Surface integral ex3 part 3: Top surface

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Kind: captions
Language: en

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- [Instructor] We're
now in the home stretch.

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We just need to evaluate
this third surface integral,

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which is over this top part
of our little chopped cylinder

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right over here.

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So let's try to think
of a parameterization,

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and let me just copy and
paste this entire drawing,

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just so that I can use it down below

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as we parameterize it.

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So let me copy it, copy, and
then go all the way down here

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and let me paste it.

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Let me paste it, okay, that is
our shape again, our surface.

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And then we go to the layer
that I wanted to get on.

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And then let me start,
let's start evaluating it.

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So what we wanna care
about is the integral

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over surface three of z d s.

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And surface three here, we
see that the x and y values,

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and since you take on all
the possible x and y values,

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inside of the unit circle,
including the boundary,

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and then the z values
are gonna be a function

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of the x values.

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We know that this plane,

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that this top surface right over here, S3,

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it is a subset of the plane z.

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Z is equal to one minus x.

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It's the subset that's kind
of above the unit circle

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in the xy-plane, or kind of the subset

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that intersects with our
cylinder and kind of chops it.

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So let's think about x and y's first.

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Let's think about it in
terms of polar coordinates

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because that's probably the
easiest way to think about it.

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I'm gonna redraw kind of a top view.

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So I'm gonna redraw a top
view, so that is my y-axis

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and this is my x-axis.

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X-axis.

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And the x's and y's can take on any value,

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they said you have to
fill the unit circle.

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So if you were to kind of
project this top surface down

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onto the xy-plane, you would
get this orange surface,

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that bottom surface,
which looked like this.

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It was essentially the
unit circle just like that.

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And I'm gonna redraw it a
little bit neater than that.

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I can do a better job, so
that would be, all right.

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So let me draw the unit
circle as neatly as I can.

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So there's my unit circle.

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And so we can have one parameter.

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We can have one parameter
that essentially says

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how far around the unit
circle we're going.

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Essentially that would be our angle

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and let's use theta, 'cause
that's, well just for fun.

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We haven't used theta as a parameter yet.

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That's theta, but if we had x's and y's

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as just a function of theta
and we had a fixed radius,

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that would essentially
just give us the points

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on the outside of the unit circle.

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But we need to be able
to have all of the xy's

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that're outside and
inside the unit circle.

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So we actually have to
have two parameters.

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We need to not only vary this angle,

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but we also need to vary the radius.

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So we would wanna trace out
the outside of that unit circle

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and maybe we'd wanna
shorten it a little bit.

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And then trace it out again.

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And then shorten it some more.

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And then trace it out again.

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And so you wanna actually
have a variable radius as well

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so that you can have how
far out you're going.

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You could call that r.

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So for example if r is fixed

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and you change your ranges of theta,

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then you would essentially
get all of those points

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right over there and you would
do that for all of the r's

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and from r zero all the way to r one

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and you would essentially fill
up the entire unit circle.

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Let's do that, so r is going
to go between zero and one.

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R is going to be between zero and one.

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And our theta, our theta's
gonna go all the way around.

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So our theta is going to
go between zero and two pi.

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This is, lemme write this down,

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I wrote zero instead of theta.

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Our theta is going to be
greater than or equal to zero,

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less than or equal to two pi.

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And now we're ready to parameterize it.

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X of r and theta is going to be equal to,

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so whatever r is, it's
going to be r cosine theta.

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So x is going to be r cosine theta.

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Y is going to be r sine theta.

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That's going to be the y value.

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R sine theta.

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And now z is essentially
just a function of x.

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Z is going to be equal to one minus x,

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but x is just r cosine theta.

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So there you have it.

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We have our parameterization

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of this surface right over here.

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The x's and y's can take all
the values on the unit circle,

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but then the z is up here
based on a function of,

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well based on really on a function of x.

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It's one minus x.

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So that would give us all
of the possible points

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right over here on the surface.

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You pick an x and y and then
the z is going to pop us

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right here someplace on that surface.

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And we can write it as a
position vector function.

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Instead of calling that
position vector function r,

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since we've already used r for radius,

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I will call it, I don't
know, let's call it,

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I'm just gonna pick a random letter here,

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let's call it p for
position vector function.

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And so p, our surface p
is, we can write it as,

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actually no why don't I
just call it surface three.

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So surface three, surface three,

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I'll do it in that same purple color too,

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just so we know we're talking about this,

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surface three as a
position vector function,

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as a function of theta and r,
maybe I'll write r and theta

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'cause that's how I think of things.

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R and theta is going to be
equal to r cosine theta i

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plus r sine of theta j plus
one minus r cosine theta,

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need to get some real estate here,

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one minus r cosine of theta k.

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One minus r cosine of theta k.

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And now we are ready to start
doing all of the business

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of evaluating the actual surface integral.

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So the first thing we need to
do is take the cross product

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of this, the partial of
this with respect to r,

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and the partial of this
with respect to theta.

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Let's just get down to business.

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Let's take the cross product,
let's take the cross product

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and so we have our i unit vector,
we have our j unit vector,

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and we have our k unit vector.

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And this might get a little bit involved

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but we'll try our best
to just work through it.

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Give myself a little bit more space.

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And so the partial of
this with respect to r,

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so let's take the partial
of this with respect to r,

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I'll do it in blue, the partial
of this with respect to r,

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the partial of this with respect
to r is just cosine theta i

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so this is just cosine theta,

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I said I was gonna do it in
blue and that's not blue,

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so this is going to be
cosine, cosine theta.

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The partial of this with
respect to r is sine theta.

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Sine theta.

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And the partial of this
term right over here

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with respect to r is
negative cosine theta.

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Negative cosine theta.

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Now let's take the partial
with respect to theta.

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The partial of this with respect

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to theta is negative r sine of theta.

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Negative r sine of theta.

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The partial of this with respect
to theta is r cosine theta.

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R cosine theta.

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And the partial here, this is zero,

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and then this would be
negative r sine theta.

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Negative r, oh no let me be careful.

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This is going to be, you have a negative r

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so the derivative of cosine
theta with respect to theta is

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negative sine theta.

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So the negatives are gonna cancel out

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and so you're going to have
r, r sine, r sine theta.

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And now we can actually
evaluate that this determinant

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to figure out the cross product

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of the partial of this with respect to r

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and the partial of this
with respect to theta.

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I'm not writing it down,

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just to kind of save
some real estate here.

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And so we have, actually
maybe I will write it down

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just to be clear what we're doing.

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The partial of s three
with respect to r crossed

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with the partial of s three

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with respect to theta is equal to,

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now our i component is gonna be

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sine theta times r sine theta,

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so that's going to be
r sine squared theta,

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minus r cosine theta times
negative cosine theta.

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So that's plus r cosine squared theta.

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All of that times i, and you already,

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a simplification might be
popping out here at you,

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and then you have minus the j component.

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The j component is gonna be
cosine theta times r sine theta.

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So it's r cosine theta sine theta.

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And then we're gonna subtract from that.

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See the negative signs cancel out.

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So you're gonna subtract
r sine theta cosine theta,

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or r cosine theta sine theta.

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Well this is interesting

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because these are the
negatives of each other.

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R cosine theta sine theta minus
r cosine theta sine theta.

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This just evaluates to zero.

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So we have no j component.

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And then finally for our k component,

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we have cosine theta times r cosine theta.

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So we have r cosine squared theta

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minus r sine theta times sine theta.

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Or minus negative r sine
theta times sine theta.

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So this would give you a negative

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but we're gonna have to subtract

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until it gives you a positive.

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So plus r sine squared theta k.

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And so this simplifies quite nicely

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because this is going to be equal to,

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this term up here you can factor out an r.

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This is r times the sine squared theta

00:09:23.690 --> 00:09:27.280
plus cosine squared theta, which is just,

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that part simplifies to one
so that's just r times i.

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So this is equal to r times i.

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And we do the same thing over here.

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This also simplifies.

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This is actually the same thing.

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This also simplifies to r.

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So this whole thing simplifies,
lemme write it this way,

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this is also r sine squared
theta plus cosine squared theta.

00:09:47.620 --> 00:09:48.930
Also simplifies to r.

00:09:48.930 --> 00:09:52.420
So you have r times k, and
so if you want the magnitude

00:09:52.420 --> 00:09:55.010
of this business, so let me make it clear,

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so the magnitude, we'll
go back to the magenta,

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the magnitude of, I don't
feel like rewriting it all,

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just copy and paste it.

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Edit copy and paste.

00:10:06.190 --> 00:10:10.740
The magnitude of all of
this business is going to be

00:10:10.740 --> 00:10:14.730
equal to the square root of this squared,

00:10:14.730 --> 00:10:18.520
which is r squared, plus this
squared, which is r squared,

00:10:18.520 --> 00:10:21.420
which simplifies to,
this is two r squared.

00:10:21.420 --> 00:10:23.781
So you take the square
root of both of those,

00:10:23.781 --> 00:10:28.781
you get the square root of two times r.

00:10:29.300 --> 00:10:34.300
So this is equal to the
square root of two times,

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it would be the absolute value of r

00:10:36.170 --> 00:10:38.740
but we know that r only
takes on positive values.

00:10:38.740 --> 00:10:40.530
R only takes on positive values

00:10:40.530 --> 00:10:44.070
so it's the square root of two times r,

00:10:44.070 --> 00:10:47.420
which is very nice because
now we can evaluate d s.

00:10:47.420 --> 00:10:52.420
D s is going to be this
business times d r d theta.

00:10:52.420 --> 00:10:53.830
So let's do it.

00:10:53.830 --> 00:10:55.040
So our surface integral,

00:10:55.040 --> 00:10:57.300
the thing that we were dealing
with from the beginning,

00:10:57.300 --> 00:10:59.340
that thing right over there.

00:10:59.340 --> 00:11:04.340
Our surface integral, the
surface s three, of z d s is now

00:11:05.963 --> 00:11:08.650
equal to, so I'm gonna
use different colors

00:11:08.650 --> 00:11:10.750
for the different
variables of integration,

00:11:10.750 --> 00:11:14.290
so one on the outside, and
then I'll do one on the inside,

00:11:14.290 --> 00:11:16.283
I'll do the inside one in pink.

00:11:17.280 --> 00:11:21.460
Z is equal to one minus r cosine theta.

00:11:21.460 --> 00:11:24.840
So the z is equal to one
minus r cosine theta.

00:11:24.840 --> 00:11:26.600
It involves both so I'll
use a different color.

00:11:26.600 --> 00:11:29.850
Minus r cosine theta.

00:11:29.850 --> 00:11:31.320
And then I just have to integrate

00:11:31.320 --> 00:11:34.580
relative to both of the variables.

00:11:34.580 --> 00:11:38.530
One minus r cosine theta, oh
no I have to do the times d s.

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D s is this thing, it's this thing,

00:11:41.870 --> 00:11:46.070
times d theta d, d r.

00:11:46.070 --> 00:11:47.410
So let's see, let's write this out.

00:11:47.410 --> 00:11:52.190
Times square root of two r,
so let's write that down,

00:11:52.190 --> 00:11:55.220
so this times the square root of two r,

00:11:55.220 --> 00:11:57.660
times the square root of two,

00:11:57.660 --> 00:11:59.160
and we can write the square
root of two out front

00:11:59.160 --> 00:12:00.750
since it's a constant
so lemme just do that,

00:12:00.750 --> 00:12:01.600
simplifies things.

00:12:01.600 --> 00:12:06.600
Square root of two times
r, and now d theta d r.

00:12:07.290 --> 00:12:09.110
Or we can write d r d theta, either way.

00:12:09.110 --> 00:12:10.380
So let's do that.

00:12:10.380 --> 00:12:12.940
D, let's do d r d theta.

