Second example of line integral of conservative vector field

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:01.800
Let's do another problem.

00:00:01.800 --> 00:00:04.430
Very similar to the last one,
but with a subtle difference.

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And that subtle difference
will make a big difference.

00:00:06.860 --> 00:00:10.050
Let's say we take the line
integral over some curve c--

00:00:10.050 --> 00:00:15.590
I'll define the curve in a
second-- of x squared plus y

00:00:15.590 --> 00:00:25.420
squared dx plus 2xy dy-- and
this might look very familiar.

00:00:25.420 --> 00:00:28.000
This was very similar to what
we saw last time, except last

00:00:28.000 --> 00:00:29.520
time we had a closed
line integral.

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This is not a closed
line integral.

00:00:31.550 --> 00:00:37.560
And our curve, c, the
parameterization is x is

00:00:37.560 --> 00:00:43.080
equal to cosine of t, y
is equal to sine of t.

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So far-- it looks like sit.

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Let me write sine of t-- so
far, it looks very similar to

00:00:49.020 --> 00:00:51.800
the closed line integral
example we did in the last

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video, but instead of t going
from 0 to 2 pi, we're going

00:00:55.120 --> 00:00:58.570
to have t go from 0 to pi.

00:00:58.570 --> 00:01:01.180
t is greater than or equal to
0, is less than or equal to pi.

00:01:01.180 --> 00:01:04.650
So now we're essentially, our
path-- if I were to draw it on

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the x-y plane-- so that is my
y-axis, that is my x-axis.

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So now our path isn't all the
way around the unit circle.

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Our path-- our curve c now--
just starts at t is equal to 0.

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You can imagine t is
almost the angle.

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t is equal to 0, and we're
going to go all the way to pi.

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So that's what our path is
right now, in this example.

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So it's not a curved path.

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It's not a closed path.

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So we can't just show that f,
in this example-- and we're

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going to re-look at what f
looks like-- hey, if that's a

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conservative vector field, if
it's a closed loop that equals

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0, this isn't a closed loop.

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So we can't apply that.

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But let's see if we can apply
some of our other tools.

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So like we saw in the last
video, this might look a

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little bit foreign to you.

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But if you say that f is equal
to that times i, plus that

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times j, then it might look
a little bit more familiar.

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If we say that f of xy-- the
vector field f is equal to x

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squared plus y squared times i
plus 2xy times j and dr-- I

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don't even have to look at this
right now. dr, you can always

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write it as dx times
i plus dy times j.

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You'll immediately see, if you
take the dot product of these 2

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things, if you take f dot dr--
they're both vector valued,

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vector valued differential,
vector valued field, or vector

00:02:35.150 --> 00:02:38.760
valued function-- if you take f
dot dr, you'll get

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this right here.

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You'll get what we have
inside of the interval.

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You'll get that
right there, right?

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That times that-- you take
the product of the i terms--

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that times that is equal to
that, and add it to the

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product of the j terms.

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2xy times dy.

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Write like that.

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So our integral, we can
rewrite it as this.

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Along this curve of f dot
dr, where this is our f.

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Now, we still might want
to ask ourselves, is this

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a conservative field?

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Or does it have a potential?

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Is f equal to the gradient of
some function, capital F?

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I guess I could write the
gradient like that, because

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it creates a vector.

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This is a vector, too.

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Is this true?

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And we saw in the
last video, it is.

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I'll redo it a little
bit fast this time.

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Because if this is true, we
can't say this is a closed

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loop and say, oh, it's just
going to be equal to 0.

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But if this is true, then we
know that this-- that the

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integral is path independent.

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And we'll know that this is
going to be equal to capital F,

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if we say that t is going
from-- well, in this case t is

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going from 0 to pi-- we could
say that this is going to be

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equal to capital F of pi
minus capital F of 0.

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Or if we want to write it in
terms of x and y-- because f is

00:04:05.910 --> 00:04:09.150
going to be a function of x and
y-- we could write-- and this

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right here, these are t's.

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We could also write that this
is equal to f of x of pi, y of

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pi, minus f of x of 0, y of 0.

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That's what I mean
when I say f of pi.

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If we were to write f
purely as a function of t.

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But we know that this capital
F is going to be a function.

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It's a scalar function
defined on xy.

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So we could say f of
x of pi, y of pi.

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These are the t's now.

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These are all
equivalent things.

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So if it is path dependent,
we can find our f.

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We can just evaluate this
thing by just taking our

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f, evaluating it at
these two points.

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At this point, and at
that point right there.

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Because it would be
path independent.

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If this is a conservative, if
this has a potential function,

00:05:04.130 --> 00:05:08.090
if this is the gradient of
another scalar field, then this

00:05:08.090 --> 00:05:11.790
is a conservative vector field,
and its line integral

00:05:11.790 --> 00:05:12.750
is path independent.

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It's only dependent on that
point and that point.

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So let's see if we
can find our f.

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So I'm going to do exactly what
we did in the last video.

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If you watch that last
video, it might be a

00:05:22.400 --> 00:05:23.090
little bit monotonous.

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But I'll do it a little
bit faster here.

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So we know that the partial of
f with respect to x is going to

00:05:28.430 --> 00:05:31.310
have to be equal to
this right here.

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So that's x squared
plus y squared.

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Which tells us, if we take the
antiderivative, with respect to

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x, then f of xy is going to
have to be equal to x to the

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third over 3 plus xy squared--
right? y squared is just a

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constant in terms of
x-- plus f of y.

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There might be some function
of y that, when you take

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the partial with respect
to x, it just disappears.

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And then we know that the
partial of f with respect to

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y has got to be equal to
that thing or that thing.

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We're saying that this
is the gradient of f.

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So this has to be the
partial with respect to y.

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2xy.

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And you might want to
watch the other video.

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I go through this just a little
bit slower in that one.

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So the antiderivative of this
with respect to y-- so we get f

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of xy-- would be equal to xy
squared plus some

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function of x.

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Now we did this in
the last video.

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These 2 things have to be the
same thing, in order for the

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gradient of capital F
to be lowercase f.

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And we have xy
squared, xy squared.

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We have a function of x, we
have a function purely of x.

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And then we don't have a
function purely of y here,

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so this thing right
here must be 0.

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So we've solved.

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Our capital F of xy must
be equal to x to the 3

00:07:02.250 --> 00:07:05.660
over 3 plus xy squared.

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So we know that lowercase f
is definitely conservative.

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It is path independent.

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It has its potential.

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It is the gradient of
this thing right here.

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And so to solve our integral--
this was a 0-- to solve our

00:07:20.970 --> 00:07:23.470
integral, we just have to
figure out x of pi, y

00:07:23.470 --> 00:07:25.970
of pi, x of 0, y of 0.

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Evaluate the bullet points,
and then subtract the 2.

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So let's do that.

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So x was cosine of
t, y is sine of t.

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Let me rewrite it down here.

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So x is equal to cosine of t.

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y is equal to sine of t.

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So x of 0 is equal to cosine
of 0, which is equal to 1.

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x of pi is equal to cosine of
pi, which is equal to minus 1.

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y of 0 is sine of
0, which is 0.

00:08:01.960 --> 00:08:07.870
y of pi, which is equal to sine
of pi, which is equal to 0.

00:08:07.870 --> 00:08:12.140
So f of x of pi, y of pi--
this is the same thing,

00:08:12.140 --> 00:08:12.970
so let me rewrite this.

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Our integral is simplified to--
our integral along that path of

00:08:17.560 --> 00:08:26.130
f dot dr-- is going to be equal
to capital F of x of pi.

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x of pi is minus 1.

00:08:29.620 --> 00:08:34.730
y of pi is equal to 0.

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Minus capital F of x of 0
is 1, comma y of 0 is 0.

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And so what is this equal to?

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Just remember, this right
here is the same thing

00:08:53.160 --> 00:08:53.810
is that right there.

00:08:53.810 --> 00:08:56.150
That is x of pi.

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That is y of pi.

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That term right there.

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You can imagine this whole f of
minus 1, 0-- that's the same

00:09:04.250 --> 00:09:07.970
thing as f of pi, if you
think in terms of just t.

00:09:07.970 --> 00:09:09.420
That could be a little
confusing, so I want

00:09:09.420 --> 00:09:10.410
to make that clear.

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So this is just
straightforward to evaluate.

00:09:12.440 --> 00:09:14.940
What is f of minus 1, 0?

00:09:14.940 --> 00:09:16.780
x is minus 1. y is 0.

00:09:16.780 --> 00:09:20.410
So it's going to be minus 1
to the third power-- right,

00:09:20.410 --> 00:09:22.700
that's our x-- over 3.

00:09:22.700 --> 00:09:23.960
So it's minus 1/3.

00:09:23.960 --> 00:09:29.700
It's going to be minus 1/3
plus minus 1 times 0 squared.

00:09:29.700 --> 00:09:31.360
So that's just going to be a 0.

00:09:31.360 --> 00:09:32.830
In both cases, the y is 0.

00:09:32.830 --> 00:09:34.540
So this term is
going to disappear.

00:09:34.540 --> 00:09:35.950
So we can ignore that.

00:09:35.950 --> 00:09:42.000
And then we have minus
f of 1, comma 0.

00:09:42.000 --> 00:09:42.760
We put a 1 here.

00:09:42.760 --> 00:09:43.730
1 to the third over 3.

00:09:43.730 --> 00:09:48.640
That is 1/3 plus 1
times 0 squared.

00:09:48.640 --> 00:09:49.430
That's just 0.

00:09:49.430 --> 00:09:51.640
So this is going to be
equal to minus 1/3.

00:09:51.640 --> 00:09:57.990
Minus 1/3 is equal
to minus 2/3.

00:09:57.990 --> 00:09:59.070
And we're done.

00:09:59.070 --> 00:10:02.220
And once again, because this is
a conservative vector field,

00:10:02.220 --> 00:10:04.620
and it's path independent, we
really didn't have to mess with

00:10:04.620 --> 00:10:07.180
the cosine of t's and sines of
t's when we actually took

00:10:07.180 --> 00:10:08.130
our antiderivative.

00:10:08.130 --> 00:10:11.425
We just have to find the
potential function and evaluate

00:10:11.425 --> 00:10:14.590
it at the 2 end points to get
the answer of our integral, of

00:10:14.590 --> 00:10:16.170
our line integral, minus 2/3.


Example of closed line integral of conservative field

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:03.860
Let's see if we can apply some
of our new tools to solve

00:00:03.860 --> 00:00:04.880
some line integrals.

00:00:04.880 --> 00:00:07.990
So let's say we have a line
integral along a closed curve

00:00:07.990 --> 00:00:12.690
-- I'm going to define the path
in a second -- of x squared

00:00:12.690 --> 00:00:24.240
plus y squared times
dx plus 2xy times dy.

