• # Paper 3, Section I, B

(a) What is meant by an antisymmetric tensor of second rank? Show that if a second rank tensor is antisymmetric in one Cartesian coordinate system, it is antisymmetric in every Cartesian coordinate system.

(b) Consider the vector field $\mathbf{F}=(y, z, x)$ and the second rank tensor defined by $T_{i j}=\partial F_{i} / \partial x_{j}$. Calculate the components of the antisymmetric part of $T_{i j}$ and verify that it equals $-(1 / 2) \epsilon_{i j k} B_{k}$, where $\epsilon_{i j k}$ is the alternating tensor and $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{F}$.

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• # Paper 3, Section I, B

(a) Prove that

\begin{aligned} &\nabla \times(\psi \mathbf{A})=\psi \mathbf{\nabla} \times \mathbf{A}+\boldsymbol{\nabla} \psi \times \mathbf{A} \\ &\nabla \cdot(\mathbf{A} \times \mathbf{B})=\mathbf{B} \cdot \boldsymbol{\nabla} \times \mathbf{A}-\mathbf{A} \cdot \boldsymbol{\nabla} \times \mathbf{B} \end{aligned}

where $\mathbf{A}$ and $\mathbf{B}$ are differentiable vector fields and $\psi$ is a differentiable scalar field.

(b) Find the solution of $\nabla^{2} u=16 r^{2}$ on the two-dimensional domain $\mathcal{D}$ when

(i) $\mathcal{D}$ is the unit disc $0 \leqslant r \leqslant 1$, and $u=1$ on $r=1$;

(ii) $\mathcal{D}$ is the annulus $1 \leqslant r \leqslant 2$, and $u=1$ on both $r=1$ and $r=2$.

[Hint: the Laplacian in plane polar coordinates is:

$\left.\nabla^{2} u=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}} . \quad\right]$

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• # Paper 3, Section II, B

For a given charge distribution $\rho(\mathbf{x}, t)$ and current distribution $\mathbf{J}(\mathbf{x}, t)$ in $\mathbb{R}^{3}$, the electric and magnetic fields, $\mathbf{E}(\mathbf{x}, t)$ and $\mathbf{B}(\mathbf{x}, t)$, satisfy Maxwell's equations, which in suitable units, read

\begin{aligned} \boldsymbol{\nabla} \cdot \mathbf{E}=\rho, & \boldsymbol{\nabla} \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ \boldsymbol{\nabla} \cdot \mathbf{B}=0, & \boldsymbol{\nabla} \times \mathbf{B}=\mathbf{J}+\frac{\partial \mathbf{E}}{\partial t} \end{aligned}

The Poynting vector $\mathbf{P}$ is defined as $\mathbf{P}=\mathbf{E} \times \mathbf{B}$.

(a) For a closed surface $\mathcal{S}$ around a volume $\mathcal{V}$, show that

$\int_{\mathcal{S}} \mathbf{P} \cdot d \mathbf{S}=-\int_{\mathcal{V}} \mathbf{E} \cdot \mathbf{J} d V-\frac{\partial}{\partial t} \int_{\mathcal{V}} \frac{|\mathbf{E}|^{2}+|\mathbf{B}|^{2}}{2} d V$

(b) Suppose $\mathbf{J}=\mathbf{0}$ and consider an electromagnetic wave

$\mathbf{E}=E_{0} \hat{\mathbf{y}} \cos (k x-\omega t) \quad \text { and } \quad \mathbf{B}=B_{0} \hat{\mathbf{z}} \cos (k x-\omega t)$

where $E_{0}, B_{0}, k$ and $\omega$ are positive constants. Show that these fields satisfy Maxwell's equations for appropriate $E_{0}, \omega$, and $\rho$.

Confirm the wave satisfies the integral identity $(*)$ by considering its propagation through a box $\mathcal{V}$, defined by $0 \leqslant x \leqslant \pi /(2 k), 0 \leqslant y \leqslant L$, and $0 \leqslant z \leqslant L$.

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• # Paper 3, Section II, B

(a) By a suitable change of variables, calculate the volume enclosed by the ellipsoid $x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1$, where $a, b$, and $c$ are constants.

(b) Suppose $T_{i j}$ is a second rank tensor. Use the divergence theorem to show that

$\int_{\mathcal{S}} T_{i j} n_{j} d S=\int_{\mathcal{V}} \frac{\partial T_{i j}}{\partial x_{j}} d V$

where $\mathcal{S}$ is a closed surface, with unit normal $n_{j}$, and $\mathcal{V}$ is the volume it encloses.

