[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.50,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.50,0:00:01.95,Default,,0000,0000,0000,,In the last video,\Nwe figured out
Dialogue: 0,0:00:01.95,0:00:05.59,Default,,0000,0000,0000,,how to construct a unit\Nnormal vector to a surface.
Dialogue: 0,0:00:05.59,0:00:08.80,Default,,0000,0000,0000,,And so now we can use that\Nback in our original surface
Dialogue: 0,0:00:08.80,0:00:10.75,Default,,0000,0000,0000,,integral to try to\Nsimplify a little bit,
Dialogue: 0,0:00:10.75,0:00:13.32,Default,,0000,0000,0000,,or at least give us a clue how\Nwe can calculate these things.
Dialogue: 0,0:00:13.32,0:00:14.82,Default,,0000,0000,0000,,And also, think\Nabout different ways
Dialogue: 0,0:00:14.82,0:00:17.95,Default,,0000,0000,0000,,to represent this type\Nof a surface integral.
Dialogue: 0,0:00:17.95,0:00:20.71,Default,,0000,0000,0000,,So if we just substitute what\Nwe came up as our normal vector,
Dialogue: 0,0:00:20.71,0:00:22.19,Default,,0000,0000,0000,,our unit normal\Nvector right here,
Dialogue: 0,0:00:22.19,0:00:26.12,Default,,0000,0000,0000,,we will get-- so\Nonce again, it's
Dialogue: 0,0:00:26.12,0:00:31.54,Default,,0000,0000,0000,,the surface integral of F dot.
Dialogue: 0,0:00:31.54,0:00:33.50,Default,,0000,0000,0000,,And F dot all of this\Nbusiness right over here.
Dialogue: 0,0:00:33.50,0:00:35.12,Default,,0000,0000,0000,,And I'm going to\Nwrite it all in white,
Dialogue: 0,0:00:35.12,0:00:36.90,Default,,0000,0000,0000,,just so it doesn't\Ntake me too much time.
Dialogue: 0,0:00:36.90,0:00:38.93,Default,,0000,0000,0000,,So the partial of\Nr with respect to u
Dialogue: 0,0:00:38.93,0:00:41.94,Default,,0000,0000,0000,,crossed with the partial\Nof r with respect
Dialogue: 0,0:00:41.94,0:00:46.28,Default,,0000,0000,0000,,to v over the magnitude\Nof the same thing, partial
Dialogue: 0,0:00:46.28,0:00:49.56,Default,,0000,0000,0000,,of r with respect to u\Ncrossed with the partial of r
Dialogue: 0,0:00:49.56,0:00:51.77,Default,,0000,0000,0000,,with respect to v.
Dialogue: 0,0:00:51.77,0:00:53.71,Default,,0000,0000,0000,,And now, we've\Nplayed with ds a lot.
Dialogue: 0,0:00:53.71,0:00:55.38,Default,,0000,0000,0000,,We know that the other\Nway to write ds--
Dialogue: 0,0:00:55.38,0:00:56.87,Default,,0000,0000,0000,,and I gave the\Nintuition, hopefully,
Dialogue: 0,0:00:56.87,0:00:58.84,Default,,0000,0000,0000,,for that several videos\Nago when we first
Dialogue: 0,0:00:58.84,0:01:01.12,Default,,0000,0000,0000,,explored what a surface\Nintegral was all about.
Dialogue: 0,0:01:01.12,0:01:05.77,Default,,0000,0000,0000,,We know that ds is--\Nit can be represented
Dialogue: 0,0:01:05.77,0:01:08.42,Default,,0000,0000,0000,,as the magnitude of the\Npartial of r with respect
Dialogue: 0,0:01:08.42,0:01:13.50,Default,,0000,0000,0000,,to u crossed with the partial\Nof r with respect to v du dv.
Dialogue: 0,0:01:13.50,0:01:17.02,Default,,0000,0000,0000,,And Obviously, the du dv, it\Ncould be written as dv du.
Dialogue: 0,0:01:17.02,0:01:20.41,Default,,0000,0000,0000,,You could write it as da,\Na little chunk of area
Dialogue: 0,0:01:20.41,0:01:23.31,Default,,0000,0000,0000,,and the uv plane or\Nin the uv domain.
