[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.52,0:00:03.24,Default,,0000,0000,0000,,In the last video, we began\Nto explore Stokes' theorem.
Dialogue: 0,0:00:03.24,0:00:04.70,Default,,0000,0000,0000,,And what I want to\Ndo in this video
Dialogue: 0,0:00:04.70,0:00:07.06,Default,,0000,0000,0000,,is to see whether it's\Nconsistent with some
Dialogue: 0,0:00:07.06,0:00:09.05,Default,,0000,0000,0000,,of what we have already seen.
Dialogue: 0,0:00:09.05,0:00:12.19,Default,,0000,0000,0000,,And to do that, let's imagine--\Nso let me draw my axes.
Dialogue: 0,0:00:12.19,0:00:14.34,Default,,0000,0000,0000,,So that's my z-axis.
Dialogue: 0,0:00:14.34,0:00:16.68,Default,,0000,0000,0000,,That is my x-axis.
Dialogue: 0,0:00:16.68,0:00:19.61,Default,,0000,0000,0000,,And then that is my y-axis.
Dialogue: 0,0:00:19.61,0:00:23.43,Default,,0000,0000,0000,,And now let's imagine a\Nregion in the xy plane.
Dialogue: 0,0:00:23.43,0:00:25.77,Default,,0000,0000,0000,,So let me draw it like this.
Dialogue: 0,0:00:25.77,0:00:30.67,Default,,0000,0000,0000,,So let's say this is my\Nregion in the xy plane.
Dialogue: 0,0:00:30.67,0:00:34.85,Default,,0000,0000,0000,,I will call that\Nregion R. And I also
Dialogue: 0,0:00:34.85,0:00:36.58,Default,,0000,0000,0000,,have a boundary of that region.
Dialogue: 0,0:00:36.58,0:00:39.47,Default,,0000,0000,0000,,And let's say we care\Nabout the direction
Dialogue: 0,0:00:39.47,0:00:40.72,Default,,0000,0000,0000,,that we traverse the boundary.
Dialogue: 0,0:00:40.72,0:00:41.65,Default,,0000,0000,0000,,And let's say we're\Ngoing to traverse it
Dialogue: 0,0:00:41.65,0:00:43.18,Default,,0000,0000,0000,,in a counterclockwise direction.
Dialogue: 0,0:00:43.18,0:00:47.15,Default,,0000,0000,0000,,So we have this path that\Ngoes around this region.
Dialogue: 0,0:00:47.15,0:00:49.89,Default,,0000,0000,0000,,We can call that c.
Dialogue: 0,0:00:49.89,0:00:51.95,Default,,0000,0000,0000,,So we'll call that\Nc, and we're going
Dialogue: 0,0:00:51.95,0:00:57.01,Default,,0000,0000,0000,,to traverse it in the\Ncounterclockwise direction.
Dialogue: 0,0:00:57.01,0:01:02.31,Default,,0000,0000,0000,,And let's say that we also\Nhave a vector field f.
Dialogue: 0,0:01:02.31,0:01:05.36,Default,,0000,0000,0000,,That essentially its\Ni component is just
Dialogue: 0,0:01:05.36,0:01:08.03,Default,,0000,0000,0000,,going to be a\Nfunction of x and y.
Dialogue: 0,0:01:08.03,0:01:10.31,Default,,0000,0000,0000,,And its j component\Nis only going
Dialogue: 0,0:01:10.31,0:01:12.53,Default,,0000,0000,0000,,to be a function of x and y.
Dialogue: 0,0:01:12.53,0:01:14.78,Default,,0000,0000,0000,,And let's say it\Nhas no k component.
Dialogue: 0,0:01:14.78,0:01:17.23,Default,,0000,0000,0000,,So the vector field\Non this region,
Dialogue: 0,0:01:17.23,0:01:18.75,Default,,0000,0000,0000,,it might look\Nsomething like this.
Dialogue: 0,0:01:18.75,0:01:20.42,Default,,0000,0000,0000,,I'm just drawing random things.
