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Question: The range of the polar coordinates are 0 <= r <= ~, 0 <= theta <= pi, and 0 <= phi <= 2pi and the range of the Cartesian coordinates are - ~ < x,y,z, < ~. In the integrals we must relate the two. This is done by the Jacobian, and the result is,
dxdydz = (r^2)drsin(theta)d(theta)d(phi)
Look up the definition of the Jacobian and prove the above result.
Motivation: Jacobians are the general way of making sure that the volume element in one coordinate system is the same in another. In the case of polar coordinates, the volume element is often called the "solid angle".
z = r*cos(theta)
theta == q, phi == p
| sin(q).cos(p), r.cos(q).cos(p), -r.sin(q).sin(p)|
| sin(q).sin(p), r.cos(q).sin(p), ...
This solution illustrates which formula is required to solve this integral-based problem and shows all of the calculations required.