Dialogue: 0,0:01:23.31,0:01:26.01,Default,,0000,0000,0000,,And actually, since now this\Nintegral's in terms of uv,
Dialogue: 0,0:01:26.01,0:01:27.76,Default,,0000,0000,0000,,we're no longer taking\Na surface integral.
Dialogue: 0,0:01:27.76,0:01:30.88,Default,,0000,0000,0000,,We're now taking a double\Nintegral over the uv domain.
Dialogue: 0,0:01:30.88,0:01:33.15,Default,,0000,0000,0000,,So you could say kind\Nof a region in uv.
Dialogue: 0,0:01:33.15,0:01:38.10,Default,,0000,0000,0000,,So I'll say R to say that's\Nit's a region in the uv plane
Dialogue: 0,0:01:38.10,0:01:39.55,Default,,0000,0000,0000,,that we're now thinking about.
Dialogue: 0,0:01:39.55,0:01:41.77,Default,,0000,0000,0000,,But there's probably a\Nhuge-- or there should be,
Dialogue: 0,0:01:41.77,0:01:43.72,Default,,0000,0000,0000,,or I'm guessing there's a\Nhuge simplification that's
Dialogue: 0,0:01:43.72,0:01:44.93,Default,,0000,0000,0000,,popping out at you right now.
Dialogue: 0,0:01:44.93,0:01:47.13,Default,,0000,0000,0000,,We're dividing by the\Nmagnitude of the cross product
Dialogue: 0,0:01:47.13,0:01:48.59,Default,,0000,0000,0000,,of these two vectors\Nand then we're
Dialogue: 0,0:01:48.59,0:01:52.04,Default,,0000,0000,0000,,multiplying by the magnitude of\Nthe cross product of these two
Dialogue: 0,0:01:52.04,0:01:52.70,Default,,0000,0000,0000,,vectors.
Dialogue: 0,0:01:52.70,0:01:54.07,Default,,0000,0000,0000,,Those are just\Nscalar quantities.
Dialogue: 0,0:01:54.07,0:01:56.14,Default,,0000,0000,0000,,You divide by something\Nand multiply by something.
Dialogue: 0,0:01:56.14,0:01:58.60,Default,,0000,0000,0000,,Well, that's just the same\Nthing as multiplying or dividing
Dialogue: 0,0:01:58.60,0:01:59.39,Default,,0000,0000,0000,,by 1.
Dialogue: 0,0:01:59.39,0:02:01.42,Default,,0000,0000,0000,,So these two\Ncharacters cancel out,
Dialogue: 0,0:02:01.42,0:02:05.93,Default,,0000,0000,0000,,and our integral simplifies\Nto the double integral
Dialogue: 0,0:02:05.93,0:02:09.25,Default,,0000,0000,0000,,over that region, the\Ncorresponding region in the uv
Dialogue: 0,0:02:09.25,0:02:14.20,Default,,0000,0000,0000,,plane, of F-- of\Nour vector field F
Dialogue: 0,0:02:14.20,0:02:16.88,Default,,0000,0000,0000,,dotted with this cross product.
Dialogue: 0,0:02:16.88,0:02:19.13,Default,,0000,0000,0000,,This is going to give us\Na vector right over here.
Dialogue: 0,0:02:19.13,0:02:20.74,Default,,0000,0000,0000,,That's going to\Ngive us a vector.
Dialogue: 0,0:02:20.74,0:02:22.28,Default,,0000,0000,0000,,It gives us actually\Na normal vector.
Dialogue: 0,0:02:22.28,0:02:24.03,Default,,0000,0000,0000,,And then when you\Ndivide by its magnitude,
Dialogue: 0,0:02:24.03,0:02:25.52,Default,,0000,0000,0000,,it gives you a\Nunit normal vector.
Dialogue: 0,0:02:25.52,0:02:28.78,Default,,0000,0000,0000,,So this, you're going to\Ntake the dot product of F
Dialogue: 0,0:02:28.78,0:02:32.70,Default,,0000,0000,0000,,with r, the partial\Nof r with respect
Dialogue: 0,0:02:32.70,0:02:41.17,Default,,0000,0000,0000,,to u crossed with the partial\Nof r with respect to v du dv.
Dialogue: 0,0:02:41.17,0:02:45.55,Default,,0000,0000,0000,,Let me scroll over to the\Nright a little bit, du dv.