Dialogue: 0,0:01:20.42,0:01:21.88,Default,,0000,0000,0000,,And then if you go\Noff that region,
Dialogue: 0,0:01:21.88,0:01:23.35,Default,,0000,0000,0000,,if you go in the\Nz direction, it's
Dialogue: 0,0:01:23.35,0:01:25.70,Default,,0000,0000,0000,,just going to look the same\Nas you go higher and higher.
Dialogue: 0,0:01:25.70,0:01:27.93,Default,,0000,0000,0000,,So that vector,\Nit wouldn't change
Dialogue: 0,0:01:27.93,0:01:29.66,Default,,0000,0000,0000,,as you change your z component.
Dialogue: 0,0:01:29.66,0:01:31.45,Default,,0000,0000,0000,,And all of the vectors\Nwould essentially
Dialogue: 0,0:01:31.45,0:01:35.79,Default,,0000,0000,0000,,be parallel to, or\Nif z is 0, actually
Dialogue: 0,0:01:35.79,0:01:39.10,Default,,0000,0000,0000,,sitting on the xy plane.
Dialogue: 0,0:01:39.10,0:01:41.48,Default,,0000,0000,0000,,Now given this, let's\Nthink about what
Dialogue: 0,0:01:41.48,0:01:46.01,Default,,0000,0000,0000,,Stokes' theorem would tell us\Nabout the value of the line
Dialogue: 0,0:01:46.01,0:01:48.98,Default,,0000,0000,0000,,integral over the\Ncontour-- let me
Dialogue: 0,0:01:48.98,0:01:51.47,Default,,0000,0000,0000,,draw that a little\Nbit neater-- the line
Dialogue: 0,0:01:51.47,0:02:00.96,Default,,0000,0000,0000,,integral over the\Ncontour c of f dot
Dialogue: 0,0:02:00.96,0:02:05.96,Default,,0000,0000,0000,,dr, f dot lowercase dr,\NWhere dr is obviously
Dialogue: 0,0:02:05.96,0:02:08.28,Default,,0000,0000,0000,,going along the contour.
Dialogue: 0,0:02:08.28,0:02:11.47,Default,,0000,0000,0000,,So if we take Stokes'\Ntheorem, then this quantity
Dialogue: 0,0:02:11.47,0:02:13.85,Default,,0000,0000,0000,,right over here should\Nbe equal to this quantity
Dialogue: 0,0:02:13.85,0:02:14.61,Default,,0000,0000,0000,,right over here.
Dialogue: 0,0:02:14.61,0:02:18.85,Default,,0000,0000,0000,,It should be equal to the double\Nintegral over the surface.
Dialogue: 0,0:02:18.85,0:02:21.27,Default,,0000,0000,0000,,Well this region is\Nreally just a surface
Dialogue: 0,0:02:21.27,0:02:23.45,Default,,0000,0000,0000,,that's sitting in the xy plane.
Dialogue: 0,0:02:23.45,0:02:26.08,Default,,0000,0000,0000,,So it should really just\Nbe the double integral--
Dialogue: 0,0:02:26.08,0:02:27.66,Default,,0000,0000,0000,,let me write that\Nin that same-- it'll
Dialogue: 0,0:02:27.66,0:02:31.31,Default,,0000,0000,0000,,be the double integral over\Nour region, which is really
Dialogue: 0,0:02:31.31,0:02:35.11,Default,,0000,0000,0000,,just the same thing\Nas our surface,
Dialogue: 0,0:02:35.11,0:02:37.84,Default,,0000,0000,0000,,of the curl of f dot n.
Dialogue: 0,0:02:37.84,0:02:40.44,Default,,0000,0000,0000,,So let's just think about\Nwhat the curl of f dot n is.
Dialogue: 0,0:02:40.44,0:02:42.27,Default,,0000,0000,0000,,And then d of s would\Njust be a little chunk
Dialogue: 0,0:02:42.27,0:02:45.51,Default,,0000,0000,0000,,of our region, a little chunk\Nof our flattened surface
Dialogue: 0,0:02:45.51,0:02:46.22,Default,,0000,0000,0000,,right over there.