00:12:12.940 --> 00:12:16.270
So d r d theta, we could do it either way,

00:12:16.270 --> 00:12:18.410
it's gonna be about the
same level of complexity.

00:12:18.410 --> 00:12:21.400
And so first we're going to
integrate with respect to r,

00:12:21.400 --> 00:12:23.680
and lemme do the colors
the same way, actually.

00:12:23.680 --> 00:12:25.633
So d r, d theta.

00:12:29.160 --> 00:12:31.400
If I've got colors I
might as well use them.

00:12:31.400 --> 00:12:33.680
And actually I just realized
that I'm way out of time.

00:12:33.680 --> 00:12:36.210
So actually, let me continue
this in the next video.

00:12:36.210 --> 00:12:38.190
We'll set up the boundaries of integration

00:12:38.190 --> 00:12:40.083
and then just evaluate it.


Surface integral ex3 part 1: Parameterizing the outside surface

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Kind: captions
Language: en

00:00:00.500 --> 00:00:02.072
Let's try another
surface integral.

00:00:02.072 --> 00:00:04.030
And the surface that
we're going to care about,

00:00:04.030 --> 00:00:08.070
s, is this shape, the outside
of this shape right over here.

00:00:08.070 --> 00:00:09.940
And you can see we can
kind of decompose it

00:00:09.940 --> 00:00:12.010
into three separate surfaces.

00:00:12.010 --> 00:00:14.760
The first surface is its
base, which is really

00:00:14.760 --> 00:00:18.080
the filled-in unit
circle down here.

00:00:18.080 --> 00:00:19.946
The second surface,
which we have in blue,

00:00:19.946 --> 00:00:21.820
is you can kind of view
it as the side of it.

00:00:21.820 --> 00:00:24.480
You can view it as kind of
like the side of a cylinder,

00:00:24.480 --> 00:00:27.550
but the cylinder has been
cut by a plane up here.

00:00:27.550 --> 00:00:31.970
And the plane that cuts
it is the plane z is equal

00:00:31.970 --> 00:00:33.350
to 1 minus x.

00:00:33.350 --> 00:00:36.820
And obviously, the plane itself
goes well beyond this shape.

00:00:36.820 --> 00:00:41.290
But where that plane
cuts the cylinder kind of

00:00:41.290 --> 00:00:43.010
defines this shape.

00:00:43.010 --> 00:00:47.250
So the blue surface is above
the boundary of the unit circle

00:00:47.250 --> 00:00:48.870
and below the plane.

00:00:48.870 --> 00:00:51.670
And then the third
surface is the subset

00:00:51.670 --> 00:00:53.900
of the plane, of that
purple plane, of z

00:00:53.900 --> 00:00:57.200
equals 1 minus x, that
overlaps, that kind of forms

00:00:57.200 --> 00:00:59.440
the top of this cylinder.

00:00:59.440 --> 00:01:04.670
And so we can rewrite
this surface integral

00:01:04.670 --> 00:01:08.250
as the sum of three
surface integrals.

00:01:08.250 --> 00:01:14.920
It's the surface
integral of z over s1

00:01:14.920 --> 00:01:22.120
plus the surface
integral of z over s2

00:01:22.120 --> 00:01:29.350
plus the surface
integral of z over s3.

00:01:29.350 --> 00:01:32.470
And we can just tackle each
of these independently.

00:01:32.470 --> 00:01:34.270
So let's start with surface 1.

00:01:34.270 --> 00:01:37.080
And you might immediately want
to start parameterizing things

00:01:37.080 --> 00:01:39.690
and all the rest, but there's
actually a very fast way

00:01:39.690 --> 00:01:42.370
to handle this surface integral,
especially because we're

00:01:42.370 --> 00:01:45.000
taking the surface
integral of z.

00:01:45.000 --> 00:01:50.130
What value does z take on
throughout this surface,

00:01:50.130 --> 00:01:55.220
throughout this filled-in
unit circle right over here?

00:01:55.220 --> 00:01:58.380
Well, that surface
is on the xy plane.

00:01:58.380 --> 00:02:02.800
When we're on the xy
plane, z is equal to 0.

00:02:02.800 --> 00:02:07.280
So for this entire surface,
z is going to be equal to 0.

00:02:07.280 --> 00:02:08.799
You're essentially
integrating 0.

00:02:08.799 --> 00:02:11.600
0 times dS is just
going to be 0.

00:02:11.600 --> 00:02:15.800
So this whole thing is
going to evaluate to 0.

00:02:15.800 --> 00:02:19.000
So that's about as simple as
evaluating a surface integral

00:02:19.000 --> 00:02:19.610
can get.

00:02:19.610 --> 00:02:21.693
But it's always important
to least keep a look out

00:02:21.693 --> 00:02:22.590
for things like that.

00:02:22.590 --> 00:02:25.400
And it'll keep you from kind
of going down this wild goose

00:02:25.400 --> 00:02:26.700
chase-- that wouldn't
be a wild goose chase.

00:02:26.700 --> 00:02:28.550
You would eventually
get the answer anyway.

00:02:28.550 --> 00:02:30.280
But you don't have
to waste so much time

00:02:30.280 --> 00:02:32.634
in parameterizing
things and all the rest.

00:02:32.634 --> 00:02:34.300
Now let's tackle the
other two surfaces,

00:02:34.300 --> 00:02:39.030
and we'll focus on surface 2.

00:02:39.030 --> 00:02:42.110
So the x and y-values,
the valid x and y-values,

00:02:42.110 --> 00:02:46.630
are the x and y-values along
the unit circle right over here.

00:02:46.630 --> 00:02:48.610
So we can really
parameterize it the way

00:02:48.610 --> 00:02:51.330
that we would parameterize
a traditional unit circle.

00:02:51.330 --> 00:02:55.660
So we could set x is
equal to-- let's set it--

00:02:55.660 --> 00:02:58.380
and our radius is 1,
so x is equal to cosine

00:02:58.380 --> 00:03:01.510
of-- I'll use the parameter u.

00:03:01.510 --> 00:03:03.330
x is equal to cosine of u.

00:03:03.330 --> 00:03:07.410
And then let's say y
is equal to sine of u.

00:03:07.410 --> 00:03:13.240
And then u is the angle
between the positive x-axis

00:03:13.240 --> 00:03:16.810
and wherever we are essentially
on that unit circle.

00:03:16.810 --> 00:03:20.570
So that right over there, that
angle right over there is u.

00:03:20.570 --> 00:03:23.070
And so this would
give us for u--

00:03:23.070 --> 00:03:27.290
so as long as u is
between 0 and 2 pi,

00:03:27.290 --> 00:03:29.880
that we're essentially going
around this unit circle.

00:03:29.880 --> 00:03:32.320
So those are all the
possible x and y-values

00:03:32.320 --> 00:03:33.940
that we can take on.

00:03:33.940 --> 00:03:37.920
And then the z-value is what
takes us up above that boundary

00:03:37.920 --> 00:03:40.250
and gets it someplace
along the surface.

00:03:40.250 --> 00:03:42.300
But this is interesting,
because the z-value,

00:03:42.300 --> 00:03:44.590
it can take on obviously a
bunch of different values,

00:03:44.590 --> 00:03:48.270
but it always has to be below
this plane right over here.

00:03:48.270 --> 00:03:51.560
So for this z-value, I'm going
to introduce a new parameter.

00:03:51.560 --> 00:03:55.280
Let's call it z v. That's
the second parameter.

00:03:55.280 --> 00:03:59.530
And v is definitely going
to be greater than 0.

00:03:59.530 --> 00:04:01.660
z and v, they're the same thing.

00:04:01.660 --> 00:04:03.640
They're always going
to be positive,

00:04:03.640 --> 00:04:07.110
so they're definitely going to
be greater than or equal to 0.

00:04:07.110 --> 00:04:09.550
But it's not less than or
equal to some constant thing.

00:04:09.550 --> 00:04:11.884
This has kind of a
variable roof on it.

00:04:11.884 --> 00:04:13.550
And so it's always
going to be less than

00:04:13.550 --> 00:04:18.959
or equal to this
plane right over here.

00:04:18.959 --> 00:04:23.020
And so we could say is less
than or equal to 1 minus x.

00:04:23.020 --> 00:04:26.140
We could have said-- so we
know that z is less than

00:04:26.140 --> 00:04:30.270
or equal to 1 minus x, but if we
use the new parameters, v is z,

00:04:30.270 --> 00:04:33.746
and 1 minus x is the same
thing as 1 minus cosine u.

00:04:38.030 --> 00:04:40.560
And so now we have
our parameterization.

00:04:40.560 --> 00:04:44.470
We are ready to actually
evaluate the surface integral.

00:04:44.470 --> 00:04:47.320
And to do that, first
let's do the cross product.

00:04:47.320 --> 00:04:49.250
We want to figure
out what dS is,

00:04:49.250 --> 00:04:51.530
and we have to take the
magnitude of the cross product

00:04:51.530 --> 00:04:53.113
of the partial of
our parameterization

00:04:53.113 --> 00:04:55.490
with respect to u, crossed
with the partial with respect

00:04:55.490 --> 00:04:56.900
to v. Actually, let me just
write all this stuff down

00:04:56.900 --> 00:04:58.800
instead of trying
to do shortcuts.

00:04:58.800 --> 00:05:04.150
So dS, and this is all--
let me do it in blue still

00:05:04.150 --> 00:05:05.900
because we're talking
about surface 2.

00:05:05.900 --> 00:05:11.530
dS is going to be equal to the
magnitude of the cross product

00:05:11.530 --> 00:05:15.770
of the partial of our
parameterization with respect

00:05:15.770 --> 00:05:18.590
to u, crossed with our partial
of our parameterization

00:05:18.590 --> 00:05:23.517
with respect to v, du dv.

00:05:23.517 --> 00:05:25.600
So let's write the partial
of our parameterization

00:05:25.600 --> 00:05:26.420
with respect to u.

00:05:26.420 --> 00:05:27.890
And you might say, wait,
where's our parameterization?

00:05:27.890 --> 00:05:28.890
Well, it's right over here.

00:05:28.890 --> 00:05:31.080
I just didn't write it in
the traditional ijk form,

00:05:31.080 --> 00:05:31.750
but I can.

00:05:31.750 --> 00:05:35.310
I could write r is
equal to-- maybe

00:05:35.310 --> 00:05:38.735
I call it r2 because we're
talking about surface 2.

00:05:38.735 --> 00:05:40.110
I shouldn't use
that orange color

00:05:40.110 --> 00:05:41.526
because I used
that for surface 1.

00:05:41.526 --> 00:05:43.360
I'll use that in a
still bluish color.

00:05:43.360 --> 00:05:49.660
So r for surface 2
is equal to cosine

00:05:49.660 --> 00:06:00.240
of ui plus the
sine of uj plus vk.

00:06:00.240 --> 00:06:03.680
And this is the range that the
u's and the v's can take on.

00:06:03.680 --> 00:06:07.095
If I want to take the partial
of r with respect to u,

00:06:07.095 --> 00:06:09.120
I would get-- let's
see, the derivative

00:06:09.120 --> 00:06:13.750
of cosine u with respect
to u is negative sine ui.

00:06:13.750 --> 00:06:19.600
The derivative of sine of u with
respect to u is plus cosine uj.

00:06:19.600 --> 00:06:22.660
And the derivative of v
with respect to u is just 0.

00:06:22.660 --> 00:06:24.740
So that's our partial
with respect to u.

00:06:24.740 --> 00:06:27.090
Our partial-- let' me do
this in a different color

00:06:27.090 --> 00:06:29.480
just for the sake of it.

00:06:29.480 --> 00:06:35.210
Our partial with respect
to v is equal to-- well,

00:06:35.210 --> 00:06:37.240
this is going to be 0,
this is going to be 0,

00:06:37.240 --> 00:06:38.870
and we're just
going to have one k.