Dialogue: 0,0:02:45.55,0:02:47.59,Default,,0000,0000,0000,,And we'll see in the\Nfew videos from now
Dialogue: 0,0:02:47.59,0:02:50.27,Default,,0000,0000,0000,,that this is essentially how we\Ngo about actually calculating
Dialogue: 0,0:02:50.27,0:02:50.86,Default,,0000,0000,0000,,these things.
Dialogue: 0,0:02:50.86,0:02:52.39,Default,,0000,0000,0000,,If you have a\Nparameterization, you
Dialogue: 0,0:02:52.39,0:02:54.64,Default,,0000,0000,0000,,can then get everything in\Nterms of a double integral,
Dialogue: 0,0:02:54.64,0:02:56.51,Default,,0000,0000,0000,,in terms of uv this way.
Dialogue: 0,0:02:56.51,0:02:58.80,Default,,0000,0000,0000,,Now, the last thing I want\Nto do is explore another way
Dialogue: 0,0:02:58.80,0:03:01.05,Default,,0000,0000,0000,,that you'll see a surface\Nintegral like this written.
Dialogue: 0,0:03:01.05,0:03:03.38,Default,,0000,0000,0000,,It all comes from,\Nreally, writing this part
Dialogue: 0,0:03:03.38,0:03:04.17,Default,,0000,0000,0000,,in a different way.
Dialogue: 0,0:03:04.17,0:03:05.63,Default,,0000,0000,0000,,But it hopefully gives you\Na little bit more intuition
Dialogue: 0,0:03:05.63,0:03:07.70,Default,,0000,0000,0000,,of what this thing\Nis even saying.
Dialogue: 0,0:03:07.70,0:03:09.24,Default,,0000,0000,0000,,So I'm just going to rewrite.
Dialogue: 0,0:03:09.24,0:03:12.57,Default,,0000,0000,0000,,I'm going to rewrite this\Nchunk right over here.
Dialogue: 0,0:03:12.57,0:03:14.12,Default,,0000,0000,0000,,I'm just going to\Nrewrite that chunk.
Dialogue: 0,0:03:14.12,0:03:16.58,Default,,0000,0000,0000,,And I'm going to use slightly\Ndifferent notation because it
Dialogue: 0,0:03:16.58,0:03:19.03,Default,,0000,0000,0000,,will hopefully help make\Na little bit more sense.
Dialogue: 0,0:03:19.03,0:03:21.10,Default,,0000,0000,0000,,So the partial of r\Nwith respect to u I
Dialogue: 0,0:03:21.10,0:03:25.93,Default,,0000,0000,0000,,can write as the partial\Nof r with respect to u.
Dialogue: 0,0:03:25.93,0:03:27.60,Default,,0000,0000,0000,,And we're taking\Nthe cross product.
Dialogue: 0,0:03:27.60,0:03:30.30,Default,,0000,0000,0000,,Let me make my u's a little bit\Nmore u-like so we confuse them
Dialogue: 0,0:03:30.30,0:03:31.19,Default,,0000,0000,0000,,with v's.
Dialogue: 0,0:03:31.19,0:03:32.94,Default,,0000,0000,0000,,And we're taking the\Ncross product of that
Dialogue: 0,0:03:32.94,0:03:36.40,Default,,0000,0000,0000,,with the partial\Nof r with respect
Dialogue: 0,0:03:36.40,0:03:40.13,Default,,0000,0000,0000,,to v. So very small\Nchanges in our vector--
Dialogue: 0,0:03:40.13,0:03:41.93,Default,,0000,0000,0000,,in our parameterization\Nright here,
Dialogue: 0,0:03:41.93,0:03:44.10,Default,,0000,0000,0000,,our position vector given\Na small change in v. Very
Dialogue: 0,0:03:44.10,0:03:47.85,Default,,0000,0000,0000,,small changes in the vector\Ngiven a small change in u.
Dialogue: 0,0:03:47.85,0:03:50.67,Default,,0000,0000,0000,,And then we're multiplying\Nthat times du dv.
Dialogue: 0,0:03:50.67,0:03:56.24,Default,,0000,0000,0000,,We're multiplying\Nthat times du dv.
Dialogue: 0,0:03:56.24,0:03:59.75,Default,,0000,0000,0000,,Now, du and dv are\Njust scalar quantities.
Dialogue: 0,0:03:59.75,0:04:01.29,Default,,0000,0000,0000,,They're infinitesimally small.