00:00:24.240 --> 00:00:29.230
And then our curve c is
going to be defined by

00:00:29.230 --> 00:00:31.726
the parameterization.

00:00:31.726 --> 00:00:39.820
x is equal to cosine of t,
and y is equal to sine of t.

00:00:39.820 --> 00:00:46.940
And this is valid for
t between 0 and 2 pi.

00:00:46.940 --> 00:00:50.110
So this is essentially a
circle, a unit circle,

00:00:50.110 --> 00:00:52.440
in the xy plane, and we
know how to solve these.

00:00:52.440 --> 00:00:54.610
Let's see if we can use some of
our discoveries in the last

00:00:54.610 --> 00:00:59.180
couple of videos to maybe
simplify this process.

00:00:59.180 --> 00:01:02.310
So the first thing you might
say, hey, this looks like a

00:01:02.310 --> 00:01:04.850
line integral, but you have a
dx and dy, I don't

00:01:04.850 --> 00:01:06.850
see a dot dr here.

00:01:06.850 --> 00:01:08.850
It's not clear to me that
this is some type of even

00:01:08.850 --> 00:01:09.870
a vector line integral.

00:01:09.870 --> 00:01:11.400
I don't see any of vectors.

00:01:11.400 --> 00:01:13.370
What I want to do first, and
the reason why I wanted to show

00:01:13.370 --> 00:01:15.150
you this example, is just to
show you that this is just

00:01:15.150 --> 00:01:18.980
another form of writing really
a vector line integral.

00:01:18.980 --> 00:01:24.160
To show you that you just have
to realize if I have some are

00:01:24.160 --> 00:01:27.900
r of t -- this is our curve.

00:01:27.900 --> 00:01:29.476
I don't even write these
functions in there.

00:01:29.476 --> 00:01:33.660
I'm just going to write
it's x of t times i

00:01:33.660 --> 00:01:37.660
plus y of t times j.

00:01:37.660 --> 00:01:43.160
We've seen several videos now
that we can write dr dt as

00:01:43.160 --> 00:01:50.910
being equal to dx dt times
i plus dy dt times j.

00:01:50.910 --> 00:01:52.960
We've seen this multiple times.

00:01:52.960 --> 00:01:55.600
And we've seen multiple times
we want to get the differential

00:01:55.600 --> 00:01:59.210
dr, we can just multiply
everything times dt.

00:01:59.210 --> 00:02:01.560
And normally I just put a
dt here and a dt there

00:02:01.560 --> 00:02:02.440
and get rid of this dt.

00:02:02.440 --> 00:02:05.940
But if you multiply everything
times dt, if you view the

00:02:05.940 --> 00:02:08.740
differentials as actual
numbers, you can multiply and

00:02:08.740 --> 00:02:10.840
normally you can treat
them like that.

00:02:10.840 --> 00:02:12.780
Then you just get rid
of all of the dt's.

00:02:12.780 --> 00:02:17.020
So dr you could imagine is
equal to dx times the unit

00:02:17.020 --> 00:02:21.200
vector i plus dy times
the unit vector j.

00:02:21.200 --> 00:02:23.020
So put that aside, and
you might already

00:02:23.020 --> 00:02:24.160
see a pattern here.

00:02:24.160 --> 00:02:31.650
So if we define our vector
field f, f of xy, as being

00:02:31.650 --> 00:02:43.500
equal to x squared plus y
squared, i plus 2xy j,

00:02:43.500 --> 00:02:44.720
what is this thing?

00:02:44.720 --> 00:02:46.600
What is this thing over here?

00:02:46.600 --> 00:02:52.920
Well, what is f dot
dr going to be?

00:02:52.920 --> 00:02:57.090
Dot products, you just multiply
the corresponding components

00:02:57.090 --> 00:02:58.710
of our vectors and
then add them up.

00:02:58.710 --> 00:03:03.620
So it's going to be if you take
this f and dot it with that dr

00:03:03.620 --> 00:03:08.790
you're going to get the i
component, x squared plus y

00:03:08.790 --> 00:03:16.480
squared times that dx plus --
I'll do it in the pink again --

00:03:16.480 --> 00:03:22.150
plus the y component, the j
component 2xy times that dy.

00:03:22.150 --> 00:03:24.590
That's the dot product.

00:03:24.590 --> 00:03:30.760
And notice, this thing
right here is identical to

00:03:30.760 --> 00:03:32.270
that thing right there.

00:03:32.270 --> 00:03:35.310
So our line integral, just to
put it in a form that we're

00:03:35.310 --> 00:03:40.880
familiar with, this is the same
exact thing as the line

00:03:40.880 --> 00:03:48.483
integral over this curve c,
this closed curve c, of this f

00:03:48.483 --> 00:03:51.820
-- maybe I'll write it in that
magenta color, or actually it's

00:03:51.820 --> 00:03:57.990
more of a purple or pink
color -- f dot this dr.

00:03:57.990 --> 00:04:00.050
That's what this line integral
is, it's just a different

00:04:00.050 --> 00:04:00.840
way of writing it.

00:04:00.840 --> 00:04:03.330
Now that you see it, in the
future if you see in kind of

00:04:03.330 --> 00:04:06.850
this differential form, you'll
immediately know OK, there's

00:04:06.850 --> 00:04:11.430
one vector field that this is
its x component, this is its y

00:04:11.430 --> 00:04:16.860
component, dotting with the dr.
This is the x component of dr

00:04:16.860 --> 00:04:19.810
or the i component, and this is
the y component or the j

00:04:19.810 --> 00:04:22.570
component of the dr. So you
immediately know what the

00:04:22.570 --> 00:04:25.010
vector field is that we're
taking a line integral of.

00:04:25.010 --> 00:04:27.210
This is the x, that's the y.

00:04:27.210 --> 00:04:29.850
Now, let's ask
ourselves a question.

00:04:29.850 --> 00:04:33.070
Is f conservative?

00:04:33.070 --> 00:04:40.660
So is f equal to the gradient
of some scalar field, we'll

00:04:40.660 --> 00:04:43.480
call it capital F --
is this the case?

00:04:43.480 --> 00:04:48.860
So let's assume it is and see
if we can solve for a scalar

00:04:48.860 --> 00:04:51.010
field whose grade
it really is f.

00:04:51.010 --> 00:04:52.770
Then we know that f
is conservative.

00:04:52.770 --> 00:04:55.470
And then if f is conservative,
and this is the whole reason we

00:04:55.470 --> 00:04:59.380
want to do it, that means that
any closed loop, any line

00:04:59.380 --> 00:05:03.090
integral over a closed curve of
f is going to be equal

00:05:03.090 --> 00:05:04.560
to 0 and we'd be done.

00:05:04.560 --> 00:05:08.060
So if we can show this then the
answer to this question or this

00:05:08.060 --> 00:05:09.480
question is going to be 0.

00:05:09.480 --> 00:05:12.180
We don't even have to mess with
the cosine of t's and the

00:05:12.180 --> 00:05:13.440
sign of t's and all that.

00:05:13.440 --> 00:05:15.620
Actually, we don't even have
to take antiderivatives.

00:05:15.620 --> 00:05:18.380
So let's see if we can find
an f whose gradient is

00:05:18.380 --> 00:05:19.740
equal to that right there.

00:05:24.120 --> 00:05:29.200
So in order for f's gradient to
be that, that means that the

00:05:29.200 --> 00:05:34.410
partial derivative of our
capital F with respect to x has

00:05:34.410 --> 00:05:37.140
got to be equal to
that right there.

00:05:37.140 --> 00:05:42.820
It's got to be equal to x
squared plus y squared.

00:05:42.820 --> 00:05:45.410
And it also tells us that the
partial derivative of capital

00:05:45.410 --> 00:05:52.175
F with respect to y has
got to be equal to 2xy.

00:05:55.310 --> 00:06:00.020
And just as a review, if I have
the gradient of any function,

00:06:00.020 --> 00:06:03.215
of any scalar field is equal to
the partial of f with respect

00:06:03.215 --> 00:06:07.460
to x times i plus the
partial of capital F with

00:06:07.460 --> 00:06:09.850
respect to y times j.

00:06:09.850 --> 00:06:11.350
So that's why I'm just
pattern matching.

00:06:11.350 --> 00:06:15.060
I'm just saying well, gee, if
this is the gradient of that,

00:06:15.060 --> 00:06:21.780
then this must be that, which I
wrote down right here, and this

00:06:21.780 --> 00:06:25.090
must be that, which
I wrote down here.

00:06:25.090 --> 00:06:28.310
So let's see if I can find
an f that satisfies both

00:06:28.310 --> 00:06:30.390
of these constraints.

00:06:30.390 --> 00:06:33.255
So we could just take the
antiderivative with respect to

00:06:33.255 --> 00:06:39.230
x on both sides -- remember,
you just treat y like a

00:06:39.230 --> 00:06:42.440
constant or y squared like a
constant -- it's just a number.

00:06:42.440 --> 00:06:46.220
So then we could say that f is
equal to the antiderivative of

00:06:46.220 --> 00:06:50.840
x squared is x to
the third over 3.

00:06:53.780 --> 00:06:56.335
And then the antiderivative
of y squared -- remember,

00:06:56.335 --> 00:06:57.980
this is with respect to x.

00:06:57.980 --> 00:06:59.000
So you just treat
it like a number.

00:06:59.000 --> 00:07:00.930
That could just be the
number k, or this

00:07:00.930 --> 00:07:01.890
could be the number 5.

00:07:01.890 --> 00:07:03.820
So this is just going
to be that times x.

00:07:03.820 --> 00:07:07.640
So plus x times y squared.

00:07:07.640 --> 00:07:10.500
And then there could be
some function of y here.

00:07:10.500 --> 00:07:16.050
So plus some, I don't know,
I'll call it g of y.

00:07:16.050 --> 00:07:18.210
Because there could have been
some function of y here.

00:07:18.210 --> 00:07:20.700
If it's a pure function of y,
when you take the derivative or

00:07:20.700 --> 00:07:23.270
the partial with respect to x,
this would have disappeared.

00:07:23.270 --> 00:07:26.790
So it would reappear when we
take the antiderivative.

00:07:26.790 --> 00:07:35.150
And just to be clear, let me
make it clear that f is going

00:07:35.150 --> 00:07:37.280
to be a function of x and y.

00:07:37.280 --> 00:07:39.415
So we just have the, I
guess you could say

00:07:39.415 --> 00:07:40.830
the antiderivative
with respect to x.

00:07:40.830 --> 00:07:42.110
Let's see if we take the
antiderivative with respect

00:07:42.110 --> 00:07:44.930
to y and then we can
reconcile the two.

