Vector representation of a surface integral | Multivariable Calculus | Khan Academy

In the last video,
we figured out

how to construct a unit
normal vector to a surface.

And so now we can use that
back in our original surface

integral to try to
simplify a little bit,

or at least give us a clue how
we can calculate these things.

And also, think
about different ways

to represent this type
of a surface integral.

So if we just substitute what
we came up as our normal vector,

our unit normal
vector right here,

we will get-- so
once again, it's

the surface integral of F dot.

And F dot all of this
business right over here.

And I'm going to
write it all in white,

just so it doesn't
take me too much time.

So the partial of
r with respect to u

crossed with the partial
of r with respect

to v over the magnitude
of the same thing, partial

of r with respect to u
crossed with the partial of r

with respect to v.

And now, we've
played with ds a lot.

We know that the other
way to write ds--

and I gave the
intuition, hopefully,

for that several videos
ago when we first

explored what a surface
integral was all about.

We know that ds is--
it can be represented

as the magnitude of the
partial of r with respect

to u crossed with the partial
of r with respect to v du dv.

And Obviously, the du dv, it
could be written as dv du.

You could write it as da,
a little chunk of area

and the uv plane or
in the uv domain.

And actually, since now this
integral's in terms of uv,

we're no longer taking
a surface integral.

We're now taking a double
integral over the uv domain.

So you could say kind
of a region in uv.

So I'll say R to say that's
it's a region in the uv plane

that we're now thinking about.

But there's probably a
huge-- or there should be,

or I'm guessing there's a
huge simplification that's

popping out at you right now.

We're dividing by the
magnitude of the cross product

of these two vectors
and then we're

multiplying by the magnitude of
the cross product of these two

vectors.

Those are just
scalar quantities.

You divide by something
and multiply by something.

Well, that's just the same
thing as multiplying or dividing

by 1.

So these two
characters cancel out,

and our integral simplifies
to the double integral

over that region, the
corresponding region in the uv

plane, of F-- of
our vector field F

dotted with this cross product.

This is going to give us
a vector right over here.

That's going to
give us a vector.

It gives us actually
a normal vector.

And then when you
divide by its magnitude,

it gives you a
unit normal vector.

So this, you're going to
take the dot product of F

with r, the partial
of r with respect

to u crossed with the partial
of r with respect to v du dv.

Let me scroll over to the
right a little bit, du dv.

And we'll see in the
few videos from now

that this is essentially how we
go about actually calculating

these things.

If you have a
parameterization, you

can then get everything in
terms of a double integral,

in terms of uv this way.

Now, the last thing I want
to do is explore another way

that you'll see a surface
integral like this written.

It all comes from,
really, writing this part

in a different way.

But it hopefully gives you
a little bit more intuition

of what this thing
is even saying.

So I'm just going to rewrite.

I'm going to rewrite this
chunk right over here.

I'm just going to
rewrite that chunk.

And I'm going to use slightly
different notation because it

will hopefully help make
a little bit more sense.

So the partial of r
with respect to u I

can write as the partial
of r with respect to u.

And we're taking
the cross product.

Let me make my u's a little bit
more u-like so we confuse them

with v's.

And we're taking the
cross product of that

with the partial
of r with respect

to v. So very small
changes in our vector--

in our parameterization
right here,

our position vector given
a small change in v. Very

small changes in the vector
given a small change in u.

And then we're multiplying
that times du dv.

We're multiplying
that times du dv.

Now, du and dv are
just scalar quantities.

They're infinitesimally small.

But for the sake
of this argument,

you can just view--
they're not vectors,

they're just scalar quantities.

And so you can
essentially include them--

if you have the cross product.

If you have a cross b times some
scalar value-- I don't know, x,

you could rewrite this
as x times a cross b,

or you could write
this as a cross x times

b, because x is
just a scalar value.

It's just a number.

So we could do the
same thing over here.