00:06:38.870 --> 00:06:41.090
So it's just going
to be equal to k.

00:06:41.090 --> 00:06:42.960
And so now we're ready
to at least evaluate

00:06:42.960 --> 00:06:44.740
this cross product.

00:06:44.740 --> 00:06:46.270
So we'll evaluate
the cross product,

00:06:46.270 --> 00:06:48.050
and then we'll take
the magnitude of that,

00:06:48.050 --> 00:06:51.790
so just this part, so
just the cross product,

00:06:51.790 --> 00:06:54.810
the inside of that, before
we take the magnitude.

00:06:54.810 --> 00:06:57.740
So copy and paste.

00:06:57.740 --> 00:07:05.940
So just that is going to be
equal to the determinant, the 3

00:07:05.940 --> 00:07:06.770
by 3 determinant.

00:07:06.770 --> 00:07:11.290
Let me write my
components, i, j, k.

00:07:11.290 --> 00:07:14.190
And then for r sub
u, we have this.

00:07:14.190 --> 00:07:16.450
Our i component is
negative sine of u.

00:07:19.164 --> 00:07:23.510
Our j component is cosine of u.

00:07:23.510 --> 00:07:26.540
And it has no k component,
so we'll put a 0 right there.

00:07:26.540 --> 00:07:30.810
And then r sub v, no i
component, no j component,

00:07:30.810 --> 00:07:34.630
and it has 1 for
its k coefficient.

00:07:34.630 --> 00:07:37.100
And so we can just
evaluate this.

00:07:37.100 --> 00:07:40.790
And so first, we think
about the i component.

00:07:40.790 --> 00:07:42.830
Well, we ignore that
column, that row.

00:07:42.830 --> 00:07:45.630
It's going to be
cosine of u minus 0.

00:07:45.630 --> 00:07:49.190
So it's going to
be cosine of ui.

00:07:49.190 --> 00:07:52.830
And then we'll have minus
j times-- ignore column,

00:07:52.830 --> 00:07:56.810
ignore row-- negative
sine of u times 1 minus 0.

00:07:56.810 --> 00:07:59.590
So it's negative j times
negative sine of u.

00:07:59.590 --> 00:08:04.670
So it's just going to
be plus sine of uj.

00:08:04.670 --> 00:08:08.711
And then for the k, we have
this times 0 minus this times 0.

00:08:08.711 --> 00:08:09.960
So we're just going to have 0.

00:08:09.960 --> 00:08:11.740
So this is the cross product.

00:08:11.740 --> 00:08:13.900
And if we want to take
the magnitude of this,

00:08:13.900 --> 00:08:16.840
if we just take the
magnitude of this--

00:08:16.840 --> 00:08:19.320
so now we're ready to take
the magnitude of this.

00:08:19.320 --> 00:08:22.650
It's going to be equal
to-- the magnitude of this

00:08:22.650 --> 00:08:26.220
is going to be equal to
the square root of cosine

00:08:26.220 --> 00:08:32.039
of u squared plus
sine of u squared.

00:08:32.039 --> 00:08:33.539
There is no k component.

00:08:33.539 --> 00:08:36.200
And from the unit circle,
this is the most basic trig

00:08:36.200 --> 00:08:36.700
identity.

00:08:36.700 --> 00:08:39.510
This is just going
to be equal to 1.

00:08:39.510 --> 00:08:44.150
And so we have this part right
over here just simplifies to 1,

00:08:44.150 --> 00:08:45.460
which is nice.

00:08:45.460 --> 00:08:50.570
And dS simplifies to du dv
for at least this surface.

00:08:50.570 --> 00:08:53.120
And so now we are
ready to attempt

00:08:53.120 --> 00:08:54.732
to evaluate the integral.

00:08:54.732 --> 00:08:56.190
But I'm almost
running out of time,

00:08:56.190 --> 00:08:58.991
so I will do that
in the next video.


Surface integral ex2 part 2: Evaluating integral

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:03.880
Now that we've set up
the parametrization,

00:00:03.880 --> 00:00:05.680
let's try to evaluate
the integral.

00:00:05.680 --> 00:00:07.430
And the next thing
we'll do is essentially

00:00:07.430 --> 00:00:12.910
try to express ds right over
here in terms of du and dv.

00:00:12.910 --> 00:00:14.170
And we've seen this before.

00:00:14.170 --> 00:00:19.160
ds is going to be equal to the
magnitude of the cross product

00:00:19.160 --> 00:00:22.470
of the partial of r with
respect to u crossed

00:00:22.470 --> 00:00:28.270
with the partial of r
with respect to v du dv.

00:00:28.270 --> 00:00:29.980
So first, let's take
the cross product,

00:00:29.980 --> 00:00:32.580
and we'll do that
with a 3 by 3 matrix.

00:00:32.580 --> 00:00:36.960
I'll do that right over here.

00:00:36.960 --> 00:00:41.630
We set up, and I'll just kind
of fill in what r sub u and r

00:00:41.630 --> 00:00:45.310
sub v is in the actual
determinant right over here.

00:00:45.310 --> 00:00:50.560
So first we have our
components, i, j, and k.

00:00:50.560 --> 00:00:54.450
And now first, let's
think about what

00:00:54.450 --> 00:00:58.852
r sub u is, the partial
of r with respect to u.

00:00:58.852 --> 00:01:01.570
Well, its i component
is going to be 1.

00:01:01.570 --> 00:01:03.800
The partial of u with
respect to u is just 1,

00:01:03.800 --> 00:01:05.980
so its i component is 1.

00:01:05.980 --> 00:01:07.690
Its j component
is going to be 0,

00:01:07.690 --> 00:01:10.490
partial of v with
respect to u is

00:01:10.490 --> 00:01:12.990
0. v does not change
with respect to u,

00:01:12.990 --> 00:01:16.010
so this is going to be 0.

00:01:16.010 --> 00:01:17.680
This should be
parentheses around here.

00:01:17.680 --> 00:01:21.470
Partial of this with respect to
u is just going to be 1 again.

00:01:21.470 --> 00:01:24.150
The partial of v squared
with respect to u is just 0.

00:01:24.150 --> 00:01:26.420
So this is just 1 again.

00:01:26.420 --> 00:01:30.860
And then r sub v, the partial
of r with respect to v,

00:01:30.860 --> 00:01:34.020
the i component
is going to be 0,

00:01:34.020 --> 00:01:36.980
j component is going to
be 1, and the partial

00:01:36.980 --> 00:01:42.530
of u plus v squared with
respect to v is going to be 2v.

00:01:42.530 --> 00:01:45.360
So it's a pretty
straightforward determinant,

00:01:45.360 --> 00:01:46.540
so let's try to evaluate it.

00:01:46.540 --> 00:01:49.275
So the i component,
it's going to be i.

00:01:49.275 --> 00:01:52.160
So we cross out this
column, this row.

00:01:52.160 --> 00:01:57.170
It's going to be 0 times
2v minus 1 times 1.

00:01:57.170 --> 00:02:00.800
So essentially, it's just
going to be negative 1 times i.

00:02:00.800 --> 00:02:02.610
So we're going to
have negative i,

00:02:02.610 --> 00:02:06.010
so this is equal to negative i.

00:02:06.010 --> 00:02:07.748
And then the j
component-- and we're

00:02:07.748 --> 00:02:10.289
going to have to put a negative
out front, because, remember,

00:02:10.289 --> 00:02:11.930
we do that checkerboard pattern.

00:02:11.930 --> 00:02:14.000
So cross out that
row, that column.

00:02:14.000 --> 00:02:16.474
1 times 2v is 2v.

00:02:16.474 --> 00:02:17.640
Let me make sure I got that.

00:02:17.640 --> 00:02:20.660
1 times 2v is 2v
minus 0 times 1.

00:02:20.660 --> 00:02:23.230
So it's just going to be 2v.

00:02:23.230 --> 00:02:25.709
But since it was the
j component, which

00:02:25.709 --> 00:02:28.000
is going to be negative, it's
going to be negative 2vj.

00:02:30.650 --> 00:02:32.290
Let me make sure
I did that right.

00:02:32.290 --> 00:02:35.820
That column, that row, 1
times 2v is 2v minus 0 is 2v.

00:02:35.820 --> 00:02:38.390
The checkerboard pattern,
you'd have a negative j.

00:02:38.390 --> 00:02:40.800
So you have negative 2vj.

00:02:40.800 --> 00:02:43.900
And then we have
find the k component.

00:02:43.900 --> 00:02:47.110
Cross out that row, that
column, 1 times 1 minus 0.

00:02:47.110 --> 00:02:50.390
So it's going to be plus k.

00:02:50.390 --> 00:02:52.996
So if we want the
magnitude of this,

00:02:52.996 --> 00:02:55.400
this whole thing right
over here is just

00:02:55.400 --> 00:03:00.070
going to be the
square root-- I'm just

00:03:00.070 --> 00:03:01.780
taking the magnitude
of this part right

00:03:01.780 --> 00:03:04.060
over here, the
actual cross product.

00:03:04.060 --> 00:03:05.980
It's going to be
negative 1 squared, which

00:03:05.980 --> 00:03:09.320
is just 1, plus negative
2v squared, which

00:03:09.320 --> 00:03:13.810
is 4v squared, plus 1
squared, which is just 1.

00:03:13.810 --> 00:03:16.230
So this whole thing
is going to evaluate

00:03:16.230 --> 00:03:29.700
to-- we have 2 plus
2v squared du dv.

00:03:29.700 --> 00:03:31.807
Or actually, let me-- I
almost made a mistake.

00:03:31.807 --> 00:03:33.140
That would have been a disaster.

00:03:33.140 --> 00:03:39.940
2 plus 4v squared du dv.

00:03:39.940 --> 00:03:41.690
And if we want,
maybe it'll help us

00:03:41.690 --> 00:03:44.630
a little bit if we factor
out a 2 right over here.

00:03:47.340 --> 00:03:54.580
This is the same thing as 2
times 1 plus 2v squared du dv.

00:03:54.580 --> 00:03:56.650
If you factor out
the 2, you get this

00:03:56.650 --> 00:04:03.960
is equal to the square root of
2 times the square root of 1

00:04:03.960 --> 00:04:08.670
plus 2v squared du dv.

00:04:08.670 --> 00:04:12.550
And I now think we are ready to
evaluate the surface integral.

00:04:12.550 --> 00:04:14.090
So let's do it.

00:04:14.090 --> 00:04:15.250
All right.

00:04:15.250 --> 00:04:18.168
So let me just write
this thing down here.

00:04:18.168 --> 00:04:20.459
So I'm going to write everything
that has to do with v,

00:04:20.459 --> 00:04:22.410
I'm going to write in purple.

00:04:22.410 --> 00:04:25.620
So I'm just going to write
the ds part right over here.

00:04:25.620 --> 00:04:32.310
It is the square root of 2
times the square root of 1

00:04:32.310 --> 00:04:35.960
plus 2v squared.

00:04:35.960 --> 00:04:39.040
And then we're going
to have du, which

00:04:39.040 --> 00:04:46.170
I'll write in green,
du, and then dv.

00:04:46.170 --> 00:04:48.210
So this is just the ds part.

00:04:48.210 --> 00:04:49.290
This is just the ds.

00:04:49.290 --> 00:04:51.370
Now, we have the
y right over here.

00:04:51.370 --> 00:04:55.260
y is just equal to v. All right,
so I'll write that in purple.

00:04:55.260 --> 00:05:00.430
So y is just equal to v. That is
y-- let me make it very clear--

00:05:00.430 --> 00:05:05.360
and all of this is ds.

00:05:05.360 --> 00:05:08.010
And now I can write the
bounds in terms of u

00:05:08.010 --> 00:05:12.225
and v. And so the u part,
u is the same thing as x.