Dialogue: 0,0:04:01.29,0:04:03.08,Default,,0000,0000,0000,,But for the sake\Nof this argument,
Dialogue: 0,0:04:03.08,0:04:04.75,Default,,0000,0000,0000,,you can just view--\Nthey're not vectors,
Dialogue: 0,0:04:04.75,0:04:06.50,Default,,0000,0000,0000,,they're just scalar quantities.
Dialogue: 0,0:04:06.50,0:04:09.73,Default,,0000,0000,0000,,And so you can\Nessentially include them--
Dialogue: 0,0:04:09.73,0:04:11.53,Default,,0000,0000,0000,,if you have the cross product.
Dialogue: 0,0:04:11.53,0:04:18.47,Default,,0000,0000,0000,,If you have a cross b times some\Nscalar value-- I don't know, x,
Dialogue: 0,0:04:18.47,0:04:23.61,Default,,0000,0000,0000,,you could rewrite this\Nas x times a cross b,
Dialogue: 0,0:04:23.61,0:04:28.53,Default,,0000,0000,0000,,or you could write\Nthis as a cross x times
Dialogue: 0,0:04:28.53,0:04:30.70,Default,,0000,0000,0000,,b, because x is\Njust a scalar value.
Dialogue: 0,0:04:30.70,0:04:31.54,Default,,0000,0000,0000,,It's just a number.
Dialogue: 0,0:04:31.54,0:04:33.21,Default,,0000,0000,0000,,So we could do the\Nsame thing over here.
Dialogue: 0,0:04:33.21,0:04:35.65,Default,,0000,0000,0000,,We can rewrite all\Nof this business as--
Dialogue: 0,0:04:35.65,0:04:39.63,Default,,0000,0000,0000,,and I'm going to group the du\Nwhere we have the partial--
Dialogue: 0,0:04:39.63,0:04:41.30,Default,,0000,0000,0000,,or with respect to u\Nin the denominator.
Dialogue: 0,0:04:41.30,0:04:42.97,Default,,0000,0000,0000,,And I'll do the same\Nthing with the v's.
Dialogue: 0,0:04:42.97,0:04:48.43,Default,,0000,0000,0000,,And so you will get the\Npartial of r with respect
Dialogue: 0,0:04:48.43,0:04:52.44,Default,,0000,0000,0000,,to u times du,\Ntimes that scalar.
Dialogue: 0,0:04:52.44,0:04:54.24,Default,,0000,0000,0000,,So that'll give us a vector.
Dialogue: 0,0:04:54.24,0:04:55.75,Default,,0000,0000,0000,,And we're going to cross that.
Dialogue: 0,0:04:55.75,0:04:58.53,Default,,0000,0000,0000,,We're going to cross that with\Nthe partial of r with respect
Dialogue: 0,0:04:58.53,0:05:06.65,Default,,0000,0000,0000,,to v dv.
Dialogue: 0,0:05:06.65,0:05:08.81,Default,,0000,0000,0000,,Now, these might\Nlook notationally
Dialogue: 0,0:05:08.81,0:05:10.48,Default,,0000,0000,0000,,like two different\Nthings, but that just
Dialogue: 0,0:05:10.48,0:05:12.47,Default,,0000,0000,0000,,comes from the\Nnecessity of when we
Dialogue: 0,0:05:12.47,0:05:16.78,Default,,0000,0000,0000,,take partial derivatives to say,\Noh, no, this vector function
Dialogue: 0,0:05:16.78,0:05:18.89,Default,,0000,0000,0000,,is defined-- it's a function\Nof multiple variables
Dialogue: 0,0:05:18.89,0:05:20.72,Default,,0000,0000,0000,,and this is taking a\Nderivative with respect
Dialogue: 0,0:05:20.72,0:05:22.01,Default,,0000,0000,0000,,to only one of them.
Dialogue: 0,0:05:22.01,0:05:23.51,Default,,0000,0000,0000,,So this is, how\Nmuch does our vector
Dialogue: 0,0:05:23.51,0:05:26.33,Default,,0000,0000,0000,,change when you have a\Nvery small change in u?
Dialogue: 0,0:05:26.33,0:05:28.88,Default,,0000,0000,0000,,But this is also an\Ninfinitesimally small change
Dialogue: 0,0:05:28.88,0:05:31.95,Default,,0000,0000,0000,,in u over here, we're just using\Nslightly different notation.