Dialogue: 0,0:02:46.22,0:02:50.18,Default,,0000,0000,0000,,So instead of ds,\NI'll just write da.
Dialogue: 0,0:02:50.18,0:02:54.00,Default,,0000,0000,0000,,But let's think of what curl\Nof f dot n would actually be.
Dialogue: 0,0:02:54.00,0:02:56.06,Default,,0000,0000,0000,,So let's work on\Ncurl of f first.
Dialogue: 0,0:02:56.06,0:02:58.91,Default,,0000,0000,0000,,So the curl of f--\Nand the way I always
Dialogue: 0,0:02:58.91,0:03:00.81,Default,,0000,0000,0000,,remember it is\Nwe're going to take
Dialogue: 0,0:03:00.81,0:03:06.85,Default,,0000,0000,0000,,the determinant of\Nthis ijk partial
Dialogue: 0,0:03:06.85,0:03:10.82,Default,,0000,0000,0000,,with respect to x,\Npartial with respect to y,
Dialogue: 0,0:03:10.82,0:03:12.33,Default,,0000,0000,0000,,partial with respect to z.
Dialogue: 0,0:03:12.33,0:03:14.45,Default,,0000,0000,0000,,This is just the definition\Nof taking the curl.
Dialogue: 0,0:03:14.45,0:03:16.82,Default,,0000,0000,0000,,We're figuring out how\Nmuch this vector field
Dialogue: 0,0:03:16.82,0:03:18.91,Default,,0000,0000,0000,,would cause something to spin.
Dialogue: 0,0:03:18.91,0:03:20.98,Default,,0000,0000,0000,,And then we want the\Ni component, which
Dialogue: 0,0:03:20.98,0:03:24.42,Default,,0000,0000,0000,,is our function p, which is\Njust a function of x and y,
Dialogue: 0,0:03:24.42,0:03:26.99,Default,,0000,0000,0000,,j component, which is\Njust the function q.
Dialogue: 0,0:03:26.99,0:03:30.72,Default,,0000,0000,0000,,And there was no z\Ncomponent over here, so 0.
Dialogue: 0,0:03:30.72,0:03:32.89,Default,,0000,0000,0000,,And so this is going\Nto be equal to-- well
Dialogue: 0,0:03:32.89,0:03:34.35,Default,,0000,0000,0000,,if we look at the\Ni component, it's
Dialogue: 0,0:03:34.35,0:03:35.96,Default,,0000,0000,0000,,going to be the\Npartial of y of 0.
Dialogue: 0,0:03:35.96,0:03:42.57,Default,,0000,0000,0000,,That's just going to be 0, minus\Nthe partial of q with respect
Dialogue: 0,0:03:42.57,0:03:43.45,Default,,0000,0000,0000,,to z.
Dialogue: 0,0:03:43.45,0:03:46.00,Default,,0000,0000,0000,,Well what's the partial\Nof q with respect to z?
Dialogue: 0,0:03:46.00,0:03:48.19,Default,,0000,0000,0000,,Well q isn't a\Nfunction of z at all.
Dialogue: 0,0:03:48.19,0:03:50.47,Default,,0000,0000,0000,,So that's also going to be\N0-- let me write this out
Dialogue: 0,0:03:50.47,0:03:52.33,Default,,0000,0000,0000,,just so it's not too confusing.
Dialogue: 0,0:03:52.33,0:03:56.18,Default,,0000,0000,0000,,So our i component, it's\Ngoing to be partial of 0
Dialogue: 0,0:03:56.18,0:03:57.13,Default,,0000,0000,0000,,with respect to y.
Dialogue: 0,0:03:57.13,0:04:01.00,Default,,0000,0000,0000,,Well that's just going to\Nbe 0 minus the partial of q
Dialogue: 0,0:04:01.00,0:04:02.29,Default,,0000,0000,0000,,with respect to z.
Dialogue: 0,0:04:02.29,0:04:04.34,Default,,0000,0000,0000,,Well the partial of\Nq with respect to z
Dialogue: 0,0:04:04.34,0:04:05.70,Default,,0000,0000,0000,,is just going to be 0.