00:07:44.930 --> 00:07:51.260
So based on this, f of xy, f of
xy is going to have to look

00:07:51.260 --> 00:07:53.280
like -- so let's take the
antiderivative with

00:07:53.280 --> 00:07:55.035
respect to y here.

00:07:55.035 --> 00:07:57.540
So remember, you just treat x
like it's just some number --

00:07:57.540 --> 00:08:00.210
it could be a k, it could
be an m, it could be a 5.

00:08:00.210 --> 00:08:01.610
It's just some number.

00:08:01.610 --> 00:08:03.260
So if x is just some
-- the antiderivative

00:08:03.260 --> 00:08:04.960
of 2y is y squared.

00:08:04.960 --> 00:08:07.230
And if x is just a number
there, the antiderivative of

00:08:07.230 --> 00:08:11.820
this with respect to y is
just going to be xy squared.

00:08:11.820 --> 00:08:12.590
Don't believe me?

00:08:12.590 --> 00:08:14.720
Take the partial of this
with respect to y.

00:08:14.720 --> 00:08:18.590
Treat x like a constant
you'll get 2 times xy

00:08:18.590 --> 00:08:20.150
with no exponent there.

00:08:20.150 --> 00:08:22.660
And, of course, if you took the
antiderivative with respect to

00:08:22.660 --> 00:08:26.580
x, there might be some
function of x here.

00:08:26.580 --> 00:08:29.130
We were just basing it
off of that information.

00:08:29.130 --> 00:08:32.850
Now given that, this
information says f of xy

00:08:32.850 --> 00:08:34.480
is going to have to look
something like this.

00:08:34.480 --> 00:08:36.940
This information tells us f
of xy's going to have to

00:08:36.940 --> 00:08:38.320
look something like that.

00:08:38.320 --> 00:08:40.430
Let's see if there is an
f of xy that looks like

00:08:40.430 --> 00:08:42.470
both of them essentially.

00:08:42.470 --> 00:08:44.780
So let's see.

00:08:44.780 --> 00:08:47.740
On this one we have xy
squared here, we have

00:08:47.740 --> 00:08:48.990
an xy squared there.

00:08:48.990 --> 00:08:49.940
So good.

00:08:49.940 --> 00:08:51.110
That looks good.

00:08:51.110 --> 00:08:53.770
And it over here we have an f
of x -- we have something

00:08:53.770 --> 00:08:56.450
that's a pure function of x.

00:08:56.450 --> 00:08:59.550
And here we have something
that is a pure function of x.

00:08:59.550 --> 00:09:01.990
So these two things could
be the same thing.

00:09:01.990 --> 00:09:05.170
Then here we have a pure
function of y that might be

00:09:05.170 --> 00:09:08.630
there, but it didn't really
show up anywhere over here.

00:09:08.630 --> 00:09:12.610
So we could just say hey,
that's going to be 0.

00:09:12.610 --> 00:09:14.460
0 is a pure function of y.

00:09:14.460 --> 00:09:16.870
You could have something
called g of y is equal to 0.

00:09:16.870 --> 00:09:24.230
And then we get that capital F
of xy is equal to x to the

00:09:24.230 --> 00:09:30.020
third over 3 plus xy squared.

00:09:30.020 --> 00:09:33.980
And the gradient of this is
going to be equal to f.

00:09:33.980 --> 00:09:35.230
And we've already
established that.

00:09:35.230 --> 00:09:37.810
But just to hit the point home,
let's take the gradient of it.

00:09:37.810 --> 00:09:40.540
Just if you don't believe this
little stuff that I did right

00:09:40.540 --> 00:09:41.790
there, let's take the gradient.

00:09:41.790 --> 00:09:46.280
The gradient of f is equal to,
and sometimes people put a

00:09:46.280 --> 00:09:47.450
little vector there
because you're getting

00:09:47.450 --> 00:09:48.460
a vector out of it.

00:09:48.460 --> 00:09:51.290
You could put a little vector
on top of that gradient sign.

00:09:51.290 --> 00:09:53.760
The gradient of f is
going to be what?

00:09:53.760 --> 00:09:57.625
The partial of this with
respect to x times i.

00:09:57.625 --> 00:09:59.320
So the partial of this
with respect to x.

00:09:59.320 --> 00:10:02.500
The derivative here is
3 divided by 3 is 1.

00:10:02.500 --> 00:10:06.590
So it's just x squared plus the
derivative of this with respect

00:10:06.590 --> 00:10:15.370
to x is y squared times i plus
the partial with respect to y.

00:10:15.370 --> 00:10:17.986
Well, the partial with respect
to y of this 0, partial with

00:10:17.986 --> 00:10:22.340
respect to y of this is
2xy or 2xy to the first.

00:10:22.340 --> 00:10:25.890
So it's 2xy times j.

00:10:25.890 --> 00:10:31.180
And this is exactly equal to f,
our f that we wrote up there.

00:10:31.180 --> 00:10:36.470
So we've established that f can
definitely be written -- f is

00:10:36.470 --> 00:10:39.790
definitely the gradient of some
potential scalar

00:10:39.790 --> 00:10:40.750
function there.

00:10:40.750 --> 00:10:49.390
So f is conservative, and that
tells us that this closed loop

00:10:49.390 --> 00:10:54.530
integral, line integral, of f
is going to be equal to 0.

00:10:54.530 --> 00:10:56.220
And we are done.

00:10:56.220 --> 00:11:00.890
We could even ignore the actual
parameterization of the path.


Closed curve line integrals of conservative vector fields

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:04.130
In the last video, we saw that
if a vector field can be

00:00:04.130 --> 00:00:12.580
written as the gradient of a
scalar field-- or another way

00:00:12.580 --> 00:00:17.080
we could say it: this would be
equal to the partial of our big

00:00:17.080 --> 00:00:23.060
f with respect to x times i
plus the partial of big f, our

00:00:23.060 --> 00:00:28.100
scalar field with respect to y
times j; and I'm just writing

00:00:28.100 --> 00:00:30.690
it in multiple ways just so you
remember what the gradient is

00:00:30.690 --> 00:00:35.490
--but we saw that if our vector
field is the gradient of

00:00:35.490 --> 00:00:38.780
a scalar field then we
call it conservative.

00:00:38.780 --> 00:00:47.825
So that tells us that f is a
conservative vector field.

00:00:50.510 --> 00:00:54.140
And it also tells us, and this
was the big take away from the

00:00:54.140 --> 00:00:58.400
last video, that the line
integral of f between two

00:00:58.400 --> 00:01:05.900
points-- let me draw two points
here; so let me draw my

00:01:05.900 --> 00:01:09.210
coordinates just so we know
we're on the xy plane.

00:01:09.210 --> 00:01:12.780
My axes: x-axis, y-axis.

00:01:12.780 --> 00:01:16.820
Let's say I have the point, I
have that point and that point,

00:01:16.820 --> 00:01:19.620
and I have two different paths
between those two points.

00:01:19.620 --> 00:01:24.190
So I have path 1, that goes
something like that, so

00:01:24.190 --> 00:01:27.005
I'll call that c1 and it
goes in that direction.

00:01:30.270 --> 00:01:32.380
And then I have, maybe in
a different shades of

00:01:32.380 --> 00:01:38.010
green, c2 goes like that.

00:01:38.010 --> 00:01:41.010
They both start here
and go to there.

00:01:41.010 --> 00:01:43.520
We learned in the last video
that the line integral

00:01:43.520 --> 00:01:48.330
is path independent
between any two points.

00:01:48.330 --> 00:01:57.930
So in this case the line
integral along c1 of f dot dr

00:01:57.930 --> 00:02:03.560
is going to be equal to the
line integral of c2, over the

00:02:03.560 --> 00:02:10.870
path c2, of f dot dr. The line,
if we have a potential in

00:02:10.870 --> 00:02:14.500
a region, and we may be
everywhere, then the line

00:02:14.500 --> 00:02:17.590
integral between any two points
is independent of the path.

00:02:17.590 --> 00:02:19.390
That's the neat thing about
a conservative field.

00:02:19.390 --> 00:02:21.190
Now what I want to do in this
video is do a little bit of

00:02:21.190 --> 00:02:23.680
an extension of the take
away of the last video.

00:02:23.680 --> 00:02:25.710
It's actually a pretty
important extension; it might

00:02:25.710 --> 00:02:27.540
already be obvious to you.

00:02:27.540 --> 00:02:29.550
I've already written this
here; I could rearrange

00:02:29.550 --> 00:02:31.410
this equation a little bit.

00:02:31.410 --> 00:02:32.745
So let me do it.

00:02:32.745 --> 00:02:35.310
So let me a rearrange this.

00:02:35.310 --> 00:02:36.976
I'll just rewrite
this in orange.

00:02:36.976 --> 00:02:42.730
So the line integral on path c1
dot dr minus-- I'll just go

00:02:42.730 --> 00:02:48.670
subtract this from both sides
--minus the line integral c2 of

00:02:48.670 --> 00:02:53.110
f dot dr is going
to be equal to 0.

00:02:53.110 --> 00:02:55.610
All I did is I took this take
away from the last video and

00:02:55.610 --> 00:02:58.430
I subtracted this
from both sides.

00:02:58.430 --> 00:03:04.370
Now we learned several videos
ago that if we're dealing with

00:03:04.370 --> 00:03:08.560
a line integral of a vector
field-- not a scalar field

00:03:08.560 --> 00:03:10.690
--with a vector field,
the direction of the

00:03:10.690 --> 00:03:12.090
path is important.

00:03:12.090 --> 00:03:20.370
We learned that the line
integral over, say, c2 of f dot

00:03:20.370 --> 00:03:26.580
dr, is equal to the negative of
the line integral of minus c2

00:03:26.580 --> 00:03:32.830
of f dot dr where we denoted
minus c2 is the same path as

00:03:32.830 --> 00:03:35.740
c2, but just in the
opposite direction.

00:03:35.740 --> 00:03:39.370
So for example, minus c2 I
would write like this-- so let

00:03:39.370 --> 00:03:42.440
me do it in a different color
--so let's say this is minus

00:03:42.440 --> 00:03:46.740
c2, it'd be a path just like
c2-- I'm going to call this

00:03:46.740 --> 00:03:49.100
minus c2 --but instead of going
in that direction, I'm now

00:03:49.100 --> 00:03:51.380
going to go in that direction.

00:03:51.380 --> 00:03:52.980
So ignore the old c2 arrows.

00:03:52.980 --> 00:03:56.030
We're now starting from
there and coming back here.

00:03:56.030 --> 00:03:58.100
So this is minus c2.