Dialogue: 0,0:05:31.95,0:05:33.28,Default,,0000,0000,0000,,So for the sake of--\Nand this is a little bit
Dialogue: 0,0:05:33.28,0:05:35.28,Default,,0000,0000,0000,,loosey-goosey mathematics,\Nbut it will hopefully
Dialogue: 0,0:05:35.28,0:05:37.86,Default,,0000,0000,0000,,give you the intuition\Nfor why this thing could
Dialogue: 0,0:05:37.86,0:05:39.79,Default,,0000,0000,0000,,be written in a different way.
Dialogue: 0,0:05:39.79,0:05:42.29,Default,,0000,0000,0000,,These are essentially\Nthe same quantity.
Dialogue: 0,0:05:42.29,0:05:44.77,Default,,0000,0000,0000,,So if you divide by something\Nand multiply by something,
Dialogue: 0,0:05:44.77,0:05:46.07,Default,,0000,0000,0000,,you can cancel them out.
Dialogue: 0,0:05:46.07,0:05:48.28,Default,,0000,0000,0000,,If you divide by something\Nand multiply by something,
Dialogue: 0,0:05:48.28,0:05:49.84,Default,,0000,0000,0000,,you can cancel them out.
Dialogue: 0,0:05:49.84,0:05:52.07,Default,,0000,0000,0000,,And all you're left\Nwith then-- all
Dialogue: 0,0:05:52.07,0:05:55.67,Default,,0000,0000,0000,,you're left with is\Nthe differential of r.
Dialogue: 0,0:05:55.67,0:05:57.08,Default,,0000,0000,0000,,And since we lost\Nthe information
Dialogue: 0,0:05:57.08,0:05:58.50,Default,,0000,0000,0000,,that it's in the\Nu-direction, I'll
Dialogue: 0,0:05:58.50,0:06:00.83,Default,,0000,0000,0000,,write here, the differential\Nof r in the u-direction.
Dialogue: 0,0:06:00.83,0:06:02.89,Default,,0000,0000,0000,,I don't want to get\Nthe notation confused.
Dialogue: 0,0:06:02.89,0:06:04.14,Default,,0000,0000,0000,,This is just the differential.
Dialogue: 0,0:06:04.14,0:06:06.00,Default,,0000,0000,0000,,This is just how much r changed.
Dialogue: 0,0:06:06.00,0:06:08.82,Default,,0000,0000,0000,,This is not the partial\Nderivative of r with respect
Dialogue: 0,0:06:08.82,0:06:09.46,Default,,0000,0000,0000,,to u.
Dialogue: 0,0:06:09.46,0:06:11.35,Default,,0000,0000,0000,,This right over here\Nis, how much does r
Dialogue: 0,0:06:11.35,0:06:15.32,Default,,0000,0000,0000,,change given per unit change,\Nper small change in u?
Dialogue: 0,0:06:15.32,0:06:19.19,Default,,0000,0000,0000,,This just says a differential\Nin the direction of-- I
Dialogue: 0,0:06:19.19,0:06:22.53,Default,,0000,0000,0000,,guess as u changes, this is\Nhow much that infinitely small
Dialogue: 0,0:06:22.53,0:06:24.06,Default,,0000,0000,0000,,change that just r changes.
Dialogue: 0,0:06:24.06,0:06:27.46,Default,,0000,0000,0000,,This isn't change in r with\Nrespect to change in u.
Dialogue: 0,0:06:27.46,0:06:28.71,Default,,0000,0000,0000,,And we're going to cross that.
Dialogue: 0,0:06:28.71,0:06:31.78,Default,,0000,0000,0000,,Now, we're going to cross\Nthat with the partial of r,
Dialogue: 0,0:06:31.78,0:06:34.90,Default,,0000,0000,0000,,the partial of r\Nin the v-direction.
Dialogue: 0,0:06:34.90,0:06:37.28,Default,,0000,0000,0000,,Now, this right over here,\Nlet's just conceptualize this.
Dialogue: 0,0:06:37.28,0:06:39.92,Default,,0000,0000,0000,,And this goes back to\Nour original visions
Dialogue: 0,0:06:39.92,0:06:42.33,Default,,0000,0000,0000,,of what a surface\Nintegral was all about.
