Posted by shylo on Wednesday, November 25, 2009 at 9:24am.
What you have to do here is show that the volume element:
dxdydz can be written as
r^2 sin(theta)dphi dtheta dr
where theta is the angle w.r.t. the z-axis and phi is the angle that corresponds to rotating around the z-axis.
It is easy to see that this is the volume element because you can see the three orthogonal length elements hee:
r dtheta
r sin(theta) dphi
Note that if you rotate around the z-axis, your radius will be
r sin(theta)
and
dr
If you want to prove this formally by direct substituton of
x = r sin(theta)cos(phi)
y = r sin(theta)sin(phi)
z = r cos(theta)
You have to write down the Jacobian, i.e. the 3x3 matrix of partial deivatives of the the three cartesian coordinates w.r.t. r, theta and phi.
Once you've got that the volume element is r^2 sin(theta)dphi dtheta dr you can integrate this straightforwadly. r ranges from zero to R, phi goes from zero to 2 pi and theta goes from zero to pi.
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