00:05:12.225 --> 00:05:16.400
It goes between 0 and 1.

00:05:16.400 --> 00:05:20.860
And then v, v is
the same thing as y,

00:05:20.860 --> 00:05:23.720
and y goes between, or
v goes between, 0 and 2.

00:05:26.950 --> 00:05:31.210
And I now think we're
ready to evaluate.

00:05:31.210 --> 00:05:33.384
And the u and v variables
are not so mixed up,

00:05:33.384 --> 00:05:35.550
so we can actually separate
out these two integrals,

00:05:35.550 --> 00:05:38.020
make this a product of
two single integrals.

00:05:38.020 --> 00:05:40.966
The first thing, if we look
at it with respect to du,

00:05:40.966 --> 00:05:44.500
all this stuff in purple is just
a constant with respect to u,

00:05:44.500 --> 00:05:47.060
so we can take it out
of the du integral.

00:05:47.060 --> 00:05:51.050
We can take all this purple
stuff out of this du integral.

00:05:51.050 --> 00:05:53.980
And so this double integral
simplifies to the integral

00:05:53.980 --> 00:06:00.750
from 0 to 2 of-- I'll write it
as the square root of 2v times

00:06:00.750 --> 00:06:05.750
the square root of
1 plus 2v squared.

00:06:05.750 --> 00:06:09.050
So I factored out
all of this stuff.

00:06:09.050 --> 00:06:16.110
And then you have times the
integral from 0 to 1 du,

00:06:16.110 --> 00:06:22.460
and then times d,
and then you have dv.

00:06:22.460 --> 00:06:24.960
And now, if this was really
complicated, I could say,

00:06:24.960 --> 00:06:26.910
OK, this is just going
to be a function of u.

00:06:26.910 --> 00:06:28.370
It's a constant
with respect to v.

00:06:28.370 --> 00:06:29.860
And you could factor
this whole thing out

00:06:29.860 --> 00:06:30.810
and separate the integrals.

00:06:30.810 --> 00:06:31.810
But this is even easier.

00:06:31.810 --> 00:06:35.130
This integral is just
going to evaluate to 1.

00:06:35.130 --> 00:06:37.660
So this whole thing
just evaluates to 1.

00:06:37.660 --> 00:06:40.930
And so we've simplified this
into one single integral.

00:06:40.930 --> 00:06:43.330
So this simplifies
to-- and I could even

00:06:43.330 --> 00:06:45.820
take the square root of 2
out front-- the square root

00:06:45.820 --> 00:06:50.050
of 2 times the integral
from v is equal to 0 to v

00:06:50.050 --> 00:06:54.890
is equal to 2 of v
times the square root

00:06:54.890 --> 00:06:59.600
of 1 plus 2v squared dv.

00:06:59.600 --> 00:07:01.430
And so we really are
in the home stretch

00:07:01.430 --> 00:07:03.630
of evaluating this
surface integral.

00:07:03.630 --> 00:07:05.280
And here, this is basic.

00:07:05.280 --> 00:07:07.360
This is actually a little
bit of u substitution

00:07:07.360 --> 00:07:08.910
that we can do in our head.

00:07:08.910 --> 00:07:11.950
If you have a function, or
kind of this embedded function

00:07:11.950 --> 00:07:14.030
right over here,
1 plus 2v squared,

00:07:14.030 --> 00:07:16.960
what's the derivative
of 1 plus 2v squared?

00:07:16.960 --> 00:07:18.370
Well, it would be 4v.

00:07:18.370 --> 00:07:20.530
And we have something
almost 4v here.

00:07:20.530 --> 00:07:24.490
We can make this 4v by
multiplying it by 4 here,

00:07:24.490 --> 00:07:26.420
and then dividing
it by 4 out here.

00:07:26.420 --> 00:07:28.790
This doesn't change the
value of the integral.

00:07:28.790 --> 00:07:30.990
And so now this part
right over here,

00:07:30.990 --> 00:07:34.340
it's pretty straightforward
to take the antiderivative.

00:07:34.340 --> 00:07:36.810
The antiderivative
of this is going

00:07:36.810 --> 00:07:41.530
to be-- we have this embedded
function's derivative right

00:07:41.530 --> 00:07:45.360
over here, so we can kind of
just treat it like an x or a v.

00:07:45.360 --> 00:07:46.860
Take the antiderivative
with respect

00:07:46.860 --> 00:07:48.670
to this thing right
over here, and we

00:07:48.670 --> 00:07:52.910
get this is essentially 1 plus
2v squared to the 1/2 power.

00:07:52.910 --> 00:07:56.930
We increment it by
1, so it's 1 plus 2v

00:07:56.930 --> 00:07:59.840
squared to the 3/2 power.

00:07:59.840 --> 00:08:04.270
And then you divide by 3/2,
or you multiply by 2/3,

00:08:04.270 --> 00:08:07.680
so times 2/3.

00:08:07.680 --> 00:08:10.292
So this is the
antiderivative of that.

00:08:10.292 --> 00:08:12.500
And then, of course, you
still have all of this stuff

00:08:12.500 --> 00:08:20.620
out front-- square
root of 2 over 4

00:08:20.620 --> 00:08:27.929
And we are going to
evaluate this from 0 to 2.

00:08:27.929 --> 00:08:30.470
And actually, just to simplify
it, let me factor out the 2/3.

00:08:30.470 --> 00:08:33.440
We don't have to worry
about that at 2 and 0.

00:08:33.440 --> 00:08:37.201
So I'm going to factor out
the 2/3, so times 2 over 3.

00:08:37.201 --> 00:08:38.659
And actually, this
will cancel out.

00:08:38.659 --> 00:08:40.120
That becomes a 1.

00:08:40.120 --> 00:08:42.020
That becomes a 2.

00:08:42.020 --> 00:08:47.900
And so we are left-- this
stuff over here is 1/6 times.

00:08:47.900 --> 00:08:51.790
And now if you evaluate this
at 2, you have 2v squared.

00:08:51.790 --> 00:08:55.240
That's going to be 2
times 4 is 8 plus 1

00:08:55.240 --> 00:08:58.930
is 9, 9 to the 3/2 power.

00:08:58.930 --> 00:09:02.340
So 9 to the 1/2 is 3,
3 to the 3rd is 27.

00:09:02.340 --> 00:09:06.110
So it's going to be equal to 27.

00:09:06.110 --> 00:09:08.050
And then minus this
thing evaluated at 0.

00:09:08.050 --> 00:09:10.305
Well, this evaluated at
0 is just going to be 1.

00:09:10.305 --> 00:09:14.110
1 to the 3/2 is
just 1, so minus 1.

00:09:14.110 --> 00:09:16.730
And so this gives us-- oh,
I almost made a mistake.

00:09:16.730 --> 00:09:20.260
There should be a square
root of 2 up here,

00:09:20.260 --> 00:09:24.330
square root of 2 over
6 times 27 minus 1.

00:09:24.330 --> 00:09:26.090
So drum roll.

00:09:26.090 --> 00:09:28.700
This gives us the value of
our surface integral is,

00:09:28.700 --> 00:09:32.130
let's see, 27 minus 1 is 26.

00:09:32.130 --> 00:09:36.872
So we get 26 times the
square root of 2 over 6.

00:09:36.872 --> 00:09:38.580
And we can simplify
it a little bit more.

00:09:38.580 --> 00:09:41.020
Divide the numerator
and denominator by 2,

00:09:41.020 --> 00:09:48.050
and you get 13 square
roots of 2 over 3.

00:09:48.050 --> 00:09:51.070
And we are done.


Surface integral ex2 part 1: Parameterizing the surface

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:02.479
Let's attempt another
surface integral,

00:00:02.479 --> 00:00:04.270
and I've changed the
notation a little bit.

00:00:04.270 --> 00:00:06.660
Instead of writing the
surface as a capital sigma,

00:00:06.660 --> 00:00:09.610
I've written it as a capital S.
Instead of writing d lowercase

00:00:09.610 --> 00:00:11.770
sigma, I wrote d uppercase
S, which is still

00:00:11.770 --> 00:00:13.797
a surface integral
of the function y.

00:00:13.797 --> 00:00:16.130
And the surface we care about
is x plus y squared mins z

00:00:16.130 --> 00:00:16.713
is equal to 0.

00:00:18.890 --> 00:00:22.390
X between 0 and 1,
y between 0 and 2.

00:00:22.390 --> 00:00:24.860
Now, this one might be a
little bit more straightforward

00:00:24.860 --> 00:00:25.970
than the last one we
did, or at least I

00:00:25.970 --> 00:00:28.370
hope it's a little bit more
straightforward, because we

00:00:28.370 --> 00:00:32.190
can explicitly define
z in terms of x and y.

00:00:32.190 --> 00:00:33.940
And actually we can
even explicitly define

00:00:33.940 --> 00:00:35.520
x in terms of y and z.

00:00:35.520 --> 00:00:36.780
But I'll do it the other way.

00:00:36.780 --> 00:00:38.860
It's a little bit easier
for me to visualize.

00:00:38.860 --> 00:00:42.000
So if you add z to both sides of
this equation right over here,

00:00:42.000 --> 00:00:45.454
you get x plus y
squared is equal to z,

00:00:45.454 --> 00:00:46.870
or z is equal to
x plus y squared.

00:00:49.927 --> 00:00:51.760
And this is actually
pretty straightforward.

00:00:51.760 --> 00:00:54.360
this surface is pretty
easy to visualize,

00:00:54.360 --> 00:00:58.050
or we can give our best
attempt at visualizing it.

00:00:58.050 --> 00:01:06.580
So if that is our z-axis,
and that is our x-axis,

00:01:06.580 --> 00:01:11.490
and that is the y-axis,
we care about the region

00:01:11.490 --> 00:01:12.690
x between 0 and 1.

00:01:12.690 --> 00:01:16.690
So maybe this is x equals
1, and y between 0 and 2.

00:01:16.690 --> 00:01:20.050
So let's say this is 1,
this is 2 in the y area.

00:01:20.050 --> 00:01:23.620
So we essentially care
about the surface over this,

00:01:23.620 --> 00:01:26.112
over this region
of the xy plane.

00:01:26.112 --> 00:01:28.320
And then we can think about
what the surface actually

00:01:28.320 --> 00:01:28.620
looks like.

00:01:28.620 --> 00:01:29.578
This isn't the surface.

00:01:29.578 --> 00:01:32.850
This is just kind of the range
of x's and y's that we actually

00:01:32.850 --> 00:01:33.920
care about.

00:01:33.920 --> 00:01:35.590
And so let's think
about the surface.

00:01:35.590 --> 00:01:38.870
When x and y are 0, z is 0.

00:01:38.870 --> 00:01:40.467
So we're going to
be sitting-- let

00:01:40.467 --> 00:01:42.300
me do this in a-- let
me do this in green. z

00:01:42.300 --> 00:01:44.120
is going to be right over there.

00:01:44.120 --> 00:01:49.530
And now as y increases, or
if when x is equal to 0,

00:01:49.530 --> 00:01:51.590
if we're just talking
about the zy plane,

00:01:51.590 --> 00:01:53.270
z is going to be
equal to y squared.

00:01:56.130 --> 00:01:58.860
So this might be
z is equal to 4.

00:01:58.860 --> 00:02:02.510
This is z is equal to 2, 1, 3.

00:02:02.510 --> 00:02:05.120
So z is going to do
something like this.

00:02:05.120 --> 00:02:07.740
It's going to be a
parabola in the zy plane.

00:02:07.740 --> 00:02:11.550
It's going to look
something like that.

00:02:11.550 --> 00:02:15.790
Now, when y is equal to
0, z is just equal to x.

00:02:15.790 --> 00:02:19.080
So as x goes to 1,
z will also go to 1.

00:02:19.080 --> 00:02:21.240
So z will go like this.