[Hint: Consider $e_{i} T_{i j}$ for a constant vector $\left.e_{i} .\right]$

(c) A half-ellipsoidal membrane $\mathcal{S}$ is described by the open surface $4 x^{2}+4 y^{2}+z^{2}=4$, with $z \geqslant 0$. At a given instant, air flows beneath the membrane with velocity $\mathbf{u}=$ $(-y, x, \alpha)$, where $\alpha$ is a constant. The flow exerts a force on the membrane given by

$F_{i}=\int_{\mathcal{S}} \beta^{2} u_{i} u_{j} n_{j} d S$

where $\beta$ is a constant parameter.

Show the vector $a_{i}=\partial\left(u_{i} u_{j}\right) / \partial x_{j}$ can be rewritten as $\mathbf{a}=-(x, y, 0)$.

Hence use $(*)$ to calculate the force $F_{i}$ on the membrane.

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• # Paper 3, Section II, B

(a) By considering an appropriate double integral, show that

$\int_{0}^{\infty} e^{-a x^{2}} d x=\sqrt{\frac{\pi}{4 a}}$

where $a>0$.

(b) Calculate $\int_{0}^{1} x^{y} d y$, treating $x$ as a constant, and hence show that

$\int_{0}^{\infty} \frac{\left(e^{-u}-e^{-2 u}\right)}{u} d u=\log 2$

(c) Consider the region $\mathcal{D}$ in the $x-y$ plane enclosed by $x^{2}+y^{2}=4, y=1$, and $y=\sqrt{3} x$ with $1.

Sketch $\mathcal{D}$, indicating any relevant polar angles.

A surface $\mathcal{S}$ is given by $z=x y /\left(x^{2}+y^{2}\right)$. Calculate the volume below this surface and above $\mathcal{D}$.

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• # Paper 3, Section II, B

(a) Given a space curve $\mathbf{r}(t)=(x(t), y(t), z(t)$, with $t$ a parameter (not necessarily arc-length), give mathematical expressions for the unit tangent, unit normal, and unit binormal vectors.

(b) Consider the closed curve given by

$x=2 \cos ^{3} t, \quad y=\sin ^{3} t, \quad z=\sqrt{3} \sin ^{3} t$

where $t \in[0,2 \pi)$.

Show that the unit tangent vector $\mathbf{T}$ may be written as

$\mathbf{T}=\pm \frac{1}{2}(-2 \cos t, \sin t, \sqrt{3} \sin t)$

with each sign associated with a certain range of $t$, which you should specify.

Calculate the unit normal and the unit binormal vectors, and hence deduce that the curve lies in a plane.

(c) A closed space curve $\mathcal{C}$ lies in a plane with unit normal $\mathbf{n}=(a, b, c)$. Use Stokes' theorem to prove that the planar area enclosed by $\mathcal{C}$ is the absolute value of the line integral

$\frac{1}{2} \int_{\mathcal{C}}(b z-c y) d x+(c x-a z) d y+(a y-b x) d z$

Hence show that the planar area enclosed by the curve given by $(*)$ is $(3 / 2) \pi$.

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• # Paper 2, Section I, B

(a) Evaluate the line integral

$\int_{(0,1)}^{(1,2)}\left(x^{2}-y\right) d x+\left(y^{2}+x\right) d y$

along

(i) a straight line from $(0,1)$ to $(1,2)$,

(ii) the parabola $x=t, y=1+t^{2}$.

(b) State Green's theorem. The curve $C_{1}$ is the circle of radius $a$ centred on the origin and traversed anticlockwise and $C_{2}$ is another circle of radius $b traversed clockwise and completely contained within $C_{1}$ but may or may not be centred on the origin. Find

$\int_{C_{1} \cup C_{2}} y(x y-\lambda) d x+x^{2} y d y$

as a function of $\lambda$.

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• # Paper 2, Section II, B

(a) State the value of $\partial x_{i} / \partial x_{j}$ and find $\partial r / \partial x_{j}$ where $r=|\boldsymbol{x}|$.

(b) A vector field $\boldsymbol{u}$ is given by

$\boldsymbol{u}=\frac{\boldsymbol{a}}{r}+\frac{(\boldsymbol{a} \cdot \boldsymbol{x}) \boldsymbol{x}}{r^{3}}$

where $\boldsymbol{a}$ is a constant vector. Calculate the second-rank tensor $d_{i j}=\partial u_{i} / \partial x_{j}$ using suffix notation and show how $d_{i j}$ splits naturally into symmetric and antisymmetric parts. Show that

$\nabla \cdot \boldsymbol{u}=0$

and

$\nabla \times u=\frac{2 a \times x}{r^{3}}$

(c) Consider the equation

$\nabla^{2} u=f$

on a bounded domain $V \subset \mathbb{R}^{3}$ subject to the mixed boundary condition

$(1-\lambda) u+\lambda \frac{d u}{d n}=0$

on the smooth boundary $S=\partial V$, where $\lambda \in[0,1)$ is a constant. Show that if a solution exists, it will be unique.

