2.11 Unit base vectors for any coordinate system can be computed using the
equation
$e_i = \frac{r_{,i}}{|r_{,i}|}$,
in which $r$ is the position vector from the origin to any point in space, and
comma denotes differentiation with respect to coordinate $x_i$. Clearly, this
relation holds in Cartesian coordinates, where
$r = xe_x + ye_y + ze_z$.
(a) Cartesian and spherical coordinates are related by
$x = r \sin \theta \cos \phi$
$y = r \sin \theta \sin \phi$
$z = r \cos \theta$.
Determine the unit base vectors for spherical coordinates in terms of $e_x$,
$e_y$, and $e_z$.
(b) Using the results from part (a), write the differential vector $dr$ in the
form
$dr = ds_r e_r + ds_\theta e_\theta + ds_\phi e_\phi$.
(c) Write the gradient operator $\nabla$ and the Laplacian $\nabla^2$ in spherical
coordinates.
