Welcome to a lesson on the Jacobian
for the change of three variables in a triple integral.
We just got done discussing the Jacobian for a change
of variables using double integrals.
We're also familiar with triple integrals using
rectangular, cylindrical, and spherical coordinates.
And we know when converting a
triple integral to cylindrical form
we add an extra factor of r in the integrand,
and when converting to spherical form, we add extra factors
of rho squared sine phi in our integrand.
So now we'll define the Jacobian
that will allow us to determine
the integrating factor we're converting
to any new coordinate system.
And here's how it works, if r and s are
regions of integration in the xyz-space
and uvw-space related by the following three equations,
they convert from a rectangular triple integral
to a triple integral in the uvw coordinate system.
There's two major steps.
First we'll convert the given function in terms of
x, y and z to a function of u, v and w.
And then differential V will be replaced
with the absolute value of the Jacobian
or the extra integrating factor
and then du, dv, dw, or any other order
of the three differentials.
And the Jacobian is defined by
the three by three determinant of
these partial derivatives with respects to u, v and w.
So what I want to do in this video,
so this video we're going to verify that differential V
is equal to rho squared sine phi when converting from
rectangular to spherical coordinates using triple integrals.
So here are the three equations that relate
rho, phi and theta to x, y and z.
So let's go and set up this three by three determinant,
and then make sure that we get rho squared sine phi.
So the partial derivative of x with respects to rho
is going to be sine phi cosine theta.
And the partial derivative of x with respects to phi,
derivative of sine phi is going to be cosine phi,
so we'll have rho cosine phi,
cosine theta,
and then we'll have the partial of x with respects to theta,
the derivative of cosine theta is going to be
negative sine theta, so we'll have
negative rho sine phi sine theta.
Then for the second row, we'll have
the partial derivative of y with respects to rho.
It's going to be sine phi sine theta.
And the partial of y with respects to phi,
or the derivative of sine phi is cosine phi, so we'll have
rho cosine phi,
sine theta,
and the partial of y with respects to theta,
derivative of sine theta is cosine theta,
so we'll have rho sine phi,
cosine theta.
And the third row will be the partial derivatives of z,
so the partial of z with respects to rho
will be cosine phi.
The partial of z with respects to phi
is going to be negative rho sine phi.
And the partial with respects to theta is going to be zero.
So let's go to the next page and evaluate
this three by three determinant.
So here's our three by three determinant.
I'm going to go ahead and use the diagonal method
to evaluate this three by three determinant
rather than the cofactor expansion method.
So remember, for the diagonal method,
we're going to determine the products
of the diagonals in this direction.
One,
two,
three.
And then we'll find the sum of these,
and then we'll subtract the products of these diagonals.
So again, we're going to sum the product
of the green diagonals,
and then we'll subtract the product of the blue diagonals.
So this first diagonal is going to be zero.
The second diagonal here is going to be rho squared,
cosine squared phi,
cosine squared theta,
sine phi.
Plus the product of this diagonal here
is going to be a positive rho squared,
sine phi cubed,
sine squared theta.
And now we'll subtract
the product of the blue diagonals.
So starting with this diagonal here,
we're going to have a negative rho squared,
cosine squared phi, sine squared theta,
sine phi.
Plus, here we're going to have a negative rho squared,
cosine squared theta,
sine cubed phi.
And then this last product here would be zero.
And to keep these organized,
let's go ahead and call this expression one,
expression two,
three and four.
Using equation two and equation four,
let's look for the greatest common factor.
They both have a rho squared in common,
as well as sine cubed phi.
Let's go ahead and factor that out of these two expressions.
So we'd be left with sine squared theta.
Notice here we're subtracting a negative,
so it's going to be a positive.
And then from here, we're left with cosine squared theta.
So we'd have this result plus,
now let's take a look at equation one and three
and do the same thing with identify
the greatest common factor.
So the greatest common factor of these two expressions
would be rho squared, cosine squared phi,
sine phi.
Looking at equation one,
we'd be left with
cosine squared theta,
again, notice here, we're subtracting a negative
so it'll be plus,
and we'll be left with sine squared theta.
Well, this is equal to one and so is this.
So we're left with rho squared, sine cubed phi,
plus rho squared, cosine squared phi,
sine phi.
Let's go ahead and continue simplifying this sum
on the next page.
Well, now we have a common factor of rho squared sine phi.
Let's factor this again.
We're left with sine squared phi
plus cosine squared phi.
And once again, we know that this
is equal to one.
We finally have our end result,
and this is our Jacobian.
So in converting a triple integral from rectangular
to spherical coordinates, we know that differential V
is equal to the Jacobian times differential rho,
differential phi, differential theta,
or any other order of these differentials.
Thank you for watching.