Chapter 17.8, Problem 36E

Radial fields Consider the radial vector field $\text{F}=\frac{r}{{\left|r\right|}^p}=\frac{\langle x,y,z\rangle }{{\left(x^2+y^2+z^2\right)}^{\frac{p}{2}}}$. Let S be the sphere of radius a centered at the origin.

a. Use a surface integral to show that the outward flux of F across S is 4πa^{3 − p}. Recall that the unit normal to the sphere is r/|r|.

b. For what values of p does F satisfy the conditions of the Divergence Theorem? For these values of p, use the fact (Theorem 17.10) that $\nabla \cdot \text{F}=\frac{3-p}{{\left|r\right|}^p}$ to compute the flux across S using the Divergence Theorem.