Multivariable Calculus Sheet 7

1. Verifying the divergence theorem

Verify the divergence theorem for the unit cube $R = [0, 1]^3$ where \[ \mb{F} = [(x-1)x^2 y, (y-1)^2 xy, z^2 - 1]. \]


2. Applying the Divergence Theorem to Poisson's equation

Let $R$ be the region $0 < a < r < b$, where $r$ is the distance from the origin in the plane. Find a solution of the boundary-value problem \begin{gather*} \nabla^2 f = -1 \quad \text{in $R$}
\pd{f}{n} + f = 0 \quad \text{on $\partial R$}, \end{gather*}

which is a function of $r$ only. SHow that this is the only solution, even within the class of not necessarily radial functions.