00:03:58.100 --> 00:04:00.220
Or we could write, we could
put, the minus on the other

00:04:00.220 --> 00:04:06.250
side and we could say that
the negative of the c2 line

00:04:06.250 --> 00:04:13.770
integral along the path of c2
of f dot dr is equal to the

00:04:13.770 --> 00:04:19.510
line integral over the reverse
path of f dot dr. All I did is

00:04:19.510 --> 00:04:21.630
I switched the negative on
the other side; multiplied

00:04:21.630 --> 00:04:23.370
both sides by negative 1.

00:04:23.370 --> 00:04:27.900
So let's replace-- in this
equation we have the minus of

00:04:27.900 --> 00:04:31.480
the c2 path; we have that right
there, and we have that right

00:04:31.480 --> 00:04:33.880
there --so we could just
replace this with

00:04:33.880 --> 00:04:34.800
this right there.

00:04:34.800 --> 00:04:35.670
So let me do that.

00:04:35.670 --> 00:04:37.670
So I'll write this
first part first.

00:04:37.670 --> 00:04:43.410
So the integral along the curve
c1 of f dot dr, instead of

00:04:43.410 --> 00:04:49.200
minus the line integral along
c2, I'm going to say plus the

00:04:49.200 --> 00:04:51.200
integral along minus c2.

00:04:51.200 --> 00:04:55.940
This-- let me switch to the
green --this we've established

00:04:55.940 --> 00:04:57.480
is the same thing as this.

00:04:57.480 --> 00:05:00.720
The negative of this curve, or
the line integral along this

00:05:00.720 --> 00:05:04.950
path, is the same thing as the
line integral, the positive of

00:05:04.950 --> 00:05:07.350
the line integral along
the reverse path.

00:05:07.350 --> 00:05:13.610
So we'll say plus the line
integral of minus c2 of

00:05:13.610 --> 00:05:19.270
f dot dr is equal to 0.

00:05:19.270 --> 00:05:20.670
Now there's something
interesting.

00:05:20.670 --> 00:05:23.550
Let's look at what the
combination of the path

00:05:23.550 --> 00:05:26.560
of c1 and minus c2 is.

00:05:26.560 --> 00:05:28.370
c1 starts over here.

00:05:28.370 --> 00:05:30.280
Let me get a nice,
vibrant color.

00:05:30.280 --> 00:05:32.580
c1 starts over here
at this point.

00:05:32.580 --> 00:05:36.540
It moves from this point
along this curve c1 and

00:05:36.540 --> 00:05:38.020
ends up at this point.

00:05:38.020 --> 00:05:39.840
And then we do the minus c2.

00:05:39.840 --> 00:05:43.590
Minus c2 starts at this point
and just goes and comes back

00:05:43.590 --> 00:05:45.810
to the original point;
it completes a loop.

00:05:45.810 --> 00:05:48.270
So this is a closed
line integral.

00:05:48.270 --> 00:05:52.440
So if you combine this,
we could rewrite this.

00:05:52.440 --> 00:05:53.660
Remember, this is just a loop.

00:05:53.660 --> 00:05:56.365
By reversing this, instead of
having two guys starting here

00:05:56.365 --> 00:05:58.450
and going there, I now can
start here, go all the way

00:05:58.450 --> 00:06:00.660
there, and then come all
the way back on this

00:06:00.660 --> 00:06:02.630
reverse path of c2.

00:06:02.630 --> 00:06:06.880
So this is equivalent to
a closed line integral.

00:06:06.880 --> 00:06:12.150
So that is the same thing as an
integral along a closed path.

00:06:12.150 --> 00:06:15.730
I mean, we could call the
closed path, maybe, c1 plus

00:06:15.730 --> 00:06:18.200
minus c2, if we wanted to be
particular about

00:06:18.200 --> 00:06:18.900
the closed path.

00:06:18.900 --> 00:06:23.390
But this could be, I drew c1
and c2 or minus c2 arbitrarily;

00:06:23.390 --> 00:06:29.600
this could be any closed path
where our vector field f has a

00:06:29.600 --> 00:06:33.000
potential, or where it is the
gradient of a scalar field,

00:06:33.000 --> 00:06:34.960
or where it is conservative.

00:06:34.960 --> 00:06:38.620
And so this can be written as
a closed path of c1 plus the

00:06:38.620 --> 00:06:45.930
reverse of c2 of f dot dr.
That's just a rewriting

00:06:45.930 --> 00:06:49.040
of that, and so that's
going to be equal to 0.

00:06:49.040 --> 00:06:53.050
And this is our take
away for this video.

00:06:53.050 --> 00:06:56.110
This is, you can view
it as a corollary.

00:06:56.110 --> 00:06:59.160
It's kind of a low-hanging
conclusion that you can make

00:06:59.160 --> 00:07:01.610
after this conclusion.

00:07:01.610 --> 00:07:05.900
So now we know that if we have
a vector field that's the

00:07:05.900 --> 00:07:09.570
gradient of a scalar field in
some region, or maybe over the

00:07:09.570 --> 00:07:13.440
entire xy plane-- and this is
called the potential of f;

00:07:13.440 --> 00:07:15.320
this is a potential function.

00:07:15.320 --> 00:07:17.370
Oftentimes it will be the
negative of it, but it's easy

00:07:17.370 --> 00:07:21.670
to mess with negatives --but if
we have a vector field that is

00:07:21.670 --> 00:07:24.650
the gradient of a scalar field,
we call that vector

00:07:24.650 --> 00:07:26.100
field conservative.

00:07:26.100 --> 00:07:29.880
That tells us that at any point
in the region where this is

00:07:29.880 --> 00:07:33.860
valid, the line integral from
one point to another is

00:07:33.860 --> 00:07:36.150
independent of the path; that's
what we got from

00:07:36.150 --> 00:07:37.130
the last video.

00:07:37.130 --> 00:07:42.890
And because of that, a closed
loop line integral, or a closed

00:07:42.890 --> 00:07:45.570
line integral, so if we take
some other place, if we take

00:07:45.570 --> 00:07:52.915
any other closed line integral
or we take the line integral of

00:07:52.915 --> 00:07:57.490
the vector field on any closed
loop, it will become 0 because

00:07:57.490 --> 00:07:58.740
it is path independent.

00:07:58.740 --> 00:08:02.240
So that's the neat take away
here, that if you know that

00:08:02.240 --> 00:08:05.370
this is conservative, if you
ever see something like this:

00:08:05.370 --> 00:08:10.580
if you see this f dot dr and
someone asks you to evaluate

00:08:10.580 --> 00:08:13.840
this given that f is
conservative, or given that f

00:08:13.840 --> 00:08:16.710
is the gradient of another
function, or given that f is

00:08:16.710 --> 00:08:19.960
path independent, you can now
immediately say, that is going

00:08:19.960 --> 00:08:24.170
to be equal to 0, which
simplifies the math a good bit.


Path independence for line integrals

WEBVTT
Kind: captions
Language: en

00:00:00.460 --> 00:00:03.540
What I want to do in this video
is establish a reasonably

00:00:03.540 --> 00:00:07.540
powerful condition in which we
can establish that at vector

00:00:07.540 --> 00:00:10.350
field, or that a line integral
of a vector field is

00:00:10.350 --> 00:00:11.670
path independent.

00:00:11.670 --> 00:00:14.420
And when I say that, I mean
that let's say I were to take

00:00:14.420 --> 00:00:21.780
this line integral along the
path c of f dot d r, and let's

00:00:21.780 --> 00:00:24.310
say my path looks like this.

00:00:28.110 --> 00:00:34.020
That's my x and y axis, and
let's say my path looks

00:00:34.020 --> 00:00:37.530
something like this: I start
there and I go over

00:00:37.530 --> 00:00:40.510
there to point c.

00:00:40.510 --> 00:00:43.050
My end point, the
curve here is c.

00:00:43.050 --> 00:00:46.210
And so I would evaluate this
line integral, this victor

00:00:46.210 --> 00:00:48.640
field along this path.

00:00:48.640 --> 00:00:52.420
This would be a path
independent vector field, or we

00:00:52.420 --> 00:00:59.890
call that a conservative vector
field, if this thing is equal

00:00:59.890 --> 00:01:06.370
to the same integral over a
different path that has

00:01:06.370 --> 00:01:07.240
the same end point.

00:01:07.240 --> 00:01:11.250
So let's call this c1, so
this is c1, and this is c2.

00:01:11.250 --> 00:01:15.680
This vector field is
conservative if I start

00:01:15.680 --> 00:01:17.530
at the same point but I
take a different path.

00:01:17.530 --> 00:01:22.050
Let's say I go something like
that: if I take a different

00:01:22.050 --> 00:01:27.600
path-- and this is my c2 --I
still get the same value.

00:01:27.600 --> 00:01:30.190
What this is telling me is that
all it cares about to evaluate

00:01:30.190 --> 00:01:33.980
these integrals is my starting
point and my ending point.

00:01:33.980 --> 00:01:35.750
It doesn't care what
I do in between.

00:01:35.750 --> 00:01:38.410
It doesn't care how I get
from my starting point

00:01:38.410 --> 00:01:40.140
to my end point.

00:01:40.140 --> 00:01:42.610
These two integrals have the
same start point and same end

00:01:42.610 --> 00:01:45.320
point, so irregardless of
their actual path, they're

00:01:45.320 --> 00:01:46.350
going to be the same.

00:01:46.350 --> 00:01:50.530
That's what it means for f to
be a conservative field, or

00:01:50.530 --> 00:01:54.670
what it means for this integral
to be path independent.

00:01:54.670 --> 00:01:58.080
So before I prove or I show you
the conditions, let's build

00:01:58.080 --> 00:02:00.560
up our tool kit a little bit.

00:02:00.560 --> 00:02:04.750
And so you may or may not
have already seen the

00:02:04.750 --> 00:02:06.590
multivariable chain rule.

00:02:11.220 --> 00:02:14.850
And I'm not going to prove
it in this video, but I

00:02:14.850 --> 00:02:16.390
think it'll be pretty
intuitive for you.

00:02:16.390 --> 00:02:19.000
So maybe it doesn't need to
have a proof, or I'll prove it

00:02:19.000 --> 00:02:21.560
eventually, but I really just
want to give you the intuition.

00:02:21.560 --> 00:02:24.610
And all that says is that if I
have some function-- let's say

00:02:24.610 --> 00:02:29.860
I have f of x and y, but x and
y are then functions of, let's

00:02:29.860 --> 00:02:37.220
say a third variable, t, so f
of x of t and y of t --that the

00:02:37.220 --> 00:02:45.720
derivative of f with respect
to t is multivariable.

00:02:45.720 --> 00:02:48.000
I have two variables
here in x and y.

00:02:48.000 --> 00:02:50.810
This is going to be equal
to the partial of f

00:02:50.810 --> 00:02:52.350
with respect to x.

