:13:36.360 --> 00:13:42.240 And so when we talk about when F dot dr is exact, 00:13:42.240 --> 00:13:47.080 the simplest case is capital F is conservative, 00:13:47.080 --> 00:13:49.810 and I'm on a simply connected domain. 00:13:49.810 --> 00:13:50.440 OK. 00:13:50.440 --> 00:13:57.320 And if you notice, capital F is defined for all values x, y, z, 00:13:57.320 --> 00:13:59.900 and is differentiable for all values of x, y, z. 00:13:59.900 --> 00:14:03.250 So F is defined and differentiable everywhere 00:14:03.250 --> 00:14:04.550 on R^3. 00:14:04.550 --> 00:14:06.040 R^3 is simply connected. 00:14:06.040 --> 00:14:08.120 So we have a conservative vector field 00:14:08.120 --> 00:14:09.650 on a simply connected region. 00:14:09.650 --> 00:14:10.700 And that's what it means. 00:14:10.700 --> 00:14:14.100 That's one way that we have of knowing F dot dr is exact. 00:14:14.100 --> 00:14:17.910 And so that actually answers the third part of the question. 00:14:17.910 --> 00:14:20.010 So again, let me just remind you what we did. 00:14:20.010 --> 00:14:22.640 We started with a vector field F, 00:14:22.640 --> 00:14:26.320 we found values for b that made that vector field conservative, 00:14:26.320 --> 00:14:27.915 and then we used one of the techniques 00:14:27.915 --> 00:14:31.950 in class to find a potential function for that value of b. 00:14:31.950 --> 00:14:34.350 So there were a number of steps involved, 00:14:34.350 --> 00:14:37.240 but ultimately, again, it's the same type of problem 00:14:37.240 --> 00:14:41.750 you've seen before, when F was a vector field in two dimensions. 00:14:41.750 --> 00:14:45.110 So it shouldn't be feeling too different from some 00:14:45.110 --> 00:14:47.120 of the stuff you saw earlier. 00:14:47.120 --> 00:14:49.520 OK, I think that's where I'll stop.