00:02:21.240 --> 00:02:25.160
The scales of the axes aren't--
they are not drawn to scale.

00:02:25.160 --> 00:02:26.720
The z is a little
bit more compressed

00:02:26.720 --> 00:02:29.100
than the x or y the
way I've drawn them.

00:02:29.100 --> 00:02:31.230
And then from this
point right over here,

00:02:31.230 --> 00:02:33.020
you add the y squared.

00:02:33.020 --> 00:02:37.110
And so you get
something that looks--

00:02:37.110 --> 00:02:38.540
so this is this point there.

00:02:38.540 --> 00:02:42.610
And then this point, when y is
equal to 2 and x is equal to 1,

00:02:42.610 --> 00:02:44.845
you have z is equal to 5.

00:02:44.845 --> 00:02:46.470
It's going to look
something like this.

00:02:49.570 --> 00:02:52.460
And then you're going to have
a straight line like that,

00:02:52.460 --> 00:02:55.010
at this point, is
right over there.

00:02:55.010 --> 00:03:00.170
And this surface is the surface
that we are going to take,

00:03:00.170 --> 00:03:05.450
or the surface over which we're
going to evaluate the surface

00:03:05.450 --> 00:03:08.250
integral of the function y.

00:03:08.250 --> 00:03:10.320
And so one way you
could think about it,

00:03:10.320 --> 00:03:13.900
y could be maybe the mass
density of this surface.

00:03:13.900 --> 00:03:16.800
And so when you multiply
y times each dS,

00:03:16.800 --> 00:03:19.890
you're essentially figuring out
the mass of that little chunk,

00:03:19.890 --> 00:03:22.769
and then you're figuring out
the mass of this entire surface.

00:03:22.769 --> 00:03:25.060
And so one way you could
imagine as we go more and more

00:03:25.060 --> 00:03:27.430
in that direction
as y is increasing,

00:03:27.430 --> 00:03:29.180
this thing is getting
more and more dense.

00:03:29.180 --> 00:03:34.450
So this part of the
surface is more dense

00:03:34.450 --> 00:03:36.540
than as y becomes
lower and lower.

00:03:36.540 --> 00:03:38.570
And then that would
actually give us the mass.

00:03:38.570 --> 00:03:41.000
With that out of the way,
let's actually evaluate it.

00:03:41.000 --> 00:03:42.590
And so, as you
know, the first step

00:03:42.590 --> 00:03:44.444
is to figure out
a parametrization.

00:03:44.444 --> 00:03:46.110
And it should be
pretty straightforward,

00:03:46.110 --> 00:03:49.520
because we can write z
explicitly in terms of x and y.

00:03:49.520 --> 00:03:53.980
And so we can actually use x
and y as the actual parameters,

00:03:53.980 --> 00:03:57.070
or if we want to just substitute
it with different parameters,

00:03:57.070 --> 00:03:57.950
we could.

00:03:57.950 --> 00:04:00.330
But let me-- so let's just
write-- let me do that.

00:04:00.330 --> 00:04:04.860
So let me just write x is equal
to-- and in the spirit of using

00:04:04.860 --> 00:04:07.840
different notation, instead
of using s and t, I'll use u

00:04:07.840 --> 00:04:12.950
and v. X is equal to u,
let's say y is equal to v,

00:04:12.950 --> 00:04:16.596
and then z is going to be
equal to u plus v squared.

00:04:20.769 --> 00:04:25.180
And so our surface, written
as a vector position function

00:04:25.180 --> 00:04:28.050
or position vector
function, our surface,

00:04:28.050 --> 00:04:30.000
we can write it as
r, which is going

00:04:30.000 --> 00:04:33.230
to be a function of
u and v. And it's

00:04:33.230 --> 00:04:46.960
going to be equal to ui plus
vj plus u plus v squared k.

00:04:46.960 --> 00:04:49.440
And then u is going
to be between 0 and 1,

00:04:49.440 --> 00:04:52.410
because x is just equal
to u or u is equal to x.

00:04:52.410 --> 00:04:57.010
So u is going to
be between 0 and 1.

00:04:57.010 --> 00:05:01.862
And then v is going
to be between 0 and 2.

00:05:01.862 --> 00:05:03.070
I'm going to leave you there.

00:05:03.070 --> 00:05:04.390
In the next video,
we'll actually

00:05:04.390 --> 00:05:06.050
set up the surface
integral now that we

00:05:06.050 --> 00:05:09.400
have the parametrization done.


Surface integral example part 3: The home stretch

WEBVTT
Kind: captions
Language: en

00:00:00.438 --> 00:00:01.953
- [Instructor] So now
that we have been able

00:00:01.953 --> 00:00:03.737
to express our d sigma,

00:00:03.737 --> 00:00:05.475
I think we're pretty close to evaluating

00:00:05.475 --> 00:00:07.202
the integral itself,

00:00:07.202 --> 00:00:08.450
and one thing I do wanna point out

00:00:08.450 --> 00:00:09.798
that might have been nagging you

00:00:09.798 --> 00:00:11.271
from the end of the last video,

00:00:11.271 --> 00:00:12.104
at the end of the last video,

00:00:12.104 --> 00:00:16.187
I took the principal root
of cosine squared of t,

00:00:17.898 --> 00:00:20.919
and I simplified that to
just being cosine of t.

00:00:20.919 --> 00:00:22.247
And you might have said,
"Wait, wait, wait, wait,

00:00:22.247 --> 00:00:25.218
"What if cosine of t evaluated
to a negative number?

00:00:25.218 --> 00:00:27.065
"If I then square it,
it would be positive,

00:00:27.065 --> 00:00:28.798
"and then if I took the
principal root of that,

00:00:28.798 --> 00:00:30.413
"I would then get the
positive version of it.

00:00:30.413 --> 00:00:32.106
"I would essentially
get the absolute value

00:00:32.106 --> 00:00:33.490
"of the cosine of t."

00:00:33.490 --> 00:00:35.836
And the reason why we were
able to do this in particular

00:00:35.836 --> 00:00:38.350
in this video or in this problem

00:00:38.350 --> 00:00:40.698
is because we saw t takes on values

00:00:40.698 --> 00:00:44.615
between negative pi over two
and positive pi over two.

00:00:44.615 --> 00:00:46.621
And so cosine of anything that's either

00:00:46.621 --> 00:00:48.446
in the first or the fourth quadrant,

00:00:48.446 --> 00:00:49.969
so this is t right over here,

00:00:49.969 --> 00:00:53.465
the cosine will always be
positive for our purposes,

00:00:53.465 --> 00:00:55.780
for the sake of this surface integral.

00:00:55.780 --> 00:00:58.472
Cosine of t is always
going to be positive,

00:00:58.472 --> 00:01:00.529
and so in this case,
we don't have to write

00:01:00.529 --> 00:01:02.039
absolute value of cosine of t.

00:01:02.039 --> 00:01:04.095
We can just write cosine of t,

00:01:04.095 --> 00:01:05.734
and so hopefully that makes you satisfied.

00:01:05.734 --> 00:01:09.002
That was just based on how
we parameterized the t.

00:01:09.002 --> 00:01:09.883
Now, that out of the way,

00:01:09.883 --> 00:01:11.871
let's actually evaluate the integral.

00:01:11.871 --> 00:01:13.502
Our original integral,

00:01:13.502 --> 00:01:15.944
the original integral, just to remind us,

00:01:15.944 --> 00:01:19.840
was the double, or I should
say the surface integral,

00:01:19.840 --> 00:01:22.490
of x squared, d sigma.

00:01:22.490 --> 00:01:24.703
We already know what d sigma is.

00:01:24.703 --> 00:01:26.654
Now we just have to write x squared

00:01:26.654 --> 00:01:28.411
in terms of the parameters.

00:01:28.411 --> 00:01:31.456
Well, we know the parameterization of x.

00:01:31.456 --> 00:01:35.204
The x, in terms of the parameters
right over here is cosine.

00:01:35.204 --> 00:01:37.334
This is our parameterization:

00:01:37.334 --> 00:01:40.811
x is going to be equal
to cosine t, cosine of s.

00:01:40.811 --> 00:01:44.978
Let me write that down:
x of s and t is equal to,

00:01:48.349 --> 00:01:51.661
I already forgot, I have a
horrible memory, is equal to,

00:01:51.661 --> 00:01:53.292
we have to go back to the
original parameterization,

00:01:53.292 --> 00:01:54.540
not these partial derivatives.

00:01:54.540 --> 00:01:56.802
Cosine t, cosine s.

00:01:56.802 --> 00:01:58.385
Cosine t, cosine s.

00:01:59.655 --> 00:02:01.988
Cosine t, and then cosine s,

00:02:05.802 --> 00:02:08.066
and we're taking the
integral of that squared.

00:02:08.066 --> 00:02:09.476
So let's think about this a little bit.

00:02:09.476 --> 00:02:11.867
So let's just do this
part right over here.

00:02:11.867 --> 00:02:15.495
If we square x, we're going to get

00:02:15.495 --> 00:02:18.412
cosine squared t, cosine squared s.

00:02:20.841 --> 00:02:22.258
Cosine squared s,

00:02:24.156 --> 00:02:26.414
that's the x squared
part right over there,

00:02:26.414 --> 00:02:27.885
and then you have the d sigma,

00:02:27.885 --> 00:02:31.468
which is this stuff,
which is times cosine,

00:02:32.687 --> 00:02:34.104
lemme do that same green,

00:02:34.104 --> 00:02:36.642
don't wanna confuse you with
different shades of colors,

00:02:36.642 --> 00:02:38.809
times cosine of t, ds, dt.

00:02:41.091 --> 00:02:42.994
And now that we have this in terms

00:02:42.994 --> 00:02:45.408
of the parameters, the
differentials of parameters,

00:02:45.408 --> 00:02:47.101
this essentially becomes a double integral

00:02:47.101 --> 00:02:49.684
with respect to these two parameters,

00:02:49.684 --> 00:02:51.310
and so, and the good thing is

00:02:51.310 --> 00:02:53.042
that the boundary is
pretty straightforward

00:02:53.042 --> 00:02:56.435
with respect to s and t:
s takes on all values,

00:02:56.435 --> 00:02:59.087
s takes on all values
between zero and two pi,

00:02:59.087 --> 00:03:01.932
t takes on all values
between negative two pi,

00:03:01.932 --> 00:03:03.913
and, sorry, negative pi over two

00:03:03.913 --> 00:03:06.098
and positive pi over two.

00:03:06.098 --> 00:03:07.723
So first, the way I wrote over here,

00:03:07.723 --> 00:03:10.154
we're gonna integrate
with respect to s first,

00:03:10.154 --> 00:03:13.737
s goes between zero
and two pi, and then t,

00:03:17.374 --> 00:03:19.622
lemme write and make clear, this is s,

00:03:19.622 --> 00:03:24.020
and then t will go between
negative pi over two

00:03:24.020 --> 00:03:26.103
and positive pi over two.

00:03:28.434 --> 00:03:30.901
And so let's see if we can
simplify this a little bit.

00:03:30.901 --> 00:03:33.209
This is equal to the double integral

00:03:33.209 --> 00:03:36.345
over that same region,
over that same area,

00:03:36.345 --> 00:03:39.733
I guess we could call it,
over that same area of,

00:03:39.733 --> 00:03:41.976
well now we have this cosine squared of t

00:03:41.976 --> 00:03:44.307
and then we have another
cosine of t right over there,

00:03:44.307 --> 00:03:45.354
so lemme just right it this way,

00:03:45.354 --> 00:03:49.521
as cosine to the third of
t times cosine squared...

00:03:51.181 --> 00:03:53.931
Cosine squared of s, and then ds,

00:03:55.271 --> 00:03:56.801
lemme color code it a little bit,

00:03:56.801 --> 00:04:01.211
ds, and so this is the
integral for the ds part,

00:04:01.211 --> 00:04:02.211
and then dt.

