Main content

## Line integrals in vector fields

Current time:0:00Total duration:16:22

# Scalar field line integral independent of path direction

## Video transcript

In the last video, we saw that
if we had some curve in the x-y plane, and we just parameterize
it in a very general sense like this, we could generate another
parameterization that essentially is the same
curve, but goes in the opposite direction. It starts here and it goes
here, as t goes from a to b, as opposed to the first
parameterization, we started with t equals a over here,
and it went up like that. And the question I want to
answer in this video is how a line integral of a scalar field
over this curve, so this is my scalar field, it's a function
of x and y, how a line integral over a scalar field over this
curve relates to, that's a line integral of that same scalar
field over the reverse curve, over the curve going in
the other direction. So the question is, does it
even matter whether we move in this direction or that
direction when we're taking the line integral of
a scalar field? And in the next video, we'll
talk about whether it matters on a vector field. And let's see if we can get
a little intuition to our answer before we even
prove our answer. So let me draw a
little diagram, here. Actually, let me do it a little
bit lower, because I think I'm going to need a little
bit more real estate. So let me draw the y-axis, that
is the x-axis, let me draw the vertical axis, just
like that, that is z. Let me draw a scalar
field, here. So I'll just draw it as some
surface, I'll draw part of it. That is my scalar field, that
is f of xy right there. For any point on the x-y plane
we can associate a height that defines this surface,
this scalar field. And let me put a
curve down there. So let's say that this is the
curve c, just like that. And the way we define it first,
we start over here and we move in that direction. That was our curve c. And we know from several videos
ago that the way to visualize what this line integral means,
is we're essentially trying to figure out the area of a
curtain that has this curve as its base, and its ceiling is
defined by this surface, by the scalar field. So we're literally just trying
to find the area of this curvy piece of paper, or wall, or
whatever you want to view it. That's what this thing is. Now, if we take the same
integral but we take it the reverse curve, instead of
going in that direction, we're now going in the
opposite direction. We're not taking a curve,
we're going from the top to the bottom. But the idea is still the same. You know, I don't know which
one is c, which one is minus c. I could have defined this path
going from that way as c, and then the minus c path would
have started here, and gone back up. So it seems in either case, no
matter what I'm doing, I'm going to try to figure out the
area of this curved piece of paper. So my intuition tells me that
the either these are going to give me the area of this curved
piece of paper, so maybe they should be equal to each other. I haven't proved anything very
rigorously yet, but it seems that they should be equal
to each other, right? In this case, let's say
I'm taking a, let me just make it very clear. I'm taking a ds. a little change in distance,
let me do it in a different color. A little change in distance,
and I'm multiplying it by the height, to find kind of a
differential of the area. And I'm going to add a
bunch of these together to get the whole area. Here I'm doing the same thing. I'm taking a little ds, and
remember, the ds is always going to be positive, the
way we've parameterized it. So here, too, we're taking
a ds, and we're going to multiply it by the height. So once again, we
should take the area. And I want to actually
differentiate that relative to, when you take a normal integral
from a to b of, say, f of x dx, we know that when we switch the
boundaries of the integration, that it makes the
integral negative. That equals the negative
of the integral from b to a of f of x dx. And the reason why this is the
case, is if you imagine this is a, this is b, that
is my f of x. When you do it this way,
your dx's are always going to be positive. When you go in that direction,
your dx's are always going to be positive, right? Each increment, the right
boundary is going to be higher than the left boundary. So your dx's are positive. In this situation, your
dx's are negative. The heights are always going to
be the same, they're always going to be f of x, but here
your change in x is a negative change in x,
when you go from b to a. And that's why you get
a negative integral. In either case here, our path
changes, but our ds's are going to be positive. And the way I've drawn this
surface, it's above the x-y plane, the f of xy is also
going to be positive. So that also kind of gives the
same intuition that this should be the exact same area. But let's prove
it to ourselves. So let's start off with our
first parameterization, just like we did in the last video. We have x is equal to x of t, y
is equal to y of t, and we're dealing with this from,
t goes from a to b. And we know we're going to need
the derivatives of these, so let write that down right now. We can write dx dt is equal to
x prime of t, and dy dt, let me write that a little bit neater,
dy dt is equal to y prime of t. This is nothing groundbreaking
I've done so far. But we know the integral
over c of f of xy. f is a scalar field,
not a vector field. ds is equal to the integral
from t is equal to a, to t is equal to b of f of x of t y of
t times the square root of dx dt squared, which is the same
thing as x prime of t squared, plus dy dt squared, the same
thing as y prime of t squared. All that under the
radical, times dt. This integral is exactly that,
given this parameterization. Now let's do the
minus c version. I'll do that in
this orange color. Actually, let me do the
minus c version down here. The minus the c version, we
have x is equal to, you remember this, actually,
just from up here, this was from the last video. x is equal to x of
a plus b minus t. y is equal to y of
a plus b minus t. And then t goes from a to b, t
goes from a to b, and this is just exactly what we did in
that last video. x is equal to x of a plus b minus t, y is
equal to y of a plus b minus t, same curve, just going in a
different direction as t increases a to b. But let's get the derivative. I'll do it in the
