0:00:00.000,0:00:00.710


0:00:00.710,0:00:04.470
In the last video I introduced[br]you to the idea of the chain

0:00:04.470,0:00:05.520
rule with partial derivatives.

0:00:05.520,0:00:10.080
And we said, well, if I have a[br]function, psi, Greek letter,

0:00:10.080,0:00:14.020
psi, it's a function[br]of x and y.

0:00:14.020,0:00:16.770
And if I wanted to take the[br]partial of this, with respect

0:00:16.770,0:00:19.360
to-- no, I want to take the[br]derivative, not the partial--

0:00:19.360,0:00:23.430
the derivative of this, with[br]respect to x, this is equal to

0:00:23.430,0:00:29.540
the partial of psi, with respect[br]to x, plus the partial

0:00:29.540,0:00:35.400
of psi, with respect[br]to y, times dy, dx.

0:00:35.400,0:00:37.630
And in the last video I didn't[br]prove it to you, but I

0:00:37.630,0:00:40.260
hopefully gave you a little bit[br]of intuition that you can

0:00:40.260,0:00:40.740
believe me.

0:00:40.740,0:00:43.030
But maybe one day I'll prove[br]it a little bit more

0:00:43.030,0:00:46.120
rigorously, but you can find[br]proofs on the web if you are

0:00:46.120,0:00:49.960
interested, for the chain rule[br]with partial derivatives.

0:00:49.960,0:00:52.760
So let's put that aside and[br]let's explore another property

0:00:52.760,0:00:55.600
of partial derivatives, and then[br]we're ready to get the

0:00:55.600,0:00:57.080
intuition behind exact[br]equations.

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Because you're going to find,[br]it's fairly straightforward to

0:00:59.070,0:01:02.210
solve exact equations, but the[br]intuition is a little bit

0:01:02.210,0:01:05.140
more-- well, I don't want to say[br]it's difficult, because if

0:01:05.140,0:01:06.890
you have the intuition,[br]you have it.

0:01:06.890,0:01:11.490
So what if I had, say, this[br]function, psi, and I were to

0:01:11.490,0:01:16.580
take the partial derivative of[br]psi, with respect to x, first.

0:01:16.580,0:01:17.510
I'll just write psi.

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I don't have to write[br]x and y every time.

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And then I were to take the[br]partial derivative with

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respect to y.

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So just as a notation, this you[br]could write as, you could

0:01:32.730,0:01:34.620
kind of view it as you're[br]multiplying the operators, so

0:01:34.620,0:01:36.050
it could be written like this.

0:01:36.050,0:01:42.400
The partial del squared times[br]psi, or del squared psi, over

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del y del, or curly d x.

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And that can also be written[br]as-- and this is my preferred

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notation, because it doesn't[br]have all this extra junk

0:01:53.040,0:01:53.800
everywhere.

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You could just say, well, the[br]partial, we took the partial,

0:01:56.350,0:02:00.050
with respect to x, first. So[br]this just means the partial of

0:02:00.050,0:02:01.240
psi, with respect to x.

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And then we took the partial,[br]with respect to y.

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So that's one situation[br]to consider.

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What happens when we take the[br]partial, with respect to x,

0:02:07.970,0:02:08.650
and then y?

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So with respect to x, you hold[br]y constant to get just the

0:02:13.100,0:02:14.190
partial, with respect to x.

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Ignore the y there.

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And then you hold the x[br]constant, and you take the

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partial, with respect to y.

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So what's the difference between[br]that and if we were to

0:02:21.480,0:02:22.370
switch the order?

0:02:22.370,0:02:24.970
So what happens if we were to--[br]I'll do it in a different

0:02:24.970,0:02:30.400
color-- if we had psi, and we[br]were to take the partial, with

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respect to y, first, and then[br]we were to take the partial,

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with respect to x?

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So just the notation, just so[br]you're comfortable with it,

0:02:40.640,0:02:44.660
that would be-- so partial[br]x, partial y.

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And this is the operator.

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And it might be a little[br]confusing that here, between

0:02:48.750,0:02:51.060
these two notations, even though[br]they're the same thing,

0:02:51.060,0:02:52.740
the order is mixed.

0:02:52.740,0:02:54.250
That's just because it's[br]just a different way of

0:02:54.250,0:02:54.910
thinking about it.

0:02:54.910,0:02:57.990
This says, OK, partial first,[br]with respect to x, then y.

0:02:57.990,0:03:00.160
This views it more as the[br]operator, so we took the

0:03:00.160,0:03:03.000
partial of x first, and then[br]we took y, like you're

0:03:03.000,0:03:04.950
multiplying the operators.

0:03:04.950,0:03:08.840
But anyway, so this can also be[br]written as the partial of

0:03:08.840,0:03:13.070
y, with respect to x-- sorry,[br]the partial of y, and then we

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took the partial of that[br]with respect to x.

