Exercise 13

Let $h(x,y,z)$ be a smooth function of three variables and let $\vF=grad(h)=\nabla{h}$.

1. Write out the component functions of $\vF$ in terms of partial derivatives of $h$.

2. Write the divergence of $\vF$ in terms of derivatives of $h$.

3. Is the divergence of $\vF$ necessarily zero? Explain why or why not.

4. Calculate $curl(\vF)$ in terms of derivatives of $h$.

5. Show that $curl(\vF)=\vec{0}$.