Find the spherically symmetric solution $u(r)$ for the choice $f=6$ in the region $r=|\boldsymbol{x}| \leqslant b$ for $b>0$, as a function of the constant $\lambda \in[0,1)$. Explain why a solution does not exist for $\lambda=1$

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• # Paper 2, Section II, B

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

Let the surface $S$ be the part of the inverted paraboloid

$z=5-x^{2}-y^{2}, \quad 1

and the vector field $\mathbf{A}(\mathbf{x})=\left(3 y,-x z, y z^{2}\right)$.

(a) Sketch the surface $S$ and directly calculate $I=\int_{S}(\nabla \times \mathbf{A}) \cdot d \mathbf{S}$.

(b) Now calculate $I$ a different way by using Stokes' theorem.

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• # Paper 3, Section I, B

Let

\begin{aligned} u &=\left(2 x+x^{2} z+z^{3}\right) \exp ((x+y) z) \\ v &=\left(x^{2} z+z^{3}\right) \exp ((x+y) z) \\ w &=\left(2 z+x^{3}+x^{2} y+x z^{2}+y z^{2}\right) \exp ((x+y) z) \end{aligned}

Show that $u d x+v d y+w d z$ is an exact differential, clearly stating any criteria that you use.

Show that for any path between $(-1,0,1)$ and $(1,0,1)$

$\int_{(-1,0,1)}^{(1,0,1)}(u d x+v d y+w d z)=4 \sinh 1$

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• # Paper 3, Section I, B

Apply the divergence theorem to the vector field $\mathbf{u}(\mathbf{x})=\mathbf{a} \phi(\mathbf{x})$ where $\mathbf{a}$ is an arbitrary constant vector and $\phi$ is a scalar field, to show that

$\int_{V} \nabla \phi d V=\int_{S} \phi d \mathbf{S}$

where $V$ is a volume bounded by the surface $S$ and $d \mathbf{S}$ is the outward pointing surface element.

Verify that this result holds when $\phi=x+y$ and $V$ is the spherical volume $x^{2}+$ $y^{2}+z^{2} \leqslant a^{2}$. [You may use the result that $d \mathbf{S}=a^{2} \sin \theta d \theta d \phi(\sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta)$, where $\theta$ and $\phi$ are the usual angular coordinates in spherical polars and the components of $d \mathbf{S}$ are with respect to standard Cartesian axes.]

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• # Paper 3, Section II, B

(a) The function $u$ satisfies $\nabla^{2} u=0$ in the volume $V$ and $u=0$ on $S$, the surface bounding $V$.

Show that $u=0$ everywhere in $V$.

The function $v$ satisfies $\nabla^{2} v=0$ in $V$ and $v$ is specified on $S$. Show that for all functions $w$ such that $w=v$ on $S$

$\int_{V} \nabla v \cdot \nabla w d V=\int_{V}|\nabla v|^{2} d V$

Hence show that

$\int_{V}|\boldsymbol{\nabla} w|^{2} d V=\int_{V}\left\{|\boldsymbol{\nabla} v|^{2}+|\boldsymbol{\nabla}(w-v)|^{2}\right\} d V \geqslant \int_{V}|\boldsymbol{\nabla} v|^{2} d V$

(b) The function $\phi$ satisfies $\nabla^{2} \phi=\rho(\mathbf{x})$ in the spherical region $|\mathbf{x}|, with $\phi=0$ on $|\mathbf{x}|=a$. The function $\rho(\mathbf{x})$ is spherically symmetric, i.e. $\rho(\mathbf{x})=\rho(|\mathbf{x}|)=\rho(r)$.

Suppose that the equation and boundary conditions are satisfied by a spherically symmetric function $\Phi(r)$. Show that

$4 \pi r^{2} \Phi^{\prime}(r)=4 \pi \int_{0}^{r} s^{2} \rho(s) d s$

Hence find the function $\Phi(r)$ when $\rho(r)$ is given by $\rho(r)=\left\{\begin{array}{ll}\rho_{0} & \text { if } 0 \leqslant r \leqslant b \\ 0 & \text { if } b, with $\rho_{0}$ constant.

Explain how the results obtained in part (a) of the question imply that $\Phi(r)$ is the only solution of $\nabla^{2} \phi=\rho(r)$ which satisfies the specified boundary condition on $|\mathbf{x}|=a$.

Use your solution and the results obtained in part (a) of the question to show that, for any function $w$ such that $w=1$ on $r=b$ and $w=0$ on $r=a$,

$\int_{U(b, a)}|\nabla w|^{2} d V \geqslant \frac{4 \pi a b}{a-b}$

where $U(b, a)$ is the region $b.