00:02:52.350 --> 00:02:58.130
How fast does f change as x
changes times the derivative of

00:02:58.130 --> 00:03:02.120
x with respect to t-- This is a
single variable function right

00:03:02.120 --> 00:03:04.780
here, so you to take a
regular derivative.

00:03:04.780 --> 00:03:09.140
So times how fast x changes
with respect to t.

00:03:09.140 --> 00:03:11.680
This is a standard derivative,
this is a partial derivative,

00:03:11.680 --> 00:03:14.290
because at that level we're
dealing with two variables.

00:03:14.290 --> 00:03:18.280
And we're not done. --plus how
fast f changes with respect to

00:03:18.280 --> 00:03:25.960
y times the derivative
y with respected t.

00:03:25.960 --> 00:03:29.110
So d y d t.

00:03:29.110 --> 00:03:31.490
And I'm not going to prove
it, but I think it makes

00:03:31.490 --> 00:03:32.730
pretty good intuition.

00:03:32.730 --> 00:03:35.950
This is saying, as I move
a little bit d t, how

00:03:35.950 --> 00:03:37.290
much of a d f do I get?

00:03:37.290 --> 00:03:40.200
Or how fast this f change
with respect to t?

00:03:40.200 --> 00:03:42.400
It says, well there's two ways
that f can change: it can

00:03:42.400 --> 00:03:45.670
change with respect to x and it
can change with respect to y.

00:03:45.670 --> 00:03:49.170
So why don't I add those two
things together as they're both

00:03:49.170 --> 00:03:51.020
changing with respect to t?

00:03:51.020 --> 00:03:53.940
That's all it's saying, and if
you kind of imagined that you

00:03:53.940 --> 00:03:57.090
could cancel out this partial x
with this d x, and this partial

00:03:57.090 --> 00:04:00.550
y with this d y, you could kind
of imagine the partial of f

00:04:00.550 --> 00:04:04.200
with respect to t on the x side
of things, and then plus the

00:04:04.200 --> 00:04:08.650
partial of f with respect
to t in the y dimension.

00:04:08.650 --> 00:04:10.930
And then that'll give you
the total change of f

00:04:10.930 --> 00:04:12.150
with respected to t.

00:04:12.150 --> 00:04:15.560
Kind of a hand-wavy argument
there, but at least to me this

00:04:15.560 --> 00:04:16.900
is a pretty intuitive formula.

00:04:16.900 --> 00:04:19.420
So that's our tool kit right
there; the multivariable

00:04:19.420 --> 00:04:20.450
chain rule.

00:04:20.450 --> 00:04:23.040
We're going to put that
aside for a second.

00:04:23.040 --> 00:04:29.090
Now let's say I have some
vector field f-- and it's

00:04:29.090 --> 00:04:31.100
different than this f, so I'll
do it in a different color;

00:04:31.100 --> 00:04:34.660
magenta --I have some
vector field f that is a

00:04:34.660 --> 00:04:36.920
function of x and y.

00:04:36.920 --> 00:04:40.380
And let's say that it happens
to be the gradient of

00:04:40.380 --> 00:04:41.705
some scalar field.

00:04:46.910 --> 00:04:49.440
I'll call that capital F.

00:04:49.440 --> 00:04:53.060
And this is gradient which
means that capital F is also

00:04:53.060 --> 00:04:59.230
function of x and y-- so I
don't want to write it on a new

00:04:59.230 --> 00:05:02.890
line, I could also write up
here; capital F is also a

00:05:02.890 --> 00:05:07.470
function of x and y --and the
gradient, and all that means is

00:05:07.470 --> 00:05:13.150
that the vector field f of xy--
lower-case f of xy, is equal to

00:05:13.150 --> 00:05:18.710
the partial derivative of
upper-case F with respect to x

00:05:18.710 --> 00:05:24.250
times the i-unit vector plus
the partial of upper-case F

00:05:24.250 --> 00:05:27.690
with respect to y times
the j-unit vector.

00:05:27.690 --> 00:05:32.170
This is the definition of
the gradient right here.

00:05:32.170 --> 00:05:35.480
And if you imagine that
upper-case F is some type of

00:05:35.480 --> 00:05:48.530
surface-- so this is uppercase
F of xy --the gradient F of xy

00:05:48.530 --> 00:05:51.390
is going to be a vector field
that tells you the direction of

00:05:51.390 --> 00:05:53.440
steepest ascent at any point.

00:05:53.440 --> 00:05:56.740
So it'll be defined
the xy plane.

00:05:56.740 --> 00:06:00.060
So on the xy plane it'll tell
you-- so let me draw; that's

00:06:00.060 --> 00:06:04.820
the vertical axis, maybe that's
the x axis, that's the y axis

00:06:04.820 --> 00:06:08.150
--so the gradient of it, if you
take any point on the xy plane,

00:06:08.150 --> 00:06:10.240
it'll tell you the direction
you need to travel to go

00:06:10.240 --> 00:06:11.610
into the deepest ascent.

00:06:11.610 --> 00:06:13.320
And for this gradient
field it's going to be

00:06:13.320 --> 00:06:14.700
something like this.

00:06:18.450 --> 00:06:20.730
And maybe over here it starts
going in that direction because

00:06:20.730 --> 00:06:23.800
you would descend towards
this little minimum

00:06:23.800 --> 00:06:24.260
point right here.

00:06:24.260 --> 00:06:25.770
Anyway, I don't want
to get too involved.

00:06:25.770 --> 00:06:28.960
And the whole point of this
isn't to really get the

00:06:28.960 --> 00:06:32.050
intuition behind gradients;
there are other videos on this.

00:06:32.050 --> 00:06:35.960
The point of this is to get
other a test to see whether

00:06:35.960 --> 00:06:39.290
something is path independent;
whether a vector field is path

00:06:39.290 --> 00:06:41.050
independent, whether
it's conservative.

00:06:41.050 --> 00:06:45.010
And it turns out that if this
exists-- and I'm going to prove

00:06:45.010 --> 00:06:57.350
it now --if f is the gradient
of some scalar field,

00:06:57.350 --> 00:07:01.630
then f is conservative.

00:07:04.570 --> 00:07:07.900
Or you could say it doesn't
matter what path we follow when

00:07:07.900 --> 00:07:11.670
we take a line integral over f,
it just matters about our

00:07:11.670 --> 00:07:14.950
starting point and
our ending point.

00:07:14.950 --> 00:07:18.030
Now let me see if I can
prove that to you.

00:07:18.030 --> 00:07:20.820
So let's start with the
assumption that f can be

00:07:20.820 --> 00:07:24.560
written this way, as the
gradient, that lower-case f can

00:07:24.560 --> 00:07:28.120
be written as the gradient
of some upper-case F.

00:07:28.120 --> 00:07:36.490
So in that case our integral--
well, let's define

00:07:36.490 --> 00:07:37.510
our path first.

00:07:37.510 --> 00:07:41.310
So our position vector
function-- we always need one

00:07:41.310 --> 00:07:45.250
of those to do a line integral
or a vector line integral --r

00:07:45.250 --> 00:07:52.890
of t is going to be equal to x
of t times i plus y of t times

00:07:52.890 --> 00:07:56.770
j 4t going between a and b.

00:07:56.770 --> 00:07:59.730
You've seen this multiple
times; this is a definition

00:07:59.730 --> 00:08:04.950
of pretty much any path
in two dimensions.

00:08:04.950 --> 00:08:13.270
And then we're going to say f
of xy is going to be equal to

00:08:13.270 --> 00:08:16.760
this: it's going to be the
partial derivative of uppercase

00:08:16.760 --> 00:08:20.740
F with respect to x-- so we're
assuming that this exists, that

00:08:20.740 --> 00:08:25.790
this is true --times i plus
the partial of upper-case F

00:08:25.790 --> 00:08:29.210
with respect to y times j.

00:08:29.210 --> 00:08:42.470
Now, given this what is
lower-case f dot dr going

00:08:42.470 --> 00:08:45.080
to equal over this
path right here?

00:08:45.080 --> 00:08:48.660
This path is defined by this
position function right there.

00:08:48.660 --> 00:08:51.970
Well, it's going to be equal
to, we need to figure out

00:08:51.970 --> 00:08:54.280
what dr is, and we've done
that in multiple videos.

00:08:54.280 --> 00:08:56.480
I'll do that on the
right over here.

00:08:56.480 --> 00:08:59.740
dr, we've seen it
multiple times.

00:08:59.740 --> 00:09:00.980
Actually, I'll solve
it out again.

00:09:00.980 --> 00:09:11.590
dr over dt by definition was
equal to dx over dt times i

00:09:11.590 --> 00:09:15.960
plus-- I don't know why it
got all fat like that

00:09:15.960 --> 00:09:20.070
--dy over dt times j.

00:09:20.070 --> 00:09:21.630
That's what dr over dt is.

00:09:21.630 --> 00:09:24.320
So if we want to figure out
what dr is, the differential of

00:09:24.320 --> 00:09:26.720
dr, if we want to play with
differentials in this way,

00:09:26.720 --> 00:09:29.490
multiply both sides times dt.

00:09:29.490 --> 00:09:31.520
And actually I'm going to
treat dt, I'll multiply

00:09:31.520 --> 00:09:32.670
it; I'll distribute it.

00:09:32.670 --> 00:09:44.030
It's dx over dt times dti
plus dy over dt times dtj.

00:09:44.030 --> 00:09:48.460
So if we're taking the dot
product of f with dr, what

00:09:48.460 --> 00:09:49.060
are we going to get?

00:09:55.080 --> 00:10:01.960
So this is going to be the
integral over the curve from--

00:10:01.960 --> 00:10:04.840
I'll write the c right there;
we could write in terms of the

00:10:04.840 --> 00:10:08.790
end points as t once we feel
good that we have everything in

00:10:08.790 --> 00:10:12.560
terms of t --but it's going to
be equal to this dot that,

00:10:12.560 --> 00:10:16.510
which is equal to-- I'll try to
stay color consistent --the

00:10:16.510 --> 00:10:23.450
partial of upper-case F with
respect to x times that, times

00:10:23.450 --> 00:10:26.520
dx over d t-- I'm going to
write this st in a different

00:10:26.520 --> 00:10:35.850
color --times dt plus the
partial of upper-case F with

00:10:35.850 --> 00:10:41.020
respect to y times--
we're multiplying the

00:10:41.020 --> 00:10:41.950
j components, right?

00:10:41.950 --> 00:10:45.170
When you take the dot product,
multiply the i components, and

00:10:45.170 --> 00:10:48.680
then add that to what you get
from the product of the j

00:10:48.680 --> 00:10:50.400
components --so this j
component's partial of

00:10:50.400 --> 00:10:54.830
upper-case F with respect to y,
and then we have times-- switch

00:10:54.830 --> 00:11:03.680
to a yellow --dy over dt times
that dt right over there.