We can rewrite all
of this business as--

and I'm going to group the du
where we have the partial--

or with respect to u
in the denominator.

And I'll do the same
thing with the v's.

And so you will get the
partial of r with respect

to u times du,
times that scalar.

So that'll give us a vector.

And we're going to cross that.

We're going to cross that with
the partial of r with respect

to v dv.

Now, these might
look notationally

like two different
things, but that just

comes from the
necessity of when we

take partial derivatives to say,
oh, no, this vector function

is defined-- it's a function
of multiple variables

and this is taking a
derivative with respect

to only one of them.

So this is, how
much does our vector

change when you have a
very small change in u?

But this is also an
infinitesimally small change

in u over here, we're just using
slightly different notation.

So for the sake of--
and this is a little bit

loosey-goosey mathematics,
but it will hopefully

give you the intuition
for why this thing could

be written in a different way.

These are essentially
the same quantity.

So if you divide by something
and multiply by something,

you can cancel them out.

If you divide by something
and multiply by something,

you can cancel them out.

And all you're left
with then-- all

you're left with is
the differential of r.

And since we lost
the information

that it's in the
u-direction, I'll

write here, the differential
of r in the u-direction.

I don't want to get
the notation confused.

This is just the differential.

This is just how much r changed.

This is not the partial
derivative of r with respect

to u.

This right over here
is, how much does r

change given per unit change,
per small change in u?

This just says a differential
in the direction of-- I

guess as u changes, this is
how much that infinitely small

change that just r changes.

This isn't change in r with
respect to change in u.

And we're going to cross that.

Now, we're going to cross
that with the partial of r,

the partial of r
in the v-direction.

Now, this right over here,
let's just conceptualize this.

And this goes back to
our original visions

of what a surface
integral was all about.

If we're on a surface--
and I'll draw surface.

Let me draw another surface.

I won't use the one that
I had already drawn on.

If we draw a surface, and for
a very small change in u--

and we're not going to
think about the rate.

We're just thinking about
kind of the change in r.

You're going in that direction.

So if that thing
looks like this,

this is actually kind of a
distance moved on the surface.

Because remember, this
isn't the derivative.

This is the differential.

So it's just a small
change along the surface,

that's that over there.

And that this is a small
change when you change v. So

it's also a change
along the surface.

When you take the cross
product of these two things,

you get a vector
that is orthogonal.

You get a vector that is
normal to the surface.

So it is normal to the
surface and its magnitude--

and we saw this when we first
learned about cross products.

Its magnitude is
equal to the area that

is defined by these two vectors.

So its magnitude
is equal to area.

So in a lot of
ways, you can really

think of it-- you
really could think

of it as a unit normal
vector times ds.

And so the way that we would,
I guess notationally do this,

is we can call this--
because this is kind of a ds,

but it's a vector
version of the ds.

Over here, this is just
an area right over here.

This is just a scalar value.

But now, we have a vector
that points normally

from the surface,
but its magnitude

is the same thing as
that ds that we were just

talking about.

So we can call this thing right
over here, we can call this ds.

And the key difference here
is this is a vector now.

So we'll call it ds with
a little vector over it

to know that this thing.

This isn't the scalar ds that
is just concerned with the area.

But when you view
things this way,

we just saw that this entire
thing simplifies to ds.

Then our whole surface
integral can be rewritten.

Instead of writing
it like this, we

can write it as the
integral or the surface

integral-- those integral
signs were too fancy.

The surface integral of F dot.

And instead of saying a
normal vector times the scalar

quantity, that little chunk
of area on the surface,

we can now just call that
the vector differential ds.

And I want to make it clear,
these are two different things.

This is a vector.

This is essentially
what we're calling it.

This right over here is a
scalar times a normal vector.

So these are three
different ways

of really representing
the same thing.

And in different contexts,
you will see different things,

depending on what the author of
whoever's trying to communicate

is trying to communicate.

This right over here is the one
that we'll use most frequently

as we actually try to calculate
these surface integrals.