Dialogue: 0,0:04:05.70,0:04:07.50,Default,,0000,0000,0000,,So we have a 0 i component.
Dialogue: 0,0:04:07.50,0:04:10.26,Default,,0000,0000,0000,,And then we want to\Nsubtract the j component.
Dialogue: 0,0:04:10.26,0:04:16.70,Default,,0000,0000,0000,,And then the j component partial\Nof 0 with respect to x is 0.
Dialogue: 0,0:04:16.70,0:04:20.08,Default,,0000,0000,0000,,And then from that you're going\Nto subtract the partial of p
Dialogue: 0,0:04:20.08,0:04:22.18,Default,,0000,0000,0000,,with respect to z.
Dialogue: 0,0:04:22.18,0:04:25.59,Default,,0000,0000,0000,,Well once again, p is not\Na function of z at all.
Dialogue: 0,0:04:25.59,0:04:28.16,Default,,0000,0000,0000,,So that's going to be 0 again.
Dialogue: 0,0:04:28.16,0:04:33.91,Default,,0000,0000,0000,,And then you have plus k times\Nthe partial of q with respect
Dialogue: 0,0:04:33.91,0:04:34.41,Default,,0000,0000,0000,,to x.
Dialogue: 0,0:04:34.41,0:04:36.32,Default,,0000,0000,0000,,Remember this is just the\Npartial derivative operator.
Dialogue: 0,0:04:36.32,0:04:38.18,Default,,0000,0000,0000,,So the partial of q\Nwith respect to x.
Dialogue: 0,0:04:41.16,0:04:43.45,Default,,0000,0000,0000,,And from that we're going\Nto subtract the partial of p
Dialogue: 0,0:04:43.45,0:04:44.57,Default,,0000,0000,0000,,with respect to y.
Dialogue: 0,0:04:49.69,0:04:56.15,Default,,0000,0000,0000,,So the curl of f just simplifies\Nto this right over here.
Dialogue: 0,0:04:56.15,0:04:58.88,Default,,0000,0000,0000,,Now what is n?
Dialogue: 0,0:04:58.88,0:05:02.25,Default,,0000,0000,0000,,What is the unit normal vector.
Dialogue: 0,0:05:02.25,0:05:04.30,Default,,0000,0000,0000,,Well we're in the xy plane.
Dialogue: 0,0:05:04.30,0:05:05.93,Default,,0000,0000,0000,,So the unit normal\Nvector is just
Dialogue: 0,0:05:05.93,0:05:07.94,Default,,0000,0000,0000,,going to be straight\Nup in the z direction.
Dialogue: 0,0:05:07.94,0:05:10.39,Default,,0000,0000,0000,,It's going to have\Na magnitude of 1.
Dialogue: 0,0:05:10.39,0:05:12.45,Default,,0000,0000,0000,,So in this case, our\Nunit normal vector
Dialogue: 0,0:05:12.45,0:05:14.66,Default,,0000,0000,0000,,is just going to\Nbe the k vector.
Dialogue: 0,0:05:14.66,0:05:18.49,Default,,0000,0000,0000,,So we're essentially just going\Nto take-- so curl of f is this.
Dialogue: 0,0:05:18.49,0:05:21.88,Default,,0000,0000,0000,,And our unit normal\Nvector is just
Dialogue: 0,0:05:21.88,0:05:24.51,Default,,0000,0000,0000,,going to be equal to the k.
Dialogue: 0,0:05:24.51,0:05:26.92,Default,,0000,0000,0000,,It's just going to\Nbe the k unit vector.
Dialogue: 0,0:05:26.92,0:05:28.23,Default,,0000,0000,0000,,It's going to go straight up.
Dialogue: 0,0:05:28.23,0:05:31.16,Default,,0000,0000,0000,,So what happens if we\Ntake the curl of f dot k?
Dialogue: 0,0:05:31.16,0:05:34.03,Default,,0000,0000,0000,,If we just dot this with k.