00:04:06.015 --> 00:04:08.297
And this is when we
integrate with respect to s,

00:04:08.297 --> 00:04:11.024
notice these two, the
t parts and the s part,

00:04:11.024 --> 00:04:12.635
they're just multiplied by each other,

00:04:12.635 --> 00:04:15.716
so when we're taking the
integral with respect to s,

00:04:15.716 --> 00:04:18.984
this cosine cubed of t
really is just a constant,

00:04:18.984 --> 00:04:23.151
we can factor it out, and it
could look something like this,

00:04:24.486 --> 00:04:28.574
so let me rewrite it, this
could be the integral from,

00:04:28.574 --> 00:04:30.570
t goes from, I'll rewrite the boundaries,

00:04:30.570 --> 00:04:33.804
negative pi over two,
to positive pi over two,

00:04:33.804 --> 00:04:37.304
cosine cubed of t, I just factor that out,

00:04:40.321 --> 00:04:42.776
and then I'll write the s part,

00:04:42.776 --> 00:04:44.544
times the integral, s is going to go

00:04:44.544 --> 00:04:47.984
between zero and two pi,
and I'll write this in blue,

00:04:47.984 --> 00:04:52.151
cosine squared of s, ds, and
then you have dt out there,

00:04:54.426 --> 00:04:57.926
you have dt, I'm gonna do the dt in green,

00:05:01.891 --> 00:05:04.835
gimme that same green, dt.

00:05:04.835 --> 00:05:07.604
And now, this outer sum we can view it,

00:05:07.604 --> 00:05:09.974
you essentially view it as the product,

00:05:09.974 --> 00:05:12.217
well, of all of this
business right over here,

00:05:12.217 --> 00:05:15.591
this thing has no t's
involved in it whatsoever,

00:05:15.591 --> 00:05:17.508
so we can rewrite this,

00:05:18.358 --> 00:05:21.245
and I'll write all the stuff
involving the t's as green.

00:05:21.245 --> 00:05:24.412
So we can rewrite this as pi over two,

00:05:25.594 --> 00:05:28.643
from negative pi over two, to pi over two,

00:05:28.643 --> 00:05:31.976
cosine cubed of t dt times the integral,

00:05:34.594 --> 00:05:36.189
and I'm really just rearranging things,

00:05:36.189 --> 00:05:37.184
I guess you could kind of view this

00:05:37.184 --> 00:05:38.598
as the associative property,

00:05:38.598 --> 00:05:40.541
or I guess the commutative property.

00:05:40.541 --> 00:05:42.544
Well those things always confuse me,

00:05:42.544 --> 00:05:45.544
times the integral of zero to two pi

00:05:48.928 --> 00:05:51.531
of cosine squared of s, ds,

00:05:51.531 --> 00:05:52.579
and you didn't have to do it this way,

00:05:52.579 --> 00:05:54.229
you could've just evaluated it

00:05:54.229 --> 00:05:55.510
while it was kind of mixed like this,

00:05:55.510 --> 00:05:58.195
but this'll help us kind
of work through the,

00:05:58.195 --> 00:06:01.622
the trigonometry a little bit easier.

00:06:01.622 --> 00:06:03.090
Now to solve these two integrals,

00:06:03.090 --> 00:06:05.875
we just have to resort
to our trigonometry.

00:06:05.875 --> 00:06:10.042
Cosine squared of s, we can
rewrite that as 1/2 plus one,

00:06:12.781 --> 00:06:14.095
actually let me do that
in that same blue color

00:06:14.095 --> 00:06:15.604
so we don't get confused.

00:06:15.604 --> 00:06:19.771
That is the same thing as
1/2 plus 1/2 cosine of two s,

00:06:23.312 --> 00:06:27.221
and cosine cubed t, well
that's the same thing,

00:06:27.221 --> 00:06:30.804
let's see, we can factor
out a cosine of t,

00:06:34.570 --> 00:06:35.856
so let me rewrite, ah let's just do it,

00:06:35.856 --> 00:06:37.742
well let me just do it,
both at the same time,

00:06:37.742 --> 00:06:39.557
just get all the trigonometry out of way.

00:06:39.557 --> 00:06:43.165
This right over here can
be rewritten as cosine of t

00:06:43.165 --> 00:06:47.332
times cosine squared of t,
and the intuition here is,

00:06:48.539 --> 00:06:51.574
if we can get a product
of a sine doing something

00:06:51.574 --> 00:06:54.468
with a cosine, because
cosine is sine's derivative,

00:06:54.468 --> 00:06:56.242
that's kind of, y'know, u-substitution,

00:06:56.242 --> 00:06:57.930
you see a function and its derivative,

00:06:57.930 --> 00:06:59.682
you can just kind of
treat it as a variable,

00:06:59.682 --> 00:07:02.283
so that's what we're trying
to get to right over here.

00:07:02.283 --> 00:07:04.535
So cosine squared of t can be rewritten

00:07:04.535 --> 00:07:06.233
as one minus sine squared of t,

00:07:06.233 --> 00:07:10.400
so this is cosine of t times
one minus sine squared of t.

00:07:11.919 --> 00:07:15.067
And so we can rewrite this as cosine of t

00:07:15.067 --> 00:07:18.150
minus cosine of t, sine squared of t,

00:07:19.263 --> 00:07:20.096
and you might say "Wait,

00:07:20.096 --> 00:07:22.133
"this looked a lot simpler
than this down here."

00:07:22.133 --> 00:07:23.902
That is true, it looks simpler,

00:07:23.902 --> 00:07:25.933
but it's easier to take
the antiderivative of this,

00:07:25.933 --> 00:07:28.136
easier to take the
antiderivative of cosine of t,

00:07:28.136 --> 00:07:30.512
and even over here, you have derivative

00:07:30.512 --> 00:07:32.513
of sine of t, which is cosine of t,

00:07:32.513 --> 00:07:34.754
and so essentially you
can do u-substitution,

00:07:34.754 --> 00:07:36.964
which you probably can
do in your head now.

00:07:36.964 --> 00:07:39.433
So let's evaluate each of these integrals.

00:07:39.433 --> 00:07:41.722
So this one, let me rewrite them

00:07:41.722 --> 00:07:42.833
just so we don't get too confused,

00:07:42.833 --> 00:07:45.754
so we have the integral
from negative pi over two,

00:07:45.754 --> 00:07:48.254
to pi over two, of cosine of t

00:07:50.206 --> 00:07:53.623
minus cosine of t, sine squared of t, dt,

00:07:58.871 --> 00:08:02.038
times the integral from zero to two pi

00:08:04.516 --> 00:08:08.683
of 1/2 plus 1/2 cosine of 2s, d s.

00:08:12.765 --> 00:08:15.448
Now we are ready to take
some antiderivatives,

00:08:15.448 --> 00:08:19.685
the antiderivative of this
right over here is going to be,

00:08:19.685 --> 00:08:23.852
the antiderivative of cosine
t, well that's just sine t,

00:08:24.779 --> 00:08:26.122
and then right over here,

00:08:26.122 --> 00:08:29.102
the derivative of sine t is cosine of t.

00:08:29.102 --> 00:08:30.592
So we can just essentially,

00:08:30.592 --> 00:08:32.126
if you wanna do
u-substitution, you would say,

00:08:32.126 --> 00:08:36.709
u is equal to sine of t, d u
is equal to cosine of t, dt,

00:08:36.709 --> 00:08:37.994
and you do all of that, but the,

00:08:37.994 --> 00:08:39.693
what we probably cannot
do in your head is,

00:08:39.693 --> 00:08:41.679
okay, I have the sine
t's derivative there,

00:08:41.679 --> 00:08:44.084
so I can treat sine t just
like I would treat a t,

00:08:44.084 --> 00:08:45.312
or I would treat an x.

00:08:45.312 --> 00:08:47.852
So this is going to be, you
still have this negative sign,

00:08:47.852 --> 00:08:51.185
minus sine to the third of t over three,

00:08:54.023 --> 00:08:56.272
if this was just a t squared,

00:08:56.272 --> 00:08:59.120
the antiderivative would be
t to the third over three,

00:08:59.120 --> 00:09:00.837
but now since we have a derivative,

00:09:00.837 --> 00:09:02.899
we can kind of treat it the same way,

00:09:02.899 --> 00:09:06.103
which is essentially doing
u-substitution in our head.

00:09:06.103 --> 00:09:08.757
So that's that, and we're
going to evaluate it

00:09:08.757 --> 00:09:12.709
from negative pi over two, to pi over two.

00:09:12.709 --> 00:09:16.193
And so this is equal to, if
you evaluate it at pi over two,

00:09:16.193 --> 00:09:18.443
sine of pi over two is one.

00:09:19.591 --> 00:09:23.758
So it's one minus 1/3, so that's just 2/3,

00:09:27.158 --> 00:09:28.117
actually lemme not write it that way,

00:09:28.117 --> 00:09:29.240
I don't wanna confuse people,

00:09:29.240 --> 00:09:33.192
and then minus sine of
negative pi over two,

00:09:33.192 --> 00:09:36.712
well that's going to
be negative one minus,

00:09:36.712 --> 00:09:38.203
sine of negative pi over two is

00:09:38.203 --> 00:09:39.837
negative one to the third
power is negative one,

00:09:39.837 --> 00:09:42.754
so this is negative 1/3.

00:09:44.762 --> 00:09:48.473
And so this is going to
be equal to, this is 2/3,

00:09:48.473 --> 00:09:52.640
and this is negative one plus
1/3, which is negative 2/3,

00:09:54.396 --> 00:09:55.736
but then you have a negative out front,

00:09:55.736 --> 00:09:57.665
so this is plus 2/3 again.

00:09:57.665 --> 00:10:01.377
So this part at least evaluates to 4/3.

00:10:01.377 --> 00:10:03.712
This part, all this, is
really the home stretch,

00:10:03.712 --> 00:10:05.860
that all evaluates to 4/3.

00:10:05.860 --> 00:10:08.080
Now this part right over here,

00:10:08.080 --> 00:10:12.247
antiderivative of 1/2 is just 1/2 t,

00:10:13.908 --> 00:10:16.491
antiderivative of cosine of 2s,

00:10:17.821 --> 00:10:21.262
well, ideally you would
have a two out front here,

00:10:21.262 --> 00:10:23.207
out front, lemme write
and make this clear,

00:10:23.207 --> 00:10:24.934
so if I were to take the antiderivative

00:10:24.934 --> 00:10:29.605
of cosine of 2s, ideally you
would want a two out here,

00:10:29.605 --> 00:10:31.681
so you have the derivative of the 2s,

00:10:31.681 --> 00:10:34.401
so you could put a two out front,

00:10:34.401 --> 00:10:36.201
but then you would have
to put a 1/2 out front

00:10:36.201 --> 00:10:38.236
so that you're not
changing the value of it,

00:10:38.236 --> 00:10:41.142
and of course you would
have a ds right over here.

00:10:41.142 --> 00:10:42.883
I'm just taking a general antiderivative,

00:10:42.883 --> 00:10:44.288
but once you have it like this,

00:10:44.288 --> 00:10:45.154
then this just like taking

00:10:45.154 --> 00:10:47.466
the antiderivative of cosine of s.

00:10:47.466 --> 00:10:50.717
This becomes, antiderivative
of cosine is sine.

00:10:50.717 --> 00:10:53.446
So this will become sine of s.