derivative color, maybe. So dx dt. For this path, it's going
to be a little different. We have to do the
chain rule now. Derivative of the inside
with respect to t. Well, these are constants. Derivative of minus t with
respect to t is minus 1. So it's minus 1 times the
derivative of the outside with respect to the inside. Well, that's just x prime
of a plus b minus t. Or, we could rewrite this as,
this is just minus x prime of a plus b minus t. dy dt, same logic. Derivative of the
inside is minus 1 with respect to t, right? Derivative minus t
is just minus 1. Times the derivative
of the outside with respect to the inside. So y prime of a plus b minus
t, same thing as minus y prime a plus b minus t. So given all of that, what is
this integral going to be equal to, the integral of minus c of
the scalar field f of xy ds? What is this going
to be equal to? Well, it's going to be the
integral from, you could almost pattern match it. t is equal
to a to t is equal to be of f of x. But now x is no longer
x of t. x now equals x of a plus b minus t. It's a little bit hairy,
but I don't think anything here is groundbreaking. Hopefully it's not
too confusing. And once again, y is no
longer y of t. y is y of a plus b minus t. And then times a square root,
I'll just switch colors, times the square root
of dx dt squared. What is dx dt squared? dx dt squared is just
this thing squared, or this thing squared. This thing, if I have minus
anything squared, that's the same thing as anything
squared, right? This is equal to minus x prime
of a plus b minus t squared, which is the same thing is
just x prime of a plus b minus t squared, right? You lose that minus information
when you square it. So that's going to be equal to
x prime of a plus b minus t squared, the whole result
function squared, plus dy dt squared. By the same logic, that's going
to be, you lose the negative when you square it. y prime of a plus b
minus t squared. Let me extend the radical. And then all of that dt. So that's the line integral
over the curve c, this is the line integral over
the curve minus c. They don't look equal just yet. This looks a lot more
convoluted than that one does. So let's see if we can
simplify it little bit. And we can simplify it,
perhaps, by making a substitution. Let's let, let me get a nice
substitution color, let's let u equal to a plus b minus t. So first we're going to have to
figure out the boundaries of our integral, well actually,
let's just figure out, what's du? so du dt, the derivative of u
with respect to t is just going to be equal to minus 1, or we
could say that du, if we multiply both sides by the
differential dt, is equal to minus dt. And let's figure out our
boundaries of integration. When t is equal to a,
what is u equal to? u is equal to a plus b minus
a, which is equal to b. And then when t is equal to b,
u is equal to a plus b minus b, which is equal to a. So if we do the substitution on
this crazy, hairy-looking interval, let's simplify a
little bit, and it changes our-- so this integral is going
to be the same thing as the integral from u, when
t is a, u is b. When t is b, u is a. And f of, x of, this thing
right here is just u. x of u. So it simplified it a good bit. And y of, this thing
right here, is just u. y of u. Times the square root-- let
me do it in the same color. Times the square root of x
prime of u squared plus y prime of u squared. Instead of a dt, we have to
write a, or could write, if we multiply both sides of this by
minus, we have dt is equal to minus du. So instead of a dt, we have
to put a minus du here. So this is times minus du, or,
just so we don't think this is a subtraction, let's just put
that negative sign out here in the front, just like that. So we're going from b to a of
this thing, right, like that. And just to make the boundaries
of integration make a little bit more sense, because we know
that a is less than b, let's swap them. And I said at the beginning
of this video, for just a standard, regular, run of the
mill integral, if you swap, if you have something going from b
to a of f of x dx, or du, maybe I should write it this way. f of u du. This is equal to the minus
of the integral from a to b of f of u du. And we did that by the logic
that I had graphed up here. That here, when you switch the
order, your du's will become the negatives of each other,
when you actually visualize it, when you're actually finding
the area under the curve. So let's do that. Let's swap the boundaries
of integration right here. And if we do that, that
will negate this negative, or make it a positive. So this is going to be equal
to the integral from a to b. I'm dropping the negative
sign, because I swapped these two things. So I'm going to take the
negative of a negative, which is a positive. Of f of x of u y of u times the
square root of x prime of u squared plus y prime
of u squared du. Now remember, everything we
just did was a substitution. This was all equal to, just to
remember what we're doing, this was the integral of the minus
curve of our scalar field, f of xy ds. Now how does this compare to
when we take the regular curve? How does this compare to that? Let me copy and
paste it to see. You know, I'm using
the wrong tool. Let me copy and paste it
to see how they compare. Copy, then let me pick to
down here, edit, paste. So how do these two
things compare? Let's take a close look. Well, they actually look
pretty similar, right? Over here, for the minus curve,
we have a bunch of u's. Over here, for the positive
curve, we have a bunch of t's, but they're in
the exact same places. These integrals are the
exact same integrals. If you make a u-substitution
here, if you just make the substitution u is equal to t,
this thing is going to be integral from a to b of,
it's going to be the exact same thing. Of f of x of u, y of u times
the square root of of x prime of u squared plus
y prime of u squared du. These two things are identical. So we did all the substitution,
everything, but we got the exact same integrals. So hopefully that satisfies you
that it doesn't matter what direction we go on the curve,
as long as the shape of the curve is the same. Doesn't matter if we go forward
or backward on the curve, we're going to get the same answer. And I think that meets our
intuition, because in either case, we're finding the
area of this curtain.