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Now, I'm going to tell you right[br]now, that if each of the

0:03:17.980,0:03:20.840
first partials are continuous--[br]and most of the

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functions we've dealt with in[br]a normal domain, as long as

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there aren't any[br]discontinuities, or holes, or

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something strange in the[br]function definition, they

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usually are continuous.

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And especially in a first-year[br]calculus or differential

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course, we're probably going to[br]be dealing with continuous

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functions in soon. our domain.

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If both of these functions are[br]continuous, if both of the

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first partials are continuous,[br]then these two are going to be

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equal to each other.

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So psi of xy is going to[br]be equal to psi of yx.

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Now, we can use this knowledge,[br]which is the chain

0:04:01.220,0:04:04.870
rule using partial derivatives,[br]and this

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knowledge to now solve a certain[br]class of differential

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equations, first order[br]differential equations, called

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exact equations.

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And what does an exact[br]equation look like?

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An exact equation[br]looks like this.

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The color picking's[br]the hard part.

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So let's say this is my[br]differential equation.

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I have some function[br]of x and y.

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So I don't know, it could[br]be x squared times

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cosine of y or something.

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I don't know, it could be[br]any function of x and y.

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Plus some function of x and y,[br]we'll call that n, times dy,

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dx is equal to 0.

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This is-- well, I don't know if[br]it's an exact equation yet,

0:04:47.520,0:04:50.880
but if you saw something of this[br]form, your first impulse

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should be, oh-- well, actually,[br]your very first

0:04:52.990,0:04:54.500
impulse is, is this separable?

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And you should try to play[br]around with the algebra a

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little bit to see if it's[br]separable, because that's

0:04:57.620,0:04:59.210
always the most straightforward[br]way.

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If it's not separable, but you[br]can still put it in this form,

0:05:01.770,0:05:04.460
you say, hey, is it[br]an exact equation?

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And what's an exact equation?

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Well, look immediately.

0:05:07.270,0:05:11.600
This pattern right here[br]looks an awful

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lot like this pattern.

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What if M was the partial of[br]psi, with respect to x?

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What if psi, with respect[br]to x, is equal to M?

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What if this was psi,[br]with respect to x?

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And what if this was psi,[br]with respect to y?

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So psi, with respect to[br]y, is equal to N.

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What if?

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I'm just saying, we don't[br]know for sure, right?

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If you just see this someplace[br]randomly, you won't know for

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sure that this is the partial[br]of, with respect to x of some

0:05:40.200,0:05:43.060
function, and this is the[br]partial, with respect to y of

0:05:43.060,0:05:43.830
some function.

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But we're just saying,[br]what if?

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If this were true, then we[br]could rewrite this as the

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partial of psi, with respect to[br]x, plus the partial of psi,

0:05:52.870,0:05:58.680
with respect to y, times[br]dy, dx, equal to 0.

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And this right here, the left[br]side right there, that's the

0:06:02.050,0:06:04.790
same thing as this, right?

0:06:04.790,0:06:09.040
This is just the derivative of[br]psi, with respect to x, using

0:06:09.040,0:06:10.940
the partial derivative[br]chain rule.

0:06:10.940,0:06:12.710
So you could rewrite it.

0:06:12.710,0:06:17.130
You could rewrite, this is just[br]the derivative of psi,

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with respect to x, inside[br]the function of x,

0:06:20.480,0:06:23.410
y, is equal to 0.

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So if you see a differential[br]equation, and it has this

0:06:27.730,0:06:31.070
form, and you say, boy, I can't[br]separate it, but maybe

0:06:31.070,0:06:32.030
it's an exact equation.

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And frankly, if that was what[br]was recently covered before

0:06:35.940,0:06:38.800
the current exam, it probably[br]is an exact equation.

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But if you see this form,[br]you say, boy, maybe

0:06:40.940,0:06:42.070
it's an exact equation.

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If it is an exact equation-- and[br]I'll show you how to test

0:06:44.580,0:06:48.350
it in a second using this[br]information-- then this can be

0:06:48.350,0:06:52.550
written as the derivative of[br]some function, psi, where this

0:06:52.550,0:06:54.840
is the partial of psi,[br]with respect to x.

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This is the partial of psi,[br]with respect to y.

0:06:57.720,0:06:59.655
And then if you could write it[br]like this, and you take the

0:06:59.655,0:07:01.370
derivative of both sides--[br]sorry, you take the

0:07:01.370,0:07:06.890
antiderivative of both sides--[br]and you would get psi of x, y

0:07:06.890,0:07:10.070
is equal to c as a solution.

0:07:10.070,0:07:12.770
So there are two things that we[br]should be caring you about.

0:07:12.770,0:07:16.470
Then you might be saying, OK,[br]Sal, you've walked through

0:07:16.470,0:07:19.550
psi's, and partials,[br]and all this.

0:07:19.550,0:07:22.020
One, how do I know that it's[br]an exact equation?

0:07:22.020,0:07:24.590
And then, if it is an exact[br]equation, which tells us that

0:07:24.590,0:07:28.290
there is some psi, then how[br]do I solve for the psi?

