| Last time | Next time |
The latter is particularly useful, as we try to visualize the various volume elements.
The volume element ($dV$) is given by $rdrd{\theta}dz$, as one can see in Figure 7, p. 1054. This is a simple generalization of the polar formula of the area element $dA$.
We'll just look at some problems that have spherical symmetry that we might exploit. Figures 5 and 8 are worth examining in some detail.... \[ x=\rho\sin{\phi}\cos{\theta} \hspace{1in} y=\rho\sin{\phi}\sin{\theta} \hspace{1in} z=\rho\cos{\phi} \] and the volume element becomes \[ dV = \rho^2\ \sin(\phi)\ d\rho\ d\theta\ d\phi \]