Dialogue: 0,0:05:34.03,0:05:36.08,Default,,0000,0000,0000,,We're just dotting\Nthis with this.
Dialogue: 0,0:05:36.08,0:05:39.73,Default,,0000,0000,0000,,Well, we're just going to end up\Nwith this part right over here.
Dialogue: 0,0:05:39.73,0:05:43.93,Default,,0000,0000,0000,,So curl of f dot the unit\Nnormal vector is just
Dialogue: 0,0:05:43.93,0:05:45.40,Default,,0000,0000,0000,,going to be equal\Nto this business.
Dialogue: 0,0:05:45.40,0:05:49.26,Default,,0000,0000,0000,,It's just going to be equal to\Nthe partial of q with respect
Dialogue: 0,0:05:49.26,0:05:54.98,Default,,0000,0000,0000,,to x minus the partial\Nof p with respect to y.
Dialogue: 0,0:05:54.98,0:05:57.94,Default,,0000,0000,0000,,And this is neat because\Nusing Stokes' theorem
Dialogue: 0,0:05:57.94,0:05:59.61,Default,,0000,0000,0000,,in the special case,\Nwhere we're dealing
Dialogue: 0,0:05:59.61,0:06:03.03,Default,,0000,0000,0000,,with a flattened-out\Nsurface in the xy plane,
Dialogue: 0,0:06:03.03,0:06:07.96,Default,,0000,0000,0000,,in this situation, this just\Nboiled down to Green's theorem.
Dialogue: 0,0:06:07.96,0:06:12.03,Default,,0000,0000,0000,,This thing right over here just\Nboiled down to Green's theorem.
Dialogue: 0,0:06:12.03,0:06:15.92,Default,,0000,0000,0000,,So we see that Green's theorem\Nis really just a special case--
Dialogue: 0,0:06:15.92,0:06:17.84,Default,,0000,0000,0000,,let me write theorem\Na little bit neater.
Dialogue: 0,0:06:17.84,0:06:20.39,Default,,0000,0000,0000,,We see that Green's\Ntheorem is really
Dialogue: 0,0:06:20.39,0:06:22.80,Default,,0000,0000,0000,,just a special case\Nof Stokes' theorem,
Dialogue: 0,0:06:22.80,0:06:27.36,Default,,0000,0000,0000,,where our surface is flattened\Nout, and it's in the xy plane.
Dialogue: 0,0:06:27.36,0:06:30.14,Default,,0000,0000,0000,,So that should make us feel\Npretty good, although we still
Dialogue: 0,0:06:30.14,0:06:32.24,Default,,0000,0000,0000,,have not proven Stokes' theorem.
Dialogue: 0,0:06:32.24,0:06:34.53,Default,,0000,0000,0000,,But the one thing that I do\Nlike about this is seeing
Dialogue: 0,0:06:34.53,0:06:36.78,Default,,0000,0000,0000,,that Green's theorem and\NStokes' theorem is consistent
Dialogue: 0,0:06:36.78,0:06:39.43,Default,,0000,0000,0000,,is now it starts to make\Nsense of this right over here.
Dialogue: 0,0:06:39.43,0:06:40.81,Default,,0000,0000,0000,,When we first learned Green's\Ntheorem, we were like,
Dialogue: 0,0:06:40.81,0:06:41.38,Default,,0000,0000,0000,,what is this?
Dialogue: 0,0:06:41.38,0:06:42.56,Default,,0000,0000,0000,,what's going on over here?
Dialogue: 0,0:06:42.56,0:06:44.19,Default,,0000,0000,0000,,But now this is\Ntelling us this is just
Dialogue: 0,0:06:44.19,0:06:47.92,Default,,0000,0000,0000,,taking the curl in this\Nregion along this surface.
Dialogue: 0,0:06:47.92,0:06:50.84,Default,,0000,0000,0000,,And now starts to make a lot\Nof sense based on the intuition
Dialogue: 0,0:06:50.84,0:06:54.09,Default,,0000,0000,0000,,that we saw in the last video.