Dialogue: 0,0:06:42.33,0:06:44.31,Default,,0000,0000,0000,,If we're on a surface--\Nand I'll draw surface.
Dialogue: 0,0:06:44.31,0:06:45.48,Default,,0000,0000,0000,,Let me draw another surface.
Dialogue: 0,0:06:45.48,0:06:47.97,Default,,0000,0000,0000,,I won't use the one that\NI had already drawn on.
Dialogue: 0,0:06:47.97,0:06:51.08,Default,,0000,0000,0000,,If we draw a surface, and for\Na very small change in u--
Dialogue: 0,0:06:51.08,0:06:52.91,Default,,0000,0000,0000,,and we're not going to\Nthink about the rate.
Dialogue: 0,0:06:52.91,0:06:56.22,Default,,0000,0000,0000,,We're just thinking about\Nkind of the change in r.
Dialogue: 0,0:06:56.22,0:06:57.79,Default,,0000,0000,0000,,You're going in that direction.
Dialogue: 0,0:06:57.79,0:07:01.22,Default,,0000,0000,0000,,So if that thing\Nlooks like this,
Dialogue: 0,0:07:01.22,0:07:05.31,Default,,0000,0000,0000,,this is actually kind of a\Ndistance moved on the surface.
Dialogue: 0,0:07:05.31,0:07:08.06,Default,,0000,0000,0000,,Because remember, this\Nisn't the derivative.
Dialogue: 0,0:07:08.06,0:07:09.35,Default,,0000,0000,0000,,This is the differential.
Dialogue: 0,0:07:09.35,0:07:11.84,Default,,0000,0000,0000,,So it's just a small\Nchange along the surface,
Dialogue: 0,0:07:11.84,0:07:13.51,Default,,0000,0000,0000,,that's that over there.
Dialogue: 0,0:07:13.51,0:07:16.71,Default,,0000,0000,0000,,And that this is a small\Nchange when you change v. So
Dialogue: 0,0:07:16.71,0:07:19.05,Default,,0000,0000,0000,,it's also a change\Nalong the surface.
Dialogue: 0,0:07:19.05,0:07:21.31,Default,,0000,0000,0000,,When you take the cross\Nproduct of these two things,
Dialogue: 0,0:07:21.31,0:07:23.43,Default,,0000,0000,0000,,you get a vector\Nthat is orthogonal.
Dialogue: 0,0:07:23.43,0:07:26.57,Default,,0000,0000,0000,,You get a vector that is\Nnormal to the surface.
Dialogue: 0,0:07:26.57,0:07:33.74,Default,,0000,0000,0000,,So it is normal to the\Nsurface and its magnitude--
Dialogue: 0,0:07:33.74,0:07:36.34,Default,,0000,0000,0000,,and we saw this when we first\Nlearned about cross products.
Dialogue: 0,0:07:36.34,0:07:40.08,Default,,0000,0000,0000,,Its magnitude is\Nequal to the area that
Dialogue: 0,0:07:40.08,0:07:42.68,Default,,0000,0000,0000,,is defined by these two vectors.
Dialogue: 0,0:07:42.68,0:07:49.30,Default,,0000,0000,0000,,So its magnitude\Nis equal to area.
Dialogue: 0,0:07:49.30,0:07:50.76,Default,,0000,0000,0000,,So in a lot of\Nways, you can really
Dialogue: 0,0:07:50.76,0:07:52.42,Default,,0000,0000,0000,,think of it-- you\Nreally could think
Dialogue: 0,0:07:52.42,0:07:57.02,Default,,0000,0000,0000,,of it as a unit normal\Nvector times ds.
Dialogue: 0,0:07:57.02,0:07:59.96,Default,,0000,0000,0000,,And so the way that we would,\NI guess notationally do this,
Dialogue: 0,0:07:59.96,0:08:02.47,Default,,0000,0000,0000,,is we can call this--\Nbecause this is kind of a ds,
Dialogue: 0,0:08:02.47,0:08:04.04,Default,,0000,0000,0000,,but it's a vector\Nversion of the ds.
Dialogue: 0,0:08:04.04,0:08:06.76,Default,,0000,0000,0000,,Over here, this is just\Nan area right over here.
Dialogue: 0,0:08:06.76,0:08:08.83,Default,,0000,0000,0000,,This is just a scalar value.