00:11:03.680 --> 00:11:05.790
And then we can
factor out the dt.

00:11:08.860 --> 00:11:11.300
Or actually, so I don't have to
even write it again, right now

00:11:11.300 --> 00:11:12.800
I wrote it without, well
let me write it again.

00:11:12.800 --> 00:11:16.035
So this is equal
to the integral.

00:11:16.035 --> 00:11:18.420
And let's say we have it in
terms of t; we've written

00:11:18.420 --> 00:11:22.760
everything in terms of t, so t
goes from a to b, and so this

00:11:22.760 --> 00:11:27.150
is going to be equal to-- I'll
write it in blue --the partial

00:11:27.150 --> 00:11:33.610
of upper case F with respect to
x times dx over dt plus-- I'm

00:11:33.610 --> 00:11:36.980
distributing this dt out --plus
the partial of uppercase

00:11:36.980 --> 00:11:39.910
F with respect to y.

00:11:39.910 --> 00:11:42.200
dy over dt.

00:11:42.200 --> 00:11:46.530
all of that times dt.

00:11:46.530 --> 00:11:49.190
This is equivalent to that.

00:11:49.190 --> 00:11:51.950
Now you might realize
why I talked about the

00:11:51.950 --> 00:11:53.640
multivariable chain rule.

00:11:53.640 --> 00:11:55.480
What is this right here?

00:11:55.480 --> 00:11:57.100
What is that right there?

00:11:57.100 --> 00:11:59.050
You can do some
pattern matching.

00:11:59.050 --> 00:12:04.180
That is the same thing as the
derivative of upper-case

00:12:04.180 --> 00:12:05.950
F with respect to t.

00:12:05.950 --> 00:12:07.980
Look at this; let me let me
copy and paste that just

00:12:07.980 --> 00:12:08.940
so you appreciate it.

00:12:17.170 --> 00:12:21.370
So this is our definition, or
this is our-- I won't say

00:12:21.370 --> 00:12:22.920
definition; one can
actually prove it.

00:12:22.920 --> 00:12:25.040
You don't have to start from
there ---but this is our

00:12:25.040 --> 00:12:27.220
multivariable chain
rule right here.

00:12:27.220 --> 00:12:29.890
The driven of any function with
respect t is the partial of

00:12:29.890 --> 00:12:32.400
that function with respect to x
times dx over dt plus the

00:12:32.400 --> 00:12:34.830
partial of that function
with respect to dy over dt.

00:12:34.830 --> 00:12:37.710
I have the partial of
upper-case F with respect to

00:12:37.710 --> 00:12:40.650
x times dx over dt plus the
partial upper-case F

00:12:40.650 --> 00:12:42.130
with respect to y.

00:12:42.130 --> 00:12:44.340
This and this are identical
if you just replace this

00:12:44.340 --> 00:12:46.420
lower-case f with
an upper-case F.

00:12:46.420 --> 00:12:49.470
So this in blue right here,
this whole expression is equal

00:12:49.470 --> 00:12:57.500
to the integral from t is equal
to a to t is equal to b of-- in

00:12:57.500 --> 00:13:07.250
blue here --the derivative
of f with respect to dt.

00:13:07.250 --> 00:13:10.030
And how do you evaluate--
let me just the dt in green

00:13:10.030 --> 00:13:11.980
--how do you evaluate
something like this?

00:13:11.980 --> 00:13:14.790
I just want to make a point:
this is just this from the

00:13:14.790 --> 00:13:16.540
multivariable chain rule.

00:13:16.540 --> 00:13:19.610
And how do you evaluate a
definite integral like this?

00:13:19.610 --> 00:13:21.430
Well, you take the
antiderivative of the

00:13:21.430 --> 00:13:25.120
inside with respect to dt.

00:13:25.120 --> 00:13:26.730
So what is this going
to be equal to?

00:13:26.730 --> 00:13:32.030
You take the antiderivative of
the inside, that's just f.

00:13:32.030 --> 00:13:36.192
So this is equal to f of t.

00:13:36.192 --> 00:13:38.410
And let me be clear.

00:13:38.410 --> 00:13:41.690
We wrote before that
f is a function.

00:13:41.690 --> 00:13:47.330
So our upper-case F is a
function of x and y, which

00:13:47.330 --> 00:13:49.390
could also be written, since
each of these are functions

00:13:49.390 --> 00:13:54.310
of t, could be written as
f of x of t of y of t.

00:13:54.310 --> 00:13:55.920
I'm just rewriting it
in different ways.

00:13:55.920 --> 00:14:00.430
And this could be just
written as f or t.

00:14:00.430 --> 00:14:02.550
These are all equivalent,
depending on whether you want

00:14:02.550 --> 00:14:04.490
to include the x's or the
y's only, or the t's

00:14:04.490 --> 00:14:06.310
only, or them both.

00:14:06.310 --> 00:14:10.370
Because both of the x's and
y's are functions of t.

00:14:10.370 --> 00:14:13.650
So this is the derivative
of f with respect to t.

00:14:13.650 --> 00:14:15.880
If this was just in terms of
t, this is the derivative

00:14:15.880 --> 00:14:17.390
of that with respect to t.

00:14:17.390 --> 00:14:20.880
We take its antiderivative,
we're left just with f, and we

00:14:20.880 --> 00:14:26.540
have to evaluate it from t is
equal to a to t is equal to b.

00:14:26.540 --> 00:14:29.730
And so this is equal to-- and
this is the home stretch

00:14:29.730 --> 00:14:35.910
--this is equal to f
of b minus f of a.

00:14:35.910 --> 00:14:38.170
And if you want to think about
it in these terms, this

00:14:38.170 --> 00:14:39.260
is the same thing.

00:14:39.260 --> 00:14:46.560
This is equal to f of x of b
over y of b-- let me make sure

00:14:46.560 --> 00:14:49.970
I got all the parentheses
--minus f of x of

00:14:49.970 --> 00:14:52.330
a over y of a.

00:14:52.330 --> 00:14:53.660
These are equivalent.

00:14:53.660 --> 00:14:58.320
You give me any point on the
xy plane, an x and a y, and

00:14:58.320 --> 00:15:00.070
it tells me where I am.

00:15:00.070 --> 00:15:03.220
This is my capital F,
it gives me a height.

00:15:03.220 --> 00:15:05.800
Just like that.

00:15:05.800 --> 00:15:09.652
This associates a value with
every point on the xy plane.

00:15:09.652 --> 00:15:13.370
But this whole exercise,
remember this is the

00:15:13.370 --> 00:15:15.280
same thing as that.

00:15:15.280 --> 00:15:18.010
This is our whole thing that we
were trying to prove: that is

00:15:18.010 --> 00:15:25.300
equal to f dot dr. f dot dr,
our vector field, which is the

00:15:25.300 --> 00:15:29.890
gradient of the capital F--
remember F was equal to the

00:15:29.890 --> 00:15:33.000
gradient of F, we assume that
it's the gradient of some

00:15:33.000 --> 00:15:36.800
function capital F, if that is
the case, then we just did a

00:15:36.800 --> 00:15:39.180
little bit of calculus or
algebra, whatever you want to

00:15:39.180 --> 00:15:43.150
call it, and we found that we
can evaluate this integral by

00:15:43.150 --> 00:15:47.170
evaluating capital F at t is
equal to b, and then

00:15:47.170 --> 00:15:50.570
subtracting from that capital
F at t is equal to a.

00:15:50.570 --> 00:15:52.860
But what that tells you is that
this integral, the value of

00:15:52.860 --> 00:15:58.980
this integral, is only
dependent at our starting

00:15:58.980 --> 00:16:06.090
point, t is equal to a, this is
the point x of a, y of a, and

00:16:06.090 --> 00:16:13.810
the ending point, t is equal to
b, which is x of b, y of b.

00:16:13.810 --> 00:16:16.020
That integral is only dependent
on these two values.

00:16:16.020 --> 00:16:17.300
How do I know that?

00:16:17.300 --> 00:16:20.340
Because to solve it-- because
I'm saying that this thing

00:16:20.340 --> 00:16:23.880
exists --I just had to evaluate
that thing at those two

00:16:23.880 --> 00:16:26.890
points; I didn't care about
the curve in between.

00:16:26.890 --> 00:16:34.450
So this shows that if F is
equal to the gradient-- this

00:16:34.450 --> 00:16:37.340
is often called a potential
function of capital F, although

00:16:37.340 --> 00:16:39.590
they're usually the negative
each other, but it's the same

00:16:39.590 --> 00:16:44.220
idea --if the vector field f is
the gradient of some scale or

00:16:44.220 --> 00:16:56.490
field upper-case F, then we can
say that f is conservative or

00:16:56.490 --> 00:17:00.980
that the line integral of f
dot dr is path independent.

00:17:00.980 --> 00:17:04.420
It doesn't matter what path we
go on as long as our starting

00:17:04.420 --> 00:17:06.420
and ending point are the same.

00:17:06.420 --> 00:17:07.580
Hopefully found that useful.

00:17:07.580 --> 00:17:08.680
And we'll some
examples with that.

00:17:08.680 --> 00:17:10.990
And actually in the next video
I'll prove another interesting

00:17:10.990 --> 00:17:13.480
outcome based on this one.


Vector field line integrals dependent on path direction

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:03.530
Let's say I have a position
vector function that looks like

00:00:03.530 --> 00:00:12.202
this. r of t is equal to x of t
times the unit vector i plus y

00:00:12.202 --> 00:00:15.075
of t times the unit vector j.

00:00:15.075 --> 00:00:18.030
And let me actually graph this.

00:00:18.030 --> 00:00:20.580
So let's say, r of t, I want
to draw it a little bit

00:00:20.580 --> 00:00:21.875
straighter than that.

00:00:21.875 --> 00:00:28.630
So that's my y-axis, that is my
x-axis, and let's say r of t,

00:00:28.630 --> 00:00:30.900
and this is for t is less
than, let me write this.

00:00:30.900 --> 00:00:35.810
So this is for t is
between a and b.

00:00:35.810 --> 00:00:39.220
So when t is equal to a, we're
at this vector right here.

00:00:39.220 --> 00:00:43.170
So if you actually substitute t
is equal to a here, you'd get

00:00:43.170 --> 00:00:46.070
a position vector that would
point to that point over there.

00:00:46.070 --> 00:00:49.670
And then, as t increases, it
traces out a curve, or the

00:00:49.670 --> 00:00:52.210
endpoints of our position
vectors trace a curve that

00:00:52.210 --> 00:00:53.730
looks something like that.

00:00:53.730 --> 00:00:58.190
So when t is equal to b, we get
a position vector that points

00:00:58.190 --> 00:00:59.510
to that point right there.

