Exercise 18

Consider the change of variables $$\begin{gathered} x(\rho, \theta, \phi) = \rho \sin(\phi) \cos(\theta) \ \ \ \ \ y(\rho, \theta, \phi) = \rho \sin(\phi) \sin(\theta) \ \ \ \ \ z(\rho, \theta, \phi) = \rho \cos(\phi), \end{gathered}$$ which is the transformation from spherical coordinates to rectangular coordinates. Determine the Jacobian of the transformation. How is the result connected to our earlier work with iterated integrals in spherical coordinates?