00:10:53.446 --> 00:10:57.596
So this right over here is just sine of s,

00:10:57.596 --> 00:11:02.532
and then you have the
1/2 out front, times 1/2,

00:11:02.532 --> 00:11:04.382
but then of course,
and then you would have

00:11:04.382 --> 00:11:06.687
plus a constant if you were
taking an indefinite integral,

00:11:06.687 --> 00:11:07.890
but we're taking a definite one,

00:11:07.890 --> 00:11:09.491
so you don't have to
worry about the constant,

00:11:09.491 --> 00:11:13.074
so just the antiderivative
of cosine of 2s,

00:11:14.127 --> 00:11:18.108
just the antiderivative of
cosine of 2s is 1/2 sine of s.

00:11:18.108 --> 00:11:19.806
And so you have this constant out front,

00:11:19.806 --> 00:11:22.243
1/2 times 1/2 is 1/4.

00:11:22.243 --> 00:11:26.410
So it's going to be plus 1/4 sine of 2s.

00:11:32.586 --> 00:11:34.017
That's the antiderivative,

00:11:34.017 --> 00:11:37.534
and now we're going to evaluate
it from zero to two pi,

00:11:37.534 --> 00:11:38.836
and in either situation,

00:11:38.836 --> 00:11:40.650
this thing's going to evaluate to zero.

00:11:40.650 --> 00:11:44.164
Sine of zero is zero,
sine of four pi is zero,

00:11:44.164 --> 00:11:48.531
and so you're gonna have 1/2
times two pi, which is just pi,

00:11:48.531 --> 00:11:51.589
plus zero, 'cause sine of four pi is zero,

00:11:51.589 --> 00:11:55.756
minus 1/2 times zero, zero;
1/4 times sine of zero, zero,

00:11:57.121 --> 00:11:59.022
so you're essentially
just gonna end up with pi,

00:11:59.022 --> 00:12:02.180
so this whole thing right
over here evaluates to pi.

00:12:02.180 --> 00:12:03.221
And so we're done!

00:12:03.221 --> 00:12:06.467
You take the product of these
two things, 4/3 times pi,

00:12:06.467 --> 00:12:10.634
our entire surface integral
evaluates to: 4/3 pi.

00:12:12.318 --> 00:12:16.795
So this is equal to 4/3 pi, which is neat!

00:12:16.795 --> 00:12:19.284
If you have a sphere of radius one,

00:12:19.284 --> 00:12:23.371
its surface area, or actually
no, I shouldn't even go that,

00:12:23.371 --> 00:12:25.913
because, let me be very careful.

00:12:25.913 --> 00:12:27.344
I shouldn't make that statement,

00:12:27.344 --> 00:12:30.237
because this wasn't just
with respect to one.

00:12:30.237 --> 00:12:33.296
But, we have at least
evaluated the surface integral,

00:12:33.296 --> 00:12:37.046
and we deserve, I think,
a bit of a rest now.


Surface integral example part 2: Calculating the surface differential

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.610
Now that we have our
parametrization right

00:00:03.610 --> 00:00:07.170
over here, let's get down to the
business of actually evaluating

00:00:07.170 --> 00:00:08.350
this surface integral.

00:00:08.350 --> 00:00:09.891
It's a little bit
involved, but we'll

00:00:09.891 --> 00:00:11.260
try to do it step by step.

00:00:11.260 --> 00:00:12.843
And so the first
thing I'm going to do

00:00:12.843 --> 00:00:17.540
is figure out what d sigma
is in terms of s and t,

00:00:17.540 --> 00:00:19.070
in terms of our parameters.

00:00:19.070 --> 00:00:21.660
So we can turn this whole
thing into a double integral

00:00:21.660 --> 00:00:25.461
with respect to-- or a double
integral in the s, t plane.

00:00:25.461 --> 00:00:27.884
And remember, d sigma
right over here,

00:00:27.884 --> 00:00:29.550
it's just a little
chunk of the surface.

00:00:29.550 --> 00:00:32.980
It's a little area of the
surface right over there.

00:00:32.980 --> 00:00:35.290
And we saw in previous
videos, the ones

00:00:35.290 --> 00:00:38.960
where we learned what
a surface integral even

00:00:38.960 --> 00:00:43.910
is, we saw that d
sigma right over there,

00:00:43.910 --> 00:00:48.190
it is equivalent to the
magnitude of the cross product

00:00:48.190 --> 00:00:51.380
of the partial of our
parametrization with respect

00:00:51.380 --> 00:00:55.100
to one parameter crossed
with the parametrization

00:00:55.100 --> 00:00:57.090
with respect to
the other parameter

00:00:57.090 --> 00:01:00.424
times the differentials
of each of the parameters.

00:01:00.424 --> 00:01:02.340
So this is what we're
going to use right here.

00:01:02.340 --> 00:01:05.810
It's a pretty simple looking
statement, but as we'll see,

00:01:05.810 --> 00:01:08.980
taking cross products tend
to get a little bit hairy.

00:01:08.980 --> 00:01:11.680
Especially cross products of
three dimensional vectors.

00:01:11.680 --> 00:01:13.080
But we'll do it step by step.

00:01:13.080 --> 00:01:14.840
But before we even
take the cross product,

00:01:14.840 --> 00:01:17.180
we first have to take the
partial of this with respect

00:01:17.180 --> 00:01:20.260
to s, and then the partial
of this with respect to t.

00:01:20.260 --> 00:01:23.950
So first, let's take the
partial with respect to s--

00:01:23.950 --> 00:01:26.830
the partial of r
with respect to s.

00:01:26.830 --> 00:01:29.120
So right over here, all
the stuff with the t in it,

00:01:29.120 --> 00:01:30.840
you can just view
that as a constant.

00:01:30.840 --> 00:01:32.510
So cosine of t isn't
going to change.

00:01:32.510 --> 00:01:35.190
The partial of-- or the
derivative of cosine of s

00:01:35.190 --> 00:01:38.337
with respect to s is
negative sine of s.

00:01:38.337 --> 00:01:39.920
So this is going to
be equal to-- I'll

00:01:39.920 --> 00:01:44.830
put the negative out front--
negative cosine of t sine of s.

00:01:44.830 --> 00:01:47.940
I'm going to keep everything
that has a t involved purple.

00:01:47.940 --> 00:01:48.900
Sine of s.

00:01:48.900 --> 00:01:50.410
And let me make-- I don't know.

00:01:50.410 --> 00:01:57.240
I'll make the vectors orange--
i, and then you we'll plus

00:01:57.240 --> 00:02:00.110
and once again, we'll take the
derivative with respect to s,

00:02:00.110 --> 00:02:02.250
cosine of t is just a
constant, derivative

00:02:02.250 --> 00:02:04.650
of sine of s with respect
to s is cosine of s.

00:02:04.650 --> 00:02:17.540
So it's going to be plus
cosine of t cosine of s j.

00:02:17.540 --> 00:02:23.670
And then plus the derivative
of this with respect to s.

00:02:23.670 --> 00:02:24.920
Well, this is just a constant.

00:02:24.920 --> 00:02:28.112
The derivative of 5 with
respect to s would just be 0.

00:02:28.112 --> 00:02:29.070
This is the same thing.

00:02:29.070 --> 00:02:30.820
It's just a constant
with-- this does not

00:02:30.820 --> 00:02:32.410
change with respect to s.

00:02:32.410 --> 00:02:36.140
So it does-- so our partial here
with respect to s is just 0.

00:02:36.140 --> 00:02:38.600
So we could write even 0k.

00:02:38.600 --> 00:02:42.827
Let me write 0-- I'll just
write 0k right over there.

00:02:42.827 --> 00:02:44.410
And that's nice to
see because that'll

00:02:44.410 --> 00:02:47.830
make our cross product a little
bit more straightforward.

00:02:47.830 --> 00:02:50.540
Now, let's take the
partial with respect to t.

00:02:59.372 --> 00:03:00.855
all right.

00:03:00.855 --> 00:03:02.980
And we get-- so the derivative
of this with respect

00:03:02.980 --> 00:03:05.020
to t-- now, cosine
of s is the constant.

00:03:05.020 --> 00:03:07.180
Derivative of cosine
t with respect to t

00:03:07.180 --> 00:03:09.800
is negative sine of t.

00:03:09.800 --> 00:03:21.540
So this is going to be negative
sine of t cosine of s i--

00:03:21.540 --> 00:03:25.630
I'll do it in that-- I'll
use this blue-- i, and then

00:03:25.630 --> 00:03:28.750
plus-- now derivative of
this with respect to t.

00:03:28.750 --> 00:03:31.240
Derivative of cosine of
t is negative sine of t.

00:03:31.240 --> 00:03:41.970
So once again, so now we have
minus sine of t sine of s--

00:03:41.970 --> 00:03:43.610
my hand is already
hurting from this.

00:03:43.610 --> 00:03:45.770
This is a painful problem.

00:03:45.770 --> 00:03:48.709
j plus derivative of sine
of t with respect to t.

00:03:48.709 --> 00:03:50.500
We're taking the partial
with respect to t.

00:03:50.500 --> 00:03:52.120
It's just cosine of t.

00:03:52.120 --> 00:03:53.840
So plus cosine of t.

00:03:57.360 --> 00:03:59.990
And now times the k unit vector.

00:03:59.990 --> 00:04:03.120
Now we're ready to take the
cross product of these two

00:04:03.120 --> 00:04:04.647
characters right over here.

00:04:04.647 --> 00:04:06.980
And to take the cross products--
let me write this down.

00:04:06.980 --> 00:04:08.396
So we're going to
say, we're going

00:04:08.396 --> 00:04:12.735
to take the cross product
of that with that.

00:04:12.735 --> 00:04:14.110
Is going to be
equal to-- and I'm

00:04:14.110 --> 00:04:16.936
going to set up
this huge matrix.

00:04:16.936 --> 00:04:18.769
Or, it's really just a
three by three matrix

00:04:18.769 --> 00:04:20.019
but it's going to
be huge because it's

00:04:20.019 --> 00:04:21.643
going to take up a
lot of space to have

00:04:21.643 --> 00:04:22.950
to write down all this stuff.

00:04:22.950 --> 00:04:26.360
So maybe I'll take up
about that much space

00:04:26.360 --> 00:04:28.800
so that I have space to work in.

00:04:28.800 --> 00:04:30.990
And I'll write my
unit vectors up here.

00:04:30.990 --> 00:04:32.740
i, j, k.

00:04:32.740 --> 00:04:34.970
Or at least this is
how I like to remember

00:04:34.970 --> 00:04:39.070
how to take cross products
of three dimensional vectors.

00:04:39.070 --> 00:04:41.160
Take the determinant
of this three

00:04:41.160 --> 00:04:42.890
by three matrix-- the
first row are just

00:04:42.890 --> 00:04:45.800
our unit vectors-- the second
row is the first vector

00:04:45.800 --> 00:04:47.690
that I'm taking the
cross product of.

00:04:47.690 --> 00:04:49.922
So this is going to be negative.

00:04:49.922 --> 00:04:51.880
I'm just going to rewrite
this right over here.

00:04:51.880 --> 00:04:55.775
So it's going to be negative
cosine of t sine of s.

00:04:58.610 --> 00:05:00.670
And then you have
cosine of t cosine of s.

00:05:07.040 --> 00:05:08.840
And then you have 0
which will hopefully

00:05:08.840 --> 00:05:10.940
simplify our calculations.

00:05:10.940 --> 00:05:13.750
And then you have the next
vector, that's the third row.

00:05:13.750 --> 00:05:19.301
Negative sine of t cosine of s.

00:05:19.301 --> 00:05:21.050
And I encourage you
to do this on your own

00:05:21.050 --> 00:05:22.320
if you already know
where this is going.

00:05:22.320 --> 00:05:23.045
It's good practice.

00:05:23.045 --> 00:05:24.794
Even if you have to
watch this whole thing

00:05:24.794 --> 00:05:27.170
to see how it's done, try to
then do it again on your own

00:05:27.170 --> 00:05:28.627
because this is
one of those things

00:05:28.627 --> 00:05:30.970
you really got to do yourself
to really have it sit in.