00:00:59.510 --> 00:01:01.800
So this defines a path.

00:01:01.800 --> 00:01:05.570
And the path is going in
this upward direction,

00:01:05.570 --> 00:01:06.950
just like that.

00:01:06.950 --> 00:01:10.770
Now let's say that we have
any another position

00:01:10.770 --> 00:01:13.440
vector function.

00:01:13.440 --> 00:01:27.630
Let me call it r of t.

00:01:27.630 --> 00:01:28.220
It's a different one.

00:01:28.220 --> 00:01:29.970
It's the green r.

00:01:29.970 --> 00:01:31.350
r of t.

00:01:31.350 --> 00:01:35.950
Instead of being x of t times
i, it's going to be x of a plus

00:01:35.950 --> 00:01:41.890
b minus t times i, and instead
of y of t, it's going to be y

00:01:41.890 --> 00:01:46.470
of a plus b minus t times i.

00:01:46.470 --> 00:01:48.640
And we've seen this in
the last two videos.

00:01:48.640 --> 00:01:54.310
This, the path defined by this
position vector function is

00:01:54.310 --> 00:01:56.020
going look more like this.

00:01:56.020 --> 00:01:58.640
Let me draw my axes.

00:01:58.640 --> 00:02:01.820
This is my y-axis,
that is my x-axis.

00:02:01.820 --> 00:02:06.430
Maybe I should label
them, y and x.

00:02:06.430 --> 00:02:08.710
This path is going to
look just like this.

00:02:08.710 --> 00:02:12.390
But instead of starting here
and going there, when t is

00:02:12.390 --> 00:02:15.780
equal to 1, let me make it
clear, this is also true for

00:02:15.780 --> 00:02:18.840
a is less than or equal to
t, which is less than b.

00:02:18.840 --> 00:02:20.750
So t is going to
go from a to b.

00:02:20.750 --> 00:02:22.900
But here, when t is equal
to a, you substitute it

00:02:22.900 --> 00:02:25.660
over here, you're going
to get this vector.

00:02:25.660 --> 00:02:30.520
You're going to start over
there, and as you increment t,

00:02:30.520 --> 00:02:33.600
as you make it larger and
larger and larger, you're going

00:02:33.600 --> 00:02:37.010
to trace out that same path,
but in the opposite direction.

00:02:39.910 --> 00:02:42.750
And so when t is equal to b,
you put that in here, you're

00:02:42.750 --> 00:02:45.550
actually going to get x of a
and y of a there, right, the

00:02:45.550 --> 00:02:48.300
b's cancel out, and so you're
going to point right like that.

00:02:48.300 --> 00:02:50.480
So these are the same, you
could imagine the shape of

00:02:50.480 --> 00:02:54.310
these paths are the same, but
we're going in the exact

00:02:54.310 --> 00:02:55.750
opposite direction.

00:02:55.750 --> 00:02:58.180
So what we're going to do in
this video is to see what

00:02:58.180 --> 00:03:02.390
happens, how, I guess you could
say, if I have some vector

00:03:02.390 --> 00:03:17.010
field f of xy equals p
of xy i plus q of xy j.

00:03:17.010 --> 00:03:17.220
Right?

00:03:17.220 --> 00:03:22.250
This is just a vector
field over the x-y plane.

00:03:22.250 --> 00:03:26.800
How the line integral of this
vector field, of this vector

00:03:26.800 --> 00:03:34.640
field over this path, compares
to the line integral the same

00:03:34.640 --> 00:03:36.500
vector field over that path.

00:03:36.500 --> 00:03:40.280
How that compares to this.

00:03:40.280 --> 00:03:41.770
We'll call this
the minus curve.

00:03:41.770 --> 00:03:46.020
So this is the positive
curve, we're going to call

00:03:46.020 --> 00:03:47.910
this the minus curve.

00:03:47.910 --> 00:03:50.400
So how does it, going over
the positive curve, compare

00:03:50.400 --> 00:03:55.330
to going over the minus
curve? f of f dot dr.

00:03:55.330 --> 00:03:57.640
So before I break into the
math, let's just think

00:03:57.640 --> 00:03:58.310
about a little bit.

00:03:58.310 --> 00:04:00.540
Let me draw this
vector field f.

00:04:00.540 --> 00:04:03.720
So maybe it looks, I'm just
going to draw random stuff.

00:04:03.720 --> 00:04:07.540
So you know, on every point in
the x-y plane, it has a vector,

00:04:07.540 --> 00:04:11.560
it defines or maps a vector, to
every point on the x-y plane.

00:04:11.560 --> 00:04:13.800
But we really care about the
points that are on the curve.

00:04:13.800 --> 00:04:17.600
So maybe on the curve, you
know, this is the vector field

00:04:17.600 --> 00:04:19.685
at the points on the curve.

00:04:19.685 --> 00:04:21.150
And let me draw it
over here, too.

00:04:21.150 --> 00:04:23.520
So all the points on the curve
where we care about, this is

00:04:23.520 --> 00:04:29.120
our vector field, that
is our vector field.

00:04:29.120 --> 00:04:32.030
And let's just get an intuition
of what's going to be going.

00:04:32.030 --> 00:04:35.990
We're summing over the dot,
we're taking each point along

00:04:35.990 --> 00:04:41.440
the line, and we're summing,
let me start over here.

00:04:41.440 --> 00:04:43.153
We're taking each point along
the line, let me do it

00:04:43.153 --> 00:04:45.450
in a different color.

00:04:45.450 --> 00:04:49.035
And we're summing the dot
product of the value of the

00:04:49.035 --> 00:04:53.910
vector field at that point, the
dot product of that, with dr,

00:04:53.910 --> 00:04:57.430
or the differential of our
position vector function.

00:04:57.430 --> 00:05:01.770
And dr, you can kind of
imagine, as an infinitesimally

00:05:01.770 --> 00:05:04.620
small vector going in the
direction of our movement.

00:05:04.620 --> 00:05:08.080
And we take this dot product
here, it's essentially, it's

00:05:08.080 --> 00:05:10.340
going to be a scalar value,
but the dot product, if you

00:05:10.340 --> 00:05:15.530
remember, it's the magnitude
of f in the direction of dr,

00:05:15.530 --> 00:05:18.350
times the magnitude of dr.

00:05:18.350 --> 00:05:21.750
So it's this, you can imagine
it's the shadow of f onto

00:05:21.750 --> 00:05:24.040
dr. Let me zoom into that,
because I think it's useful.

00:05:24.040 --> 00:05:26.880
So this little thing that I'm
drawing right here, let's

00:05:26.880 --> 00:05:29.070
say that this is my path.

00:05:29.070 --> 00:05:31.470
This is f at that point.

00:05:31.470 --> 00:05:34.260
f at that point looks
something like that.

00:05:34.260 --> 00:05:37.500
And then dr at this point
looks something like that.

00:05:37.500 --> 00:05:40.840
Let me do it in different
color. dr looks

00:05:40.840 --> 00:05:42.230
something like that.

00:05:42.230 --> 00:05:43.930
So that is f.

00:05:43.930 --> 00:05:47.300
And so the dot product of these
two says, ok, how much of f is

00:05:47.300 --> 00:05:48.670
going in the same
direction as dr?

00:05:48.670 --> 00:05:50.940
And you can kind of
imagine, there's a shadow.

00:05:50.940 --> 00:05:54.800
If you take the f that's going
in the same direction as dr,

00:05:54.800 --> 00:05:58.410
the magnitude of that times the
magnitude of dr, that

00:05:58.410 --> 00:05:59.400
is the dot product.

00:05:59.400 --> 00:06:01.170
In this case, we're going
to get a positive number.

00:06:01.170 --> 00:06:03.360
Because this length is
positive, this length is

00:06:03.360 --> 00:06:06.030
positive, that's going to
be a positive number.

00:06:06.030 --> 00:06:08.180
Now what if our dr was going
in the opposite direction,

00:06:08.180 --> 00:06:09.600
as it is in this case?

00:06:09.600 --> 00:06:12.445
So let me draw maybe that
same part of the curve.

00:06:15.360 --> 00:06:21.500
We have our f, our f will
look something like that.

00:06:21.500 --> 00:06:23.520
I'm drawing this exact
same part of the curve.

00:06:23.520 --> 00:06:27.470
But now our dr isn't
going in that direction.

00:06:27.470 --> 00:06:31.470
Our dr that at this point
is going to be going in

00:06:31.470 --> 00:06:32.730
the other direction.

00:06:32.730 --> 00:06:34.800
We're tracing the curve in
the opposite direction.

00:06:34.800 --> 00:06:37.540
Our dr is now going to be
going in that direction.

00:06:41.110 --> 00:06:44.430
So if you do f dot dr, you're
taking the shadow, or how much

00:06:44.430 --> 00:06:46.550
of f is going in the direction
of dr, you take the shadow

00:06:46.550 --> 00:06:50.550
down here, it's going in the
opposite direction of dr. So

00:06:50.550 --> 00:06:52.840
you can imagine that when you
multiply the magnitudes, you

00:06:52.840 --> 00:06:53.930
should get a negative number.

00:06:53.930 --> 00:06:56.870
Our direction is now opposite,
they're not going in the-- the

00:06:56.870 --> 00:07:01.320
shadow of f onto the same
direction is dr is going in the

00:07:01.320 --> 00:07:03.540
opposite direction as dr.
In this case, it's going in

00:07:03.540 --> 00:07:05.090
the same direction as dr.

00:07:05.090 --> 00:07:08.010
So the intuition is that maybe
these two things are the

00:07:08.010 --> 00:07:09.480
negative of each other.

00:07:09.480 --> 00:07:12.490
And now we can do some
math and try to see if

00:07:12.490 --> 00:07:16.090
that is definitely,
definitely the case.

00:07:16.090 --> 00:07:20.170
So let us first figure out,
let's write an expression for

00:07:20.170 --> 00:07:29.830
the differential dr. So in this
case, dr, dr dt is going to be

00:07:29.830 --> 00:07:40.990
equal to x prime of t times i
plus y prime of t times j.

00:07:40.990 --> 00:07:48.130
In this other example, in the
reverse case, our dr, dr

00:07:48.130 --> 00:07:51.350
dt, is going to be, what's
it going to be equal to?

00:07:51.350 --> 00:07:54.540
It's the derivative of
x with respect to t.

00:07:54.540 --> 00:07:57.530
The derivative of this term
with respect to t, that's the

00:07:57.530 --> 00:08:01.260
derivative of the inside, which
is minus 1, or minus, times

00:08:01.260 --> 00:08:03.840
the derivative of the outside
with respect to the inside.