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• # Paper 3, Section II, B

Show that for a vector field $\mathbf{A}$

$\nabla \times(\boldsymbol{\nabla} \times \mathbf{A})=\boldsymbol{\nabla}(\boldsymbol{\nabla} \cdot \mathbf{A})-\nabla^{2} \mathbf{A}$

Hence find an $\mathbf{A}(\mathbf{x})$, with $\boldsymbol{\nabla} \cdot \mathbf{A}=0$, such that $\mathbf{u}=\left(y^{2}, z^{2}, x^{2}\right)=\nabla \times \mathbf{A}$. [Hint: Note that $\mathbf{A}(\mathbf{x})$ is not defined uniquely. Choose your expression for $\mathbf{A}(\mathbf{x})$ to be as simple as possible.

Now consider the cone $x^{2}+y^{2} \leqslant z^{2} \tan ^{2} \alpha, 0 \leqslant z \leqslant h$. Let $S_{1}$ be the curved part of the surface of the cone $\left(x^{2}+y^{2}=z^{2} \tan ^{2} \alpha, 0 \leqslant z \leqslant h\right)$ and $S_{2}$ be the flat part of the surface of the cone $\left(x^{2}+y^{2} \leqslant h^{2} \tan ^{2} \alpha, z=h\right)$.

Using the variables $z$ and $\phi$ as used in cylindrical polars $(r, \phi, z)$ to describe points on $S_{1}$, give an expression for the surface element $d \mathbf{S}$ in terms of $d z$ and $d \phi$.

Evaluate $\int_{S_{1}} \mathbf{u} \cdot d \mathbf{S}$.

What does the divergence theorem predict about the two surface integrals $\int_{S_{1}} \mathbf{u} \cdot d \mathbf{S}$ and $\int_{S_{2}} \mathbf{u} \cdot d \mathbf{S}$ where in each case the vector $d \mathbf{S}$ is taken outwards from the cone?

What does Stokes theorem predict about the integrals $\int_{S_{1}} \mathbf{u} \cdot d \mathbf{S}$ and $\int_{S_{2}} \mathbf{u} \cdot d \mathbf{S}$ (defined as in the previous paragraph) and the line integral $\int_{C} \mathbf{A} \cdot d \mathbf{l}$ where $C$ is the circle $x^{2}+y^{2}=h^{2} \tan ^{2} \alpha, z=h$ and the integral is taken in the anticlockwise sense, looking from the positive $z$ direction?

Evaluate $\int_{S_{2}} \mathbf{u} \cdot d \mathbf{S}$ and $\int_{C} \mathbf{A} \cdot d \mathbf{l}$, making your method clear and verify that each of these predictions holds.

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• # Paper 3, Section II, B

For a given set of coordinate axes the components of a 2 nd rank tensor $T$ are given by $T_{i j}$.

(a) Show that if $\lambda$ is an eigenvalue of the matrix with elements $T_{i j}$ then it is also an eigenvalue of the matrix of the components of $T$ in any other coordinate frame.

Show that if $T$ is a symmetric tensor then the multiplicity of the eigenvalues of the matrix of components of $T$ is independent of coordinate frame.

A symmetric tensor $T$ in three dimensions has eigenvalues $\lambda, \lambda, \mu$, with $\mu \neq \lambda$.

Show that the components of $T$ can be written in the form

$T_{i j}=\alpha \delta_{i j}+\beta n_{i} n_{j}$

where $n_{i}$ are the components of a unit vector.

(b) The tensor $T$ is defined by

$T_{i j}(\mathbf{y})=\int_{S} x_{i} x_{j} \exp \left(-c|\mathbf{y}-\mathbf{x}|^{2}\right) d A(\mathbf{x})$

where $S$ is the surface of the unit sphere, $\mathbf{y}$ is the position vector of a point on $S$, and $c$ is a constant.

Deduce, with brief reasoning, that the components of $T$ can be written in the form (1) with $n_{i}=y_{i}$. [You may quote any results derived in part (a).]

Using suitable spherical polar coordinates evaluate $T_{k k}$ and $T_{i j} y_{i} y_{j}$.

Explain how to deduce the values of $\alpha$ and $\beta$ from $T_{k k}$ and $T_{i j} y_{i} y_{j}$. [You do not need to write out the detailed formulae for these quantities.]

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• # Paper 3, Section II, B

Define the Jacobian, $J$, of the one-to-one transformation

$(x, y, z) \rightarrow(u, v, w)$

Give a careful explanation of the result

$\int_{D} f(x, y, z) d x d y d z=\int_{\Delta}|J| \phi(u, v, w) d u d v d w$

where

$\phi(u, v, w)=f(x(u, v, w), y(u, v, w), z(u, v, w))$

and the region $D$ maps under the transformation to the region $\Delta$.