00:05:30.970 --> 00:05:37.330
Negative sine of t sine of s.

00:05:37.330 --> 00:05:41.040
And then finally, cosine of t.

00:05:41.040 --> 00:05:44.360
So let's take the
determinant now.

00:05:44.360 --> 00:05:46.330
So first we'll think
about our i component.

00:05:46.330 --> 00:05:48.110
Are i component, you
would essentially

00:05:48.110 --> 00:05:50.910
ignore this column-- the first
column in the first row--

00:05:50.910 --> 00:05:54.870
and then take the determinant of
this submatrix right over here.

00:05:54.870 --> 00:05:56.380
So it's going to
be i-- so this is

00:05:56.380 --> 00:05:59.417
going to be equal to
i times something.

00:05:59.417 --> 00:06:01.250
I'll put the something
in parentheses there.

00:06:01.250 --> 00:06:03.250
Normally you see the
something in front of the i

00:06:03.250 --> 00:06:04.520
but you can swap them there.

00:06:04.520 --> 00:06:07.030
So let me just-- so it's
going to be i times something.

00:06:07.030 --> 00:06:08.750
Ignore this column, that row.

00:06:08.750 --> 00:06:12.310
This determinant is cosine of t
cosine of s times cosine of t,

00:06:12.310 --> 00:06:16.140
which is going to be
cosine squared of t.

00:06:16.140 --> 00:06:18.390
Let me write it a little neater.

00:06:18.390 --> 00:06:23.960
Cosine squared of t cosine of s.

00:06:23.960 --> 00:06:26.340
And then from that we need
to subtract 0 times this.

00:06:26.340 --> 00:06:28.947
But that's just going to be 0
so we're just left with that.

00:06:28.947 --> 00:06:30.530
Now we're going to
do the j component.

00:06:30.530 --> 00:06:32.529
But you probably remember
the checkerboard thing

00:06:32.529 --> 00:06:35.000
when you have to evaluate
three by three matrices.

00:06:35.000 --> 00:06:36.910
Positive, negative, positive.

00:06:36.910 --> 00:06:39.490
So you will have a negative--
you write a negative j.

00:06:39.490 --> 00:06:43.180
A negative coefficient, I
guess, in front of the j times

00:06:43.180 --> 00:06:44.170
something.

00:06:44.170 --> 00:06:46.890
And so you ignore
j's column, j's row.

00:06:46.890 --> 00:06:49.460
And so you have negative
cosine of t sine

00:06:49.460 --> 00:06:51.220
of s times cosine of t.

00:06:51.220 --> 00:06:55.250
Well, that's going
to be negative cosine

00:06:55.250 --> 00:07:00.657
squared of t times sine of s.

00:07:00.657 --> 00:07:02.240
Let me make sure I'm
doing that right.

00:07:02.240 --> 00:07:03.410
Ignore that and that.

00:07:03.410 --> 00:07:05.170
It's going to be
that times that.

00:07:05.170 --> 00:07:06.190
So yep.

00:07:06.190 --> 00:07:11.430
Negative cosine sine of
s minus 0 times that.

00:07:11.430 --> 00:07:13.756
And so that's just going to
be 0, so we can ignore it.

00:07:13.756 --> 00:07:15.630
And you have a negative
times a negative here

00:07:15.630 --> 00:07:17.900
so they can both
become positive.

00:07:17.900 --> 00:07:19.770
And then finally, you
have the k component.

00:07:19.770 --> 00:07:21.853
And once again, you can
go back to positive there.

00:07:21.853 --> 00:07:24.120
Positive, negative, positive
on the coefficients.

00:07:24.120 --> 00:07:26.420
That's just evaluating
a three by three matrix.

00:07:26.420 --> 00:07:30.530
And then you have plus
k times-- and now this

00:07:30.530 --> 00:07:32.030
might get a little
bit more involved

00:07:32.030 --> 00:07:33.540
because we won't have
the 0 to help us out.

00:07:33.540 --> 00:07:35.180
Ignore this row,
ignore this column,

00:07:35.180 --> 00:07:37.840
take the determinant
of this sub two by two.

00:07:37.840 --> 00:07:41.350
You have negative
cosine t sine s times

00:07:41.350 --> 00:07:43.004
negative sine of t sine s.

00:07:43.004 --> 00:07:45.170
Well, that's going to be--
the negatives cancel out.

00:07:45.170 --> 00:07:50.420
So it's going to
be cosine of t sine

00:07:50.420 --> 00:07:54.340
of t times sine squared of s.

00:07:58.620 --> 00:08:00.800
And then, from that
we're going to subtract

00:08:00.800 --> 00:08:02.214
the product of these two things.

00:08:02.214 --> 00:08:03.880
But that product is
going to be negative

00:08:03.880 --> 00:08:05.338
so you subtract a
negative-- that's

00:08:05.338 --> 00:08:07.030
the same thing as
adding a positive.

00:08:07.030 --> 00:08:10.880
So plus-- and you're going to
have cosine t sine t again--

00:08:10.880 --> 00:08:16.410
plus cosine t-- let me scroll
to the right a little bit--

00:08:16.410 --> 00:08:21.230
plus cosine t sine t again.

00:08:21.230 --> 00:08:23.560
And that's times
cosine squared of s.

00:08:27.520 --> 00:08:29.720
Now this is already
looking pretty hairy

00:08:29.720 --> 00:08:31.883
but it already looks like
a simplification there.

00:08:31.883 --> 00:08:33.549
And that's where the
colors are helpful.

00:08:33.549 --> 00:08:35.360
Actually, now I have
trouble doing math

00:08:35.360 --> 00:08:37.909
in anything other than
multiple pastel colors

00:08:37.909 --> 00:08:39.659
because this actually
makes it much easier

00:08:39.659 --> 00:08:41.524
to see some patterns.

00:08:41.524 --> 00:08:42.940
And so what we can
do over here is

00:08:42.940 --> 00:08:45.520
we can factor out
the cosine t sine t.

00:08:45.520 --> 00:08:52.720
And so this is equal to
cosine t sine t times sine

00:08:52.720 --> 00:08:56.810
squared s plus cosine squared s.

00:08:56.810 --> 00:08:59.700
And this we know the
definition of the unit circle.

00:08:59.700 --> 00:09:01.500
This is just going
to be equal to 1.

00:09:01.500 --> 00:09:04.270
So that was a significant
simplification.

00:09:04.270 --> 00:09:07.860
And so now we get
our cross product

00:09:07.860 --> 00:09:11.050
being equal to-- let
me just rewrite it all.

00:09:11.050 --> 00:09:12.560
Our cross product.

00:09:12.560 --> 00:09:19.950
r sub s crossed with r sub t
is going to be equal to cosine

00:09:19.950 --> 00:09:33.000
squared t cosine s times our i
unit vector plus cosine squared

00:09:33.000 --> 00:09:45.150
t cosine sine of s
times our j unit vector

00:09:45.150 --> 00:09:51.130
plus-- all we have left because
this is just 1-- cosine t sine

00:09:51.130 --> 00:10:01.900
t times our k unit vector.

00:10:01.900 --> 00:10:04.567
So that was pretty good
but we're still not done.

00:10:04.567 --> 00:10:06.650
We need to figure out the
magnitude of this thing.

00:10:06.650 --> 00:10:09.610
Remember, d sigma
simplified to the magnitude

00:10:09.610 --> 00:10:10.987
of this thing times ds dt.

00:10:10.987 --> 00:10:13.070
So let's figure out what
the magnitude of that is.

00:10:13.070 --> 00:10:14.528
And this is really
the home stretch

00:10:14.528 --> 00:10:16.100
so I'm really
crossing my fingers

00:10:16.100 --> 00:10:18.870
that I don't make any
careless mistakes now.

00:10:18.870 --> 00:10:24.550
So the magnitude of
all of this business

00:10:24.550 --> 00:10:27.750
is going to be equal
to the square root--

00:10:27.750 --> 00:10:29.850
and I'm just going to
square each of these terms

00:10:29.850 --> 00:10:31.433
and then add them
up-- the square root

00:10:31.433 --> 00:10:34.370
of the sum of the squares
of each of those terms.

00:10:34.370 --> 00:10:36.800
So the square of
this is going to be--

00:10:36.800 --> 00:10:39.680
cosine squared squared is
cosine to the fourth-- cosine

00:10:39.680 --> 00:10:47.760
to the fourth t cosine
squared of s plus cosine

00:10:47.760 --> 00:10:51.670
to the fourth t sine
squared s-- and I already

00:10:51.670 --> 00:10:58.560
see a pattern jumping
out-- plus cosine

00:10:58.560 --> 00:11:02.370
squared t sine squared t.

00:11:02.370 --> 00:11:04.530
Now, the first pattern
I see is this--

00:11:04.530 --> 00:11:06.420
is just this first
part right over here.

00:11:06.420 --> 00:11:09.680
We can factor out a
cosine to the fourth t.

00:11:09.680 --> 00:11:11.970
Then we get something
like this happening again.

00:11:11.970 --> 00:11:12.700
So let's do that.

00:11:12.700 --> 00:11:15.350
So these first two terms
are equal to cosine

00:11:15.350 --> 00:11:24.090
to the fourth t times cosine
squared s plus sine squared s.

00:11:24.090 --> 00:11:27.050
Which, once again,
we know is just 1.

00:11:27.050 --> 00:11:28.690
So this whole expression
has simplified

00:11:28.690 --> 00:11:33.390
to cosine to the fourth t
plus cosine squared t sine

00:11:33.390 --> 00:11:35.380
squared t.

00:11:35.380 --> 00:11:37.400
Now we can attempt to
simplify this again

00:11:37.400 --> 00:11:38.900
because this term
and this term both

00:11:38.900 --> 00:11:41.310
have a cosine squared t in them.

00:11:41.310 --> 00:11:42.627
So let's factor those out.

00:11:42.627 --> 00:11:44.460
So this is going to be
equal to-- everything

00:11:44.460 --> 00:11:46.050
I'm doing is under
the radical sign.

00:11:46.050 --> 00:11:51.670
So this is going to be
equal to a cosine squared

00:11:51.670 --> 00:11:55.450
t times cosine squared t.

00:11:55.450 --> 00:11:57.450
And when you factor out
a cosine squared t here,

00:11:57.450 --> 00:12:01.230
you just have a plus
a sine squared t.

00:12:01.230 --> 00:12:04.410
And that's nice because that,
once again, simplified to 1.

00:12:04.410 --> 00:12:06.024
All of this is under
the radical sign.

00:12:06.024 --> 00:12:07.940
Maybe I'll keep drawing
the radical signs here

00:12:07.940 --> 00:12:09.300
to make it clear that
all this is still

00:12:09.300 --> 00:12:10.520
under the radical sign.

00:12:10.520 --> 00:12:12.580
And then this is really,
really useful for us

00:12:12.580 --> 00:12:15.180
because the square root
of cosine squared of t

00:12:15.180 --> 00:12:17.470
is just going to be cosine of t.

00:12:17.470 --> 00:12:20.770
So all of that business
actually finally simplified

00:12:20.770 --> 00:12:22.510
to something pretty
straightforward.

00:12:22.510 --> 00:12:30.230
So all of this is just going
to be equal to cosine of t.

00:12:30.230 --> 00:12:32.260
So going back to what
we wanted before,

00:12:32.260 --> 00:12:34.130
we want to rewrite what d sigma.

00:12:34.130 --> 00:12:37.180
It's just cosine t ds dt.

00:12:37.180 --> 00:12:38.830
so let me write that down.

00:12:38.830 --> 00:12:42.050
So d sigma-- and then we can
use this for the next part-- d

00:12:42.050 --> 00:12:48.130
sigma is equal to
cosine of t ds dt.

00:12:48.130 --> 00:12:50.680
And I'll see you
in the next part.