00:08:03.840 --> 00:08:07.080
So that's going to be,
derivative of the inside is

00:08:07.080 --> 00:08:10.350
minus 1, times the derivative
of the outside with respect to

00:08:10.350 --> 00:08:15.260
the inside. x prime of a
plus b minus t times i.

00:08:15.260 --> 00:08:17.640
And then same thing
for the second term.

00:08:17.640 --> 00:08:20.930
Derivative of y of this term
with respect to the inside,

00:08:20.930 --> 00:08:23.960
which is minus 1, times the
derivative of the outside with

00:08:23.960 --> 00:08:27.470
respect to the inside, which is
y prime of a plus b minus t.

00:08:27.470 --> 00:08:30.240
So this is going to be the
derivative of the inside,

00:08:30.240 --> 00:08:36.650
times y prime of a
plus b minus is t j.

00:08:36.650 --> 00:08:40.660
So this is dr dt in this case,
this is dr dt in that case.

00:08:40.660 --> 00:08:45.620
And if we wanted to write the
differential dr in the forward

00:08:45.620 --> 00:08:51.510
curve example, it's going to be
equal to x prime of t times i

00:08:51.510 --> 00:08:58.870
plus y prime of t times
j times the scalar dt.

00:08:58.870 --> 00:09:01.220
I could multiply it down into
each of these terms, but

00:09:01.220 --> 00:09:04.060
it keeps it simple, just
leaving it on the outside.

00:09:04.060 --> 00:09:06.150
Same logic over here.

00:09:06.150 --> 00:09:10.510
dr is equal to minus x.

00:09:10.510 --> 00:09:12.850
I changed my shade of green,
but at least it's still green.

00:09:12.850 --> 00:09:21.100
a plus b minus t i minus y
prime a plus b minus t j,

00:09:21.100 --> 00:09:25.320
and I'm multiplying
both sides by dt.

00:09:25.320 --> 00:09:27.940
Now we're ready to express
this as a function of t.

00:09:27.940 --> 00:09:31.870
So this curve right here, I'll
do it in pink, the pink one is

00:09:31.870 --> 00:09:36.340
going to be equal to the
integral from t is equal to a

00:09:36.340 --> 00:09:53.630
to t is equal to b of f of f of
x of f of x of t y of t dot

00:09:53.630 --> 00:09:56.820
this thing over here, which is,
I'll just write out here, I

00:09:56.820 --> 00:09:59.002
could simplify it later.

00:09:59.002 --> 00:10:06.260
x prime of t i plus
y prime of t j.

00:10:06.260 --> 00:10:08.930
And then all of that
times the scalar dt.

00:10:08.930 --> 00:10:11.230
This'll be a scalar value, and
then we'll have another scalar

00:10:11.230 --> 00:10:13.190
value of dt over there.

00:10:13.190 --> 00:10:16.490
Now, what is this going
to be equal to if I take

00:10:16.490 --> 00:10:18.630
this reverse integral?

00:10:18.630 --> 00:10:21.275
The reverse integral is going
to be the integral from, I'm

00:10:21.275 --> 00:10:26.880
going to need a little more
space, from a to b, of f of not

00:10:26.880 --> 00:10:34.220
x of t, but x of a plus b minus
t y of a plus b minus t.

00:10:34.220 --> 00:10:36.550
I'm writing it small
so I have some space.

00:10:36.550 --> 00:10:41.160
Dot, this is a vector, so dot
this guy right here, dot dr.

00:10:41.160 --> 00:10:53.800
Dot minus x prime of a plus b
minus t i, minus y prime of,

00:10:53.800 --> 00:10:55.110
I'm using up too much space.

00:10:55.110 --> 00:10:57.770
Let me scroll, go
back a little bit.

00:10:57.770 --> 00:10:59.680
Actually, let me take it
make it even simpler.

00:10:59.680 --> 00:11:01.690
Let me take this minus
sign out of it.

00:11:01.690 --> 00:11:04.400
Let me put a plus, and
then I'll put the

00:11:04.400 --> 00:11:06.790
minus sign out front.

00:11:06.790 --> 00:11:09.260
So the minus sign is just a
scalar value, so we could put

00:11:09.260 --> 00:11:11.710
that minus sign out, you know,
when you take a dot product,

00:11:11.710 --> 00:11:15.180
and if you multiply a scalar
times a dot product, you could

00:11:15.180 --> 00:11:17.680
just take the scalar out,
that's all I'm saying.

00:11:17.680 --> 00:11:20.880
So we take that minus sign
out to this part right here.

00:11:20.880 --> 00:11:30.030
And then you have x prime of a
plus b minus t i plus y prime

00:11:30.030 --> 00:11:39.970
of a plus b minus t, scroll
over a little bit, t j dt.

00:11:39.970 --> 00:11:43.130
So the this is the forward,
this is when we're following it

00:11:43.130 --> 00:11:45.930
along the forward curve, this
is when we're following it

00:11:45.930 --> 00:11:48.490
along the reverse curve.

00:11:48.490 --> 00:11:51.230
Now like we did with the
scalar example, let's

00:11:51.230 --> 00:11:51.990
make a substitution.

00:11:51.990 --> 00:11:53.100
I want to make it very
clear what I did.

00:11:53.100 --> 00:11:55.330
All I did here, is I just
took the dot product,

00:11:55.330 --> 00:11:57.600
but this negative sign,
I just took it out.

00:11:57.600 --> 00:12:00.290
I just said, this is the same
thing as negative 1 times this

00:12:00.290 --> 00:12:02.200
thing, or negative 1 times this
thing is the same

00:12:02.200 --> 00:12:03.230
thing as that.

00:12:03.230 --> 00:12:05.610
So let's make a substitution on
this side, because I really

00:12:05.610 --> 00:12:08.900
just want to show you that this
is the negative of that, right

00:12:08.900 --> 00:12:10.880
there, because that's what
our intuition was going for.

00:12:10.880 --> 00:12:13.220
So let me just focus
on that side.

00:12:13.220 --> 00:12:15.980
So let me make a substitution.

00:12:15.980 --> 00:12:21.530
u is equal to a plus b minus t.

00:12:21.530 --> 00:12:24.720
Then we get du is equal
to minus dt, right?

00:12:24.720 --> 00:12:26.770
Just take the derivative
of both sides.

00:12:26.770 --> 00:12:31.230
Or you get dt is
equal to minus du.

00:12:31.230 --> 00:12:37.890
And then you get, when t
is equal to a, u is equal

00:12:37.890 --> 00:12:40.050
to a plus b minus a.

00:12:40.050 --> 00:12:43.070
So then, u is equal to b.

00:12:43.070 --> 00:12:49.140
And then when t is equal to
b, u is equal to a, right?

00:12:49.140 --> 00:12:52.530
Which is equal to b, a
plus b minus b is a.

00:12:52.530 --> 00:12:53.230
u is equal to a.

00:12:53.230 --> 00:12:57.813
So this thing, using that
substitution simplifies to, and

00:12:57.813 --> 00:13:03.380
this is the whole point, that
simplifies to minus integral

00:13:03.380 --> 00:13:07.010
from u is, when t is a, u is b.

00:13:07.010 --> 00:13:09.570
From b, when t is b, u is a.

00:13:09.570 --> 00:13:13.930
The integral from u is equal
to b to u is equal to a of

00:13:13.930 --> 00:13:21.140
f of x of u y of u, right?

00:13:21.140 --> 00:13:24.000
That is u, that is u.

00:13:24.000 --> 00:13:39.330
Dot x prime of u times i,
that's u right there, plus

00:13:39.330 --> 00:13:44.240
y prime of u times j.

00:13:44.240 --> 00:13:47.360
And then, instead of a
dt, I need to put a du.

00:13:47.360 --> 00:13:49.970
dt is equal to minus du.

00:13:49.970 --> 00:13:53.190
So I could write minus du
here, or just to not make it

00:13:53.190 --> 00:13:55.780
confusing, I'll put the du
here, and take the

00:13:55.780 --> 00:13:56.900
minus out front.

00:13:56.900 --> 00:14:00.230
I already have a minus out
there, so they cancel out.

00:14:00.230 --> 00:14:03.730
They will cancel out,
just like that.

00:14:03.730 --> 00:14:06.480
And so you might say, hey Sal,
these two things look pretty

00:14:06.480 --> 00:14:07.320
similar to each other.

00:14:07.320 --> 00:14:08.770
They don't like they're
negative of each other.

00:14:08.770 --> 00:14:12.080
And I'd say, well, you're
almost right, except this guy's

00:14:12.080 --> 00:14:15.240
limits of integration are
reversed from this guy.

00:14:15.240 --> 00:14:17.810
So this thing right here, if
we reverse the limits of

00:14:17.810 --> 00:14:20.730
integration, we have to
then make it negative.

00:14:20.730 --> 00:14:29.390
So this is equal to minus the
integral from a to b of the

00:14:29.390 --> 00:14:40.920
vector f of x of u y of u dot
x prime of u i plus

00:14:40.920 --> 00:14:45.540
y prime of u j du.

00:14:45.540 --> 00:14:47.900
And now this is identical.

00:14:47.900 --> 00:14:50.510
This integral, this definite
integral, is identical to

00:14:50.510 --> 00:14:51.620
that definite integral.

00:14:51.620 --> 00:14:52.870
We just have a
different variable.

00:14:52.870 --> 00:14:55.620
We're doing dt here, we have du
here, but we're going to get

00:14:55.620 --> 00:14:59.220
the same exact number for any a
or b, and given this

00:14:59.220 --> 00:15:06.050
vector f and the position
vector path r of t.

00:15:06.050 --> 00:15:11.430
So just to summarize everything
up, when you're dealing with

00:15:11.430 --> 00:15:16.500
line integrals over vector
fields, the direction matters.

00:15:16.500 --> 00:15:18.530
If you go in the reverse
direction, you're going to get

00:15:18.530 --> 00:15:20.520
the negative version of that.

00:15:20.520 --> 00:15:23.610
And that's because at any point
we take the dot product, you're

00:15:23.610 --> 00:15:27.060
not going in the, necessarily,
you're going in the opposite

00:15:27.060 --> 00:15:29.330
direction, so it'll be the
negative of each other.

00:15:29.330 --> 00:15:31.730
But when you're dealing with
the scalar field, we saw on the

00:15:31.730 --> 00:15:37.430
last video, we saw that it
doesn't matter which direction

00:15:37.430 --> 00:15:39.100
that you traverse the path in.

00:15:39.100 --> 00:15:41.490
That the positive path
has the same value as

00:15:41.490 --> 00:15:42.210
the negative path.

00:15:42.210 --> 00:15:44.600
And that's just because we're
just trying to find the

00:15:44.600 --> 00:15:46.590
area of that curtain.

00:15:46.590 --> 00:15:49.670
Hopefully you found
that mildly amusing.