The vector field $\mathbf{F}(\mathbf{x})$ is given in terms of cylindrical polar coordinates $(\rho, \phi, z)$ by

$$\mathbf{F}(\mathbf{x})=f(\rho) \mathbf{e}_{\rho}$$

where $f$ is a differentiable function of $\rho$, and $\mathbf{e}_{\rho}=\cos \phi \mathbf{e}_{x}+\sin \phi \mathbf{e}_{y}$ is the unit basis vector with respect to the coordinate $\rho$. Compute the partial derivatives $\partial F_{1} / \partial x, \partial F_{2} / \partial y$, $\partial F_{3} / \partial z$ and hence find the divergence $\nabla \cdot \mathbf{F}$ in terms of $\rho$ and $\phi$.

The domain $V$ is bounded by the surface $z=\left(x^{2}+y^{2}\right)^{-1}$, by the cylinder $x^{2}+y^{2}=1$, and by the planes $z=\frac{1}{4}$ and $z=1$. Sketch $V$ and compute its volume.

Find the most general function $f(\rho)$ such that $\nabla \cdot \mathbf{F}=0$, and verify the divergence theorem for the corresponding vector field $\mathbf{F}(\mathbf{x})$ in $V$.