Consider the region $D$ defined by

$x^{2}+\frac{y^{2}}{k^{2}}+z^{2} \leqslant 1$

and

$\frac{x^{2}}{\alpha^{2}}+\frac{y^{2}}{k^{2} \alpha^{2}}-\frac{z^{2}}{\gamma^{2}} \geqslant 1$

where $\alpha, \gamma$ and $k$ are positive constants.

Let $\tilde{D}$ be the intersection of $D$ with the plane $y=0$. Write down the conditions for $\tilde{D}$ to be non-empty. Sketch the geometry of $\tilde{D}$ in $x>0$, clearly specifying the curves that define its boundaries and points that correspond to minimum and maximum values of $x$ and of $z$ on the boundaries.

Use a suitable change of variables to evaluate the volume of the region $D$, clearly explaining the steps in your calculation.

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• # Paper 3, Section I, $4 \mathbf{C}$

In plane polar coordinates $(r, \theta)$, the orthonormal basis vectors $\mathbf{e}_{r}$ and $\mathbf{e}_{\theta}$ satisfy

$\frac{\partial \mathbf{e}_{r}}{\partial r}=\frac{\partial \mathbf{e}_{\theta}}{\partial r}=\mathbf{0}, \quad \frac{\partial \mathbf{e}_{r}}{\partial \theta}=\mathbf{e}_{\theta}, \quad \frac{\partial \mathbf{e}_{\theta}}{\partial \theta}=-\mathbf{e}_{r}, \quad \text { and } \quad \boldsymbol{\nabla}=\mathbf{e}_{r} \frac{\partial}{\partial r}+\mathbf{e}_{\theta} \frac{1}{r} \frac{\partial}{\partial \theta}$

Hence derive the expression $\nabla \cdot \nabla \phi=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial \phi}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} \phi}{\partial \theta^{2}}$ for the Laplacian operator $\nabla^{2}$.

Calculate the Laplacian of $\phi(r, \theta)=\alpha r^{\beta} \cos (\gamma \theta)$, where $\alpha, \beta$ and $\gamma$ are constants. Hence find all solutions to the equation

$\nabla^{2} \phi=0 \quad \text { in } \quad 0 \leqslant r \leqslant a, \quad \text { with } \quad \partial \phi / \partial r=\cos (2 \theta) \text { on } r=a$

Explain briefly how you know that there are no other solutions.

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• # Paper 3, Section I, C

Derive a formula for the curvature of the two-dimensional curve $\mathbf{x}(u)=(u, f(u))$.

Verify your result for the semicircle with radius $a$ given by $f(u)=\sqrt{a^{2}-u^{2}}$.

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• # Paper 3, Section II, C

(a) Suppose that a tensor $T_{i j}$ can be decomposed as

$T_{i j}=S_{i j}+\epsilon_{i j k} V_{k}$

where $S_{i j}$ is symmetric. Obtain expressions for $S_{i j}$ and $V_{k}$ in terms of $T_{i j}$, and check that $(*)$ is satisfied.

(b) State the most general form of an isotropic tensor of rank $k$ for $k=0,1,2,3$, and verify that your answers are isotropic.

(c) The general form of an isotropic tensor of rank 4 is

$T_{i j k l}=\alpha \delta_{i j} \delta_{k l}+\beta \delta_{i k} \delta_{j l}+\gamma \delta_{i l} \delta_{j k}$

Suppose that $A_{i j}$ and $B_{i j}$ satisfy the linear relationship $A_{i j}=T_{i j k l} B_{k l}$, where $T_{i j k l}$ is isotropic. Express $B_{i j}$ in terms of $A_{i j}$, assuming that $\beta^{2} \neq \gamma^{2}$ and $3 \alpha+\beta+\gamma \neq 0$. If instead $\beta=-\gamma \neq 0$ and $\alpha \neq 0$, find all $B_{i j}$ such that $A_{i j}=0$.

(d) Suppose that $C_{i j}$ and $D_{i j}$ satisfy the quadratic relationship $C_{i j}=T_{i j k l m n} D_{k l} D_{m n}$, where $T_{i j k l m n}$ is an isotropic tensor of rank 6 . If $C_{i j}$ is symmetric and $D_{i j}$ is antisymmetric, find the most general non-zero form of $T_{i j k l m n} D_{k l} D_{m n}$ and prove that there are only two independent terms. [Hint: You do not need to use the general form of an isotropic tensor of rank 6.]

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• # Paper 3, Section II, C

Use Maxwell's equations,

$\boldsymbol{\nabla} \cdot \mathbf{E}=\rho, \quad \boldsymbol{\nabla} \cdot \mathbf{B}=0, \quad \boldsymbol{\nabla} \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}, \quad \boldsymbol{\nabla} \times \mathbf{B}=\mathbf{J}+\frac{\partial \mathbf{E}}{\partial t}$

to derive expressions for $\frac{\partial^{2} \mathbf{E}}{\partial t^{2}}-\nabla^{2} \mathbf{E}$ and $\frac{\partial^{2} \mathbf{B}}{\partial t^{2}}-\nabla^{2} \mathbf{B}$ in terms of $\rho$ and $\mathbf{J}$.

