Let $V$ be a bounded region of $\mathbb{R}^{3}$ and $S$ be its boundary. Let $\phi$ be the unique solution to $\nabla^{2} \phi=0$ in $V$, with $\phi=f(\mathbf{x})$ on $S$, where $f$ is a given function. Consider any smooth function $w$ also equal to $f(\mathbf{x})$ on $S$. Show, by using Green's first theorem or otherwise, that

$$\int_{V}|\nabla w|^{2} d V \geqslant \int_{V}|\nabla \phi|^{2} d V$$

[Hint: Set $w=\phi+\delta .]$

Consider the partial differential equation

$$\frac{\partial}{\partial t} w=\nabla^{2} w$$

for $w(t, \mathbf{x})$, with initial condition $w(0, \mathbf{x})=w_{0}(\mathbf{x})$ in $V$, and boundary condition $w(t, \mathbf{x})=$ $f(\mathbf{x})$ on $S$ for all $t \geqslant 0$. Show that

$$\frac{\partial}{\partial t} \int_{V}|\nabla w|^{2} d V \leqslant 0$$

with equality holding only when $w(t, \mathbf{x})=\phi(\mathbf{x})$.

Show that $(*)$ remains true with the boundary condition

$$\frac{\partial w}{\partial t}+\alpha(\mathbf{x}) \frac{\partial w}{\partial n}=0$$

on $S$, provided $\alpha(\mathbf{x}) \geqslant 0$.

3/II/10A Vector Calculus

Write down Stokes' theorem for a vector field $\mathbf{B}(\mathbf{x})$ on $\mathbb{R}^{3}$.

Consider the bounded surface $S$ defined by

$$z=x^{2}+y^{2}, \quad \frac{1}{4} \leqslant z \leqslant 1$$

Sketch the surface and calculate the surface element $d \mathbf{S}$. For the vector field

$$\mathbf{B}=\left(-y^{3}, x^{3}, z^{3}\right)$$

calculate $I=\int_{S}(\nabla \times \mathbf{B}) \cdot d \mathbf{S}$ directly.

Show using Stokes' theorem that $I$ may be rewritten as a line integral and verify this yields the same result.