0:07:28.290,0:07:32.380
So the way to figure out is it[br]an exact equation, is to use

0:07:32.380,0:07:34.690
this information right here.

0:07:34.690,0:07:38.150
We know that if psi and its[br]derivatives are continuous

0:07:38.150,0:07:42.100
over some domain, that when[br]you take the partial, with

0:07:42.100,0:07:45.760
respect to x and then y, that's[br]the same thing as doing

0:07:45.760,0:07:46.980
it in the other order.

0:07:46.980,0:07:48.930
So we said, this is[br]the partial, with

0:07:48.930,0:07:50.180
respect to x, right?

0:07:50.180,0:07:52.610


0:07:52.610,0:07:55.920
And this is the partial,[br]with respect to y.

0:07:55.920,0:07:59.880
So if this is an exact equation,[br]if this is the exact

0:07:59.880,0:08:03.250
equation, if we were take the[br]partial of this, with respect

0:08:03.250,0:08:05.330
to y, right?

0:08:05.330,0:08:11.600
If we were to take the partial[br]of M, with respect to y-- so

0:08:11.600,0:08:15.560
the partial of psi, with respect[br]to x, is equal to M.

0:08:15.560,0:08:18.490
If we were to take the partial[br]of those, with respect to y--

0:08:18.490,0:08:22.450
so we could just rewrite that as[br]that-- then that should be

0:08:22.450,0:08:28.090
equal to the partial of N,[br]with respect to x, right?

0:08:28.090,0:08:31.976
The partial of psi, with respect[br]to y, is equal to N.

0:08:31.976,0:08:34.760
So if we take the partial, with[br]respect to x, of both of

0:08:34.760,0:08:40.964
these, we know from this that[br]these should be equal, if psi

0:08:40.964,0:08:44.400
and its partials are continuous[br]over that domain.

0:08:44.400,0:08:49.320
So then this will[br]also be equal.

0:08:49.320,0:08:51.990
So that is actually the[br]test to test if

0:08:51.990,0:08:53.930
this is an exact equation.

0:08:53.930,0:08:56.300
So let me rewrite all of that[br]again and summarize it a

0:08:56.300,0:08:56.690
little bit.

0:08:56.690,0:09:04.870
So if you see something of the[br]form, M of x, y plus N of x,

0:09:04.870,0:09:09.580
y, times dy, dx is equal to 0.

0:09:09.580,0:09:13.110
And then you take the partial[br]derivative of M, with respect

0:09:13.110,0:09:18.280
to y, and then you take the[br]partial derivative of N, with

0:09:18.280,0:09:24.030
respect to x, and they are equal[br]to each other, then--

0:09:24.030,0:09:26.410
and it's actually if and only[br]if, so it goes both ways--

0:09:26.410,0:09:30.930
this is an exact equation, an[br]exact differential equation.

0:09:30.930,0:09:32.410
This is an exact equation.

0:09:32.410,0:09:35.510
And if it's an exact equation,[br]that tells us that there

0:09:35.510,0:09:47.140
exists a psi, such that the[br]derivative of psi of x, y is

0:09:47.140,0:09:52.200
equal to 0, or psi of x, y is[br]equal to c, is a solution of

0:09:52.200,0:09:53.050
this equation.

0:09:53.050,0:09:58.480
And the partial derivative of[br]psi, with respect to x, is

0:09:58.480,0:09:59.740
equal to M.

0:09:59.740,0:10:03.760
And the partial derivative of[br]psi, with respect to y, is

0:10:03.760,0:10:05.340
equal to N.

0:10:05.340,0:10:07.550
And I'll show you in the next[br]video how to actually use this

0:10:07.550,0:10:09.810
information to solve for psi.

0:10:09.810,0:10:11.640
So here are some things[br]I want to point out.

0:10:11.640,0:10:13.720
This is going to be the partial[br]derivative of psi,

0:10:13.720,0:10:17.620
with respect to x, but when we[br]take the kind of exact test,

0:10:17.620,0:10:19.590
we take it with respect to y,[br]because we want to get that

0:10:19.590,0:10:21.080
mixed derivative.

0:10:21.080,0:10:23.410
Similarly, this is going to be[br]the partial derivative of psi,

0:10:23.410,0:10:27.030
with respect to y, but when we[br]do the test, we take the

0:10:27.030,0:10:29.500
partial of it with respect[br]to x so we get that mixed

0:10:29.500,0:10:30.730
derivative.

0:10:30.730,0:10:32.570
This is with respect to y,[br]and then with respect to

0:10:32.570,0:10:33.920
x, so you get this.

0:10:33.920,0:10:36.300
Anyway, I know that might be a[br]little bit involved, but if

0:10:36.300,0:10:38.360
you understood everything I did,[br]I think you'll have the

0:10:38.360,0:10:41.390
intuition behind why[br]the methodology of

0:10:41.390,0:10:43.470
exact equations works.

0:10:43.470,0:10:45.950
I will see you in the next[br]video, where we will actually

0:10:45.950,0:10:49.400
solve some exact equations See

0:10:49.400,0:10:50.500