Chapter 17.3, Problem 37E

To determine

(a)

To prove:

$\Delta \phi =div\left(\nabla \phi \right)$

To determine

(b)

To show:

That $\phi$ is harmonic if and only if $div\left(\nabla \phi \right)=0$.

To determine

(c)

To show:

That if F is the gradient of a harmonic function, then $Curl\left(F\right)=0$ and $Div\left(F\right)=0$.

To determine

(d)

To prove:

That $F\left(x,y,z\right)=\langle xz,-yz,\frac{1}{2}\left(x^2-y^2\right)\rangle$ is gradient of a harmonic function.