Now suppose that there exists a scalar potential $\phi$ such that $\mathbf{E}=-\nabla \phi$, and $\phi \rightarrow 0$ as $r \rightarrow \infty$. If $\rho=\rho(r)$ is spherically symmetric, calculate $\mathbf{E}$ using Gauss's flux method, i.e. by integrating a suitable equation inside a sphere centred at the origin. Use your result to find $\mathbf{E}$ and $\phi$ in the case when $\rho=1$ for $r and $\rho=0$ otherwise.

For each integer $n \geqslant 0$, let $S_{n}$ be the sphere of radius $4^{-n}$ centred at the point $\left(1-4^{-n}, 0,0\right)$. Suppose that $\rho$ vanishes outside $S_{0}$, and has the constant value $2^{n}$ in the volume between $S_{n}$ and $S_{n+1}$ for $n \geqslant 0$. Calculate $\mathbf{E}$ and $\phi$ at the point $(1,0,0)$.

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• # Paper 3, Section II, C

State the formula of Stokes's theorem, specifying any orientation where needed.

Let $\mathbf{F}=\left(y^{2} z, x z+2 x y z, 0\right)$. Calculate $\boldsymbol{\nabla} \times \mathbf{F}$ and verify that $\boldsymbol{\nabla} \cdot \boldsymbol{\nabla} \times \mathbf{F}=0$.

Sketch the surface $S$ defined as the union of the surface $z=-1,1 \leqslant x^{2}+y^{2} \leqslant 4$ and the surface $x^{2}+y^{2}+z=3,1 \leqslant x^{2}+y^{2} \leqslant 4$.

Verify Stokes's theorem for $\mathbf{F}$ on $S$.

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• # Paper 3, Section II, C

Given a one-to-one mapping $u=u(x, y)$ and $v=v(x, y)$ between the region $D$ in the $(x, y)$-plane and the region $D^{\prime}$ in the $(u, v)$-plane, state the formula for transforming the integral $\iint_{D} f(x, y) d x d y$ into an integral over $D^{\prime}$, with the Jacobian expressed explicitly in terms of the partial derivatives of $u$ and $v$.

Let $D$ be the region $x^{2}+y^{2} \leqslant 1, y \geqslant 0$ and consider the change of variables $u=x+y$ and $v=x^{2}+y^{2}$. Sketch $D$, the curves of constant $u$ and the curves of constant $v$ in the $(x, y)$-plane. Find and sketch the image $D^{\prime}$ of $D$ in the $(u, v)$-plane.

Calculate $I=\iint_{D}(x+y) d x d y$ using this change of variables. Check your answer by calculating $I$ directly.

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• # Paper 3, Section $I$, B

(a) The two sets of basis vectors $\mathbf{e}_{i}$ and $\mathbf{e}_{i}^{\prime}$ (where $i=1,2,3$ ) are related by

$\mathbf{e}_{i}^{\prime}=R_{i j} \mathbf{e}_{j}$

where $R_{i j}$ are the entries of a rotation matrix. The components of a vector $\mathbf{v}$ with respect to the two bases are given by

$\mathbf{v}=v_{i} \mathbf{e}_{i}=v_{i}^{\prime} \mathbf{e}_{i}^{\prime}$

Derive the relationship between $v_{i}$ and $v_{i}^{\prime}$.

(b) Let $\mathbf{T}$ be a $3 \times 3$ array defined in each (right-handed orthonormal) basis. Using part (a), state and prove the quotient theorem as applied to $\mathbf{T}$.

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• # Paper 3, Section I, B

Use the change of variables $x=r \cosh \theta, y=r \sinh \theta$ to evaluate

$\int_{A} y d x d y$

where $A$ is the region of the $x y$-plane bounded by the two line segments:

\begin{aligned} &y=0, \quad 0 \leqslant x \leqslant 1 \\ &5 y=3 x, \quad 0 \leqslant x \leqslant \frac{5}{4} \end{aligned}

and the curve

$x^{2}-y^{2}=1, \quad x \geqslant 1$

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• # Paper 3, Section II, B

Let $S$ be a piecewise smooth closed surface in $\mathbb{R}^{3}$ which is the boundary of a volume $V$.