Dialogue: 0,0:08:08.83,0:08:12.25,Default,,0000,0000,0000,,But now, we have a vector\Nthat points normally
Dialogue: 0,0:08:12.25,0:08:14.93,Default,,0000,0000,0000,,from the surface,\Nbut its magnitude
Dialogue: 0,0:08:14.93,0:08:17.30,Default,,0000,0000,0000,,is the same thing as\Nthat ds that we were just
Dialogue: 0,0:08:17.30,0:08:18.34,Default,,0000,0000,0000,,talking about.
Dialogue: 0,0:08:18.34,0:08:22.07,Default,,0000,0000,0000,,So we can call this thing right\Nover here, we can call this ds.
Dialogue: 0,0:08:22.07,0:08:25.14,Default,,0000,0000,0000,,And the key difference here\Nis this is a vector now.
Dialogue: 0,0:08:25.14,0:08:28.24,Default,,0000,0000,0000,,So we'll call it ds with\Na little vector over it
Dialogue: 0,0:08:28.24,0:08:29.57,Default,,0000,0000,0000,,to know that this thing.
Dialogue: 0,0:08:29.57,0:08:33.20,Default,,0000,0000,0000,,This isn't the scalar ds that\Nis just concerned with the area.
Dialogue: 0,0:08:33.20,0:08:34.97,Default,,0000,0000,0000,,But when you view\Nthings this way,
Dialogue: 0,0:08:34.97,0:08:39.29,Default,,0000,0000,0000,,we just saw that this entire\Nthing simplifies to ds.
Dialogue: 0,0:08:39.29,0:08:43.42,Default,,0000,0000,0000,,Then our whole surface\Nintegral can be rewritten.
Dialogue: 0,0:08:43.42,0:08:45.20,Default,,0000,0000,0000,,Instead of writing\Nit like this, we
Dialogue: 0,0:08:45.20,0:08:50.48,Default,,0000,0000,0000,,can write it as the\Nintegral or the surface
Dialogue: 0,0:08:50.48,0:08:54.05,Default,,0000,0000,0000,,integral-- those integral\Nsigns were too fancy.
Dialogue: 0,0:08:54.05,0:08:57.79,Default,,0000,0000,0000,,The surface integral of F dot.
Dialogue: 0,0:08:57.79,0:09:00.75,Default,,0000,0000,0000,,And instead of saying a\Nnormal vector times the scalar
Dialogue: 0,0:09:00.75,0:09:03.04,Default,,0000,0000,0000,,quantity, that little chunk\Nof area on the surface,
Dialogue: 0,0:09:03.04,0:09:07.73,Default,,0000,0000,0000,,we can now just call that\Nthe vector differential ds.
Dialogue: 0,0:09:07.73,0:09:10.23,Default,,0000,0000,0000,,And I want to make it clear,\Nthese are two different things.
Dialogue: 0,0:09:10.23,0:09:12.28,Default,,0000,0000,0000,,This is a vector.
Dialogue: 0,0:09:12.28,0:09:14.13,Default,,0000,0000,0000,,This is essentially\Nwhat we're calling it.
Dialogue: 0,0:09:14.13,0:09:16.66,Default,,0000,0000,0000,,This right over here is a\Nscalar times a normal vector.
Dialogue: 0,0:09:16.66,0:09:18.03,Default,,0000,0000,0000,,So these are three\Ndifferent ways
Dialogue: 0,0:09:18.03,0:09:19.80,Default,,0000,0000,0000,,of really representing\Nthe same thing.
Dialogue: 0,0:09:19.80,0:09:22.18,Default,,0000,0000,0000,,And in different contexts,\Nyou will see different things,
Dialogue: 0,0:09:22.18,0:09:25.11,Default,,0000,0000,0000,,depending on what the author of\Nwhoever's trying to communicate
Dialogue: 0,0:09:25.11,0:09:26.82,Default,,0000,0000,0000,,is trying to communicate.
Dialogue: 0,0:09:26.82,0:09:29.99,Default,,0000,0000,0000,,This right over here is the one\Nthat we'll use most frequently
Dialogue: 0,0:09:29.99,0:09:34.19,Default,,0000,0000,0000,,as we actually try to calculate\Nthese surface integrals.
Dialogue: 0,0:09:34.19,0:09:34.69,Default,,0000,0000,0000,,