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Deriving the volume element in spherical coordinates

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A differential of volume is given by:

dV = r^2*sin(theta)*dr*d(theta)*d(phi) can you derive this for me i.e. using a differential of volume and spherical coordinates show how this equation is arrived at?

r is a radius, theta is the angle in the x-y plane and psi is the the z-y plane

Please show a diagram

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Solution Preview

Here is the derivation, or should I say the two derivations.

The first is the straight hard-core multivariable calculus approach in which one has to calculate the Jacobian of the transformation.

The ...

Solution Summary

The 3 pages solution demonstrates how to utilize the Cartesian to Spherical coordinates conversion to obtain the infinitesimal volume element in an analytic manner.
then it demonstrates how to get the same expression using graphics.

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