(a) The smooth functions $\phi$ and $\phi_{1}$ defined on $\mathbb{R}^{3}$ satisfy

$\nabla^{2} \phi=\nabla^{2} \phi_{1}=0$

in $V$ and $\phi(\mathbf{x})=\phi_{1}(\mathbf{x})=f(\mathbf{x})$ on $S$. By considering an integral of $\boldsymbol{\nabla} \psi \cdot \boldsymbol{\nabla} \psi$, where $\psi=\phi-\phi_{1}$, show that $\phi_{1}=\phi$.

(b) The smooth function $u$ defined on $\mathbb{R}^{3}$ satisfies $u(\mathbf{x})=f(\mathbf{x})+C$ on $S$, where $f$ is the function in part (a) and $C$ is constant. Show that

$\int_{V} \nabla u \cdot \nabla u d V \geqslant \int_{V} \nabla \phi \cdot \nabla \phi d V$

where $\phi$ is the function in part (a). When does equality hold?

(c) The smooth function $w(\mathbf{x}, t)$ satisfies

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

in $V$ and $\frac{\partial w}{\partial t}=0$ on $S$ for all $t$. Show that

$\frac{d}{d t} \int_{V} \nabla w \cdot \nabla w d V \leqslant 0$

with equality only if $\nabla^{2} w=0$ in $V$.

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• # Paper 3, Section II, B

(a) Let $\mathbf{x}=\mathbf{r}(s)$ be a smooth curve parametrised by arc length $s$. Explain the meaning of the terms in the equation

$\frac{d \mathbf{t}}{d s}=\kappa \mathbf{n},$

where $\kappa(s)$ is the curvature of the curve.

Now let $\mathbf{b}=\mathbf{t} \times \mathbf{n}$. Show that there is a scalar $\tau(s)$ (the torsion) such that

$\frac{d \mathbf{b}}{d s}=-\tau \mathbf{n}$

and derive an expression involving $\kappa$ and $\tau$ for $\frac{d \mathbf{n}}{d s}$.

(b) Given a (nowhere zero) vector field $\mathbf{F}$, the field lines, or integral curves, of $\mathbf{F}$ are the curves parallel to $\mathbf{F}(\mathbf{x})$ at each point $\mathbf{x}$. Show that the curvature $\kappa$ of the field lines of $\mathbf{F}$ satisfies

$\frac{\mathbf{F} \times(\mathbf{F} \cdot \boldsymbol{\nabla}) \mathbf{F}}{F^{3}}=\pm \kappa \mathbf{b}$

where $F=|\mathbf{F}|$.

(c) Use $(*)$ to find an expression for the curvature at the point $(x, y, z)$ of the field lines of $\mathbf{F}(x, y, z)=(x, y,-z)$.

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• # Paper 3, Section II, B

By a suitable choice of $\mathbf{u}$ in the divergence theorem

$\int_{V} \nabla \cdot \mathbf{u} d V=\int_{S} \mathbf{u} \cdot d \mathbf{S}$

show that

$\int_{V} \nabla \phi d V=\int_{S} \phi d \mathbf{S}$

for any continuously differentiable function $\phi$.

For the curved surface of the cone

$\mathbf{x}=(r \cos \theta, r \sin \theta, \sqrt{3} r), \quad 0 \leqslant \sqrt{3} r \leqslant 1, \quad 0 \leqslant \theta \leqslant 2 \pi$

show that $d \mathbf{S}=(\sqrt{3} \cos \theta, \sqrt{3} \sin \theta,-1) r d r d \theta$.

Verify that $(*)$ holds for this cone and $\phi(x, y, z)=z^{2}$.

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• # Paper 3, Section II, B

(a) The time-dependent vector field $\mathbf{F}$ is related to the vector field $\mathbf{B}$ by

$\mathbf{F}(\mathbf{x}, t)=\mathbf{B}(\mathbf{z})$

where $\mathbf{z}=t \mathbf{x}$. Show that

$(\mathbf{x} \cdot \nabla) \mathbf{F}=t \frac{\partial \mathbf{F}}{\partial t} \text {. }$

(b) The vector fields $\mathbf{B}$ and $\mathbf{A}$ satisfy $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}$. Show that $\boldsymbol{\nabla} \cdot \mathbf{B}=0$.

(c) The vector field $\mathbf{B}$ satisfies $\boldsymbol{\nabla} \cdot \mathbf{B}=0$. Show that

$\mathbf{B}(\mathbf{x})=\nabla \times(\mathbf{D}(\mathbf{x}) \times \mathbf{x})$

where

$\mathbf{D}(\mathbf{x})=\int_{0}^{1} t \mathbf{B}(t \mathbf{x}) d t$

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• # Paper 3, Section I, C

If $\mathbf{v}=\left(v_{1}, v_{2}, v_{3}\right)$ and $\mathbf{w}=\left(w_{1}, w_{2}, w_{3}\right)$ are vectors in $\mathbb{R}^{3}$, show that $T_{i j}=v_{i} w_{j}$ defines a rank 2 tensor. For which choices of the vectors $\mathbf{v}$ and $\mathbf{w}$ is $T_{i j}$ isotropic?

