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In the last video,
we figured out

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how to construct a unit
normal vector to a surface.

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

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integral to try to
simplify a little bit,

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

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And also, think
about different ways

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to represent this type
of a surface integral.

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

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our unit normal
vector right here,

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

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the surface integral of F dot.

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And F dot all of this
business right over here.

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

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

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So the partial of
r with respect to u

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crossed with the partial
of r with respect

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to v over the magnitude
of the same thing, partial

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

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with respect to v.

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

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We know that the other
way to write ds--

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and I gave the
intuition, hopefully,

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for that several videos
ago when we first

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explored what a surface
integral was all about.

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We know that ds is--
it can be represented

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as the magnitude of the
partial of r with respect

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

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

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

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and the uv plane or
in the uv domain.

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

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we're no longer taking
a surface integral.

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

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So you could say kind
of a region in uv.

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

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that we're now thinking about.

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

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

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popping out at you right now.

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We're dividing by the
magnitude of the cross product

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of these two vectors
and then we're

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multiplying by the magnitude of
the cross product of these two

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vectors.

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Those are just
scalar quantities.

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You divide by something
and multiply by something.

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

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by 1.

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So these two
characters cancel out,

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and our integral simplifies
to the double integral

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over that region, the
corresponding region in the uv

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plane, of F-- of
our vector field F

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dotted with this cross product.

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

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That's going to
give us a vector.

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It gives us actually
a normal vector.

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And then when you
divide by its magnitude,

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it gives you a
unit normal vector.

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

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with r, the partial
of r with respect

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

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

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And we'll see in the
few videos from now

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that this is essentially how we
go about actually calculating

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these things.

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If you have a
parameterization, you

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can then get everything in
terms of a double integral,

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in terms of uv this way.

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

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

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It all comes from,
really, writing this part

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in a different way.

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But it hopefully gives you
a little bit more intuition

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of what this thing
is even saying.

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So I'm just going to rewrite.

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

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I'm just going to
rewrite that chunk.

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

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will hopefully help make
a little bit more sense.

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So the partial of r
with respect to u I

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

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And we're taking
the cross product.

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

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with v's.

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And we're taking the
cross product of that

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with the partial
of r with respect

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

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in our parameterization
right here,

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our position vector given
a small change in v. Very

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

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And then we're multiplying
that times du dv.

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We're multiplying
that times du dv.

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Now, du and dv are
just scalar quantities.

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They're infinitesimally small.

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But for the sake
of this argument,

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

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they're just scalar quantities.

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And so you can
essentially include them--

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if you have the cross product.

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

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you could rewrite this
as x times a cross b,

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or you could write
this as a cross x times

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b, because x is
just a scalar value.

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It's just a number.

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So we could do the
same thing over here.

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We can rewrite all
of this business as--

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

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or with respect to u
in the denominator.

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

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And so you will get the
partial of r with respect

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to u times du,
times that scalar.

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So that'll give us a vector.

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And we're going to cross that.

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

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to v dv.

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Now, these might
look notationally

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like two different
things, but that just

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comes from the
necessity of when we

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

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

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and this is taking a
derivative with respect

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to only one of them.

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So this is, how
much does our vector

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change when you have a
very small change in u?

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But this is also an
infinitesimally small change

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

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

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loosey-goosey mathematics,
but it will hopefully

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give you the intuition
for why this thing could

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be written in a different way.

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These are essentially
the same quantity.

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So if you divide by something
and multiply by something,

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you can cancel them out.

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If you divide by something
and multiply by something,

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you can cancel them out.

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And all you're left
with then-- all

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you're left with is
the differential of r.

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And since we lost
the information

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

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

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I don't want to get
the notation confused.

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This is just the differential.

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This is just how much r changed.

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This is not the partial
derivative of r with respect

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to u.

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This right over here
is, how much does r

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

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

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

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change that just r changes.

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

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And we're going to cross that.

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

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the partial of r
in the v-direction.

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

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And this goes back to
our original visions

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of what a surface
integral was all about.

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

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Let me draw another surface.

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

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

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

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

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You're going in that direction.

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So if that thing
looks like this,

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

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Because remember, this
isn't the derivative.

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This is the differential.

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

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that's that over there.

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

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it's also a change
along the surface.

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When you take the cross
product of these two things,

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you get a vector
that is orthogonal.

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You get a vector that is
normal to the surface.

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

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

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Its magnitude is
equal to the area that

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is defined by these two vectors.

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So its magnitude
is equal to area.

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So in a lot of
ways, you can really

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think of it-- you
really could think

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of it as a unit normal
vector times ds.

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

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

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but it's a vector
version of the ds.

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

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This is just a scalar value.

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But now, we have a vector
that points normally

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from the surface,
but its magnitude

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is the same thing as
that ds that we were just

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talking about.

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

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

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

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to know that this thing.

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

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But when you view
things this way,

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we just saw that this entire
thing simplifies to ds.

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Then our whole surface
integral can be rewritten.

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Instead of writing
it like this, we

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can write it as the
integral or the surface

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integral-- those integral
signs were too fancy.

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The surface integral of F dot.

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And instead of saying a
normal vector times the scalar

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quantity, that little chunk
of area on the surface,

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we can now just call that
the vector differential ds.

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

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This is a vector.

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This is essentially
what we're calling it.

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

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So these are three
different ways

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of really representing
the same thing.

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And in different contexts,
you will see different things,

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

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is trying to communicate.

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

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as we actually try to calculate
these surface integrals.

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