Chapter 18.1, Problem 44E

To determine

To prove:

a. $\frac{\partial \phi }{\partial n}\left(a+r\cos \theta ,b+r\sin \theta \right)=\frac{\partial \phi }{\partial r}\left(a+r\cos \theta ,b+r\sin \theta \right)$

b. Use differentiation under the integral sign to prove that

$\frac{d}{dr}{I}_{\phi }\left(r\right)=\frac{1}{2\pi r}{\displaystyle {\int }_{C_r}\frac{\partial \phi }{\partial n}}ds$

c. $\frac{d}{dr}{I}_{\phi }\left(r\right)=\frac{1}{2\pi r}{\displaystyle {\iint }_{D\left(r\right)}\Delta \phi dA}$,where $D\left(r\right)$ is the interior of $C_r$.