Write down the most general isotropic tensor of rank 2 .

Prove that $\epsilon_{i j k}$ defines an isotropic rank 3 tensor.

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• # Paper 3, Section I, C

State the chain rule for the derivative of a composition $t \mapsto f(\mathbf{X}(t))$, where $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ and $\mathbf{X}: \mathbb{R} \rightarrow \mathbb{R}^{n}$ are smooth $.$

Consider parametrized curves given by

$\mathbf{x}(t)=(x(t), y(t))=(a \cos t, a \sin t) .$

Calculate the tangent vector $\frac{d \mathbf{x}}{d t}$ in terms of $x(t)$ and $y(t)$. Given that $u(x, y)$ is a smooth function in the upper half-plane $\left\{(x, y) \in \mathbb{R}^{2} \mid y>0\right\}$ satisfying

$x \frac{\partial u}{\partial y}-y \frac{\partial u}{\partial x}=u$

deduce that

$\frac{d}{d t} u(x(t), y(t))=u(x(t), y(t))$

If $u(1,1)=10$, find $u(-1,1)$.

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• # Paper 3, Section II, C

(a) Let

$\mathbf{F}=(z, x, y)$

and let $C$ be a circle of radius $R$ lying in a plane with unit normal vector $(a, b, c)$. Calculate $\nabla \times \mathbf{F}$ and use this to compute $\oint_{C} \mathbf{F} \cdot d \mathbf{x}$. Explain any orientation conventions which you use.

(b) Let $\mathbf{F}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ be a smooth vector field such that the matrix with entries $\frac{\partial F_{j}}{\partial x_{i}}$ is symmetric. Prove that $\oint_{C} \mathbf{F} \cdot d \mathbf{x}=0$ for every circle $C \subset \mathbb{R}^{3}$.

(c) Let $\mathbf{F}=\frac{1}{r}(x, y, z)$, where $r=\sqrt{x^{2}+y^{2}+z^{2}}$ and let $C$ be the circle which is the intersection of the sphere $(x-5)^{2}+(y-3)^{2}+(z-2)^{2}=1$ with the plane $3 x-5 y-z=2$. Calculate $\oint_{C} \mathbf{F} \cdot d \mathbf{x}$.

(d) Let $\mathbf{F}$ be the vector field defined, for $x^{2}+y^{2}>0$, by

$\mathbf{F}=\left(\frac{-y}{x^{2}+y^{2}}, \frac{x}{x^{2}+y^{2}}, z\right)$

Show that $\nabla \times \mathbf{F}=\mathbf{0}$. Let $C$ be the curve which is the intersection of the cylinder $x^{2}+y^{2}=1$ with the plane $z=x+200$. Calculate $\oint_{C} \mathbf{F} \cdot d \mathbf{x}$.

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• # Paper 3, Section II, C

(a) For smooth scalar fields $u$ and $v$, derive the identity

$\nabla \cdot(u \nabla v-v \nabla u)=u \nabla^{2} v-v \nabla^{2} u$

and deduce that

\begin{aligned} \int_{\rho \leqslant|\mathbf{x}| \leqslant r}\left(v \nabla^{2} u-u \nabla^{2} v\right) d V=\int_{|\mathbf{x}|=r}\left(v \frac{\partial u}{\partial n}-u \frac{\partial v}{\partial n}\right) d S \\ &-\int_{|\mathbf{x}|=\rho}\left(v \frac{\partial u}{\partial n}-u \frac{\partial v}{\partial n}\right) d S \end{aligned}

Here $\nabla^{2}$ is the Laplacian, $\frac{\partial}{\partial n}=\mathbf{n} \cdot \nabla$ where $\mathbf{n}$ is the unit outward normal, and $d S$ is the scalar area element.

(b) Give the expression for $(\nabla \times \mathbf{V})_{i}$ in terms of $\epsilon_{i j k}$. Hence show that

$\nabla \times(\nabla \times \mathbf{V})=\nabla(\nabla \cdot \mathbf{V})-\nabla^{2} \mathbf{V}$

(c) Assume that if $\nabla^{2} \varphi=-\rho$, where $\varphi(\mathbf{x})=O\left(|\mathbf{x}|^{-1}\right)$ and $\nabla \varphi(\mathbf{x})=O\left(|\mathbf{x}|^{-2}\right)$ as $|\mathbf{x}| \rightarrow \infty$, then

$\varphi(\mathbf{x})=\int_{\mathbb{R}^{3}} \frac{\rho(\mathbf{y})}{4 \pi|\mathbf{x}-\mathbf{y}|} d V .$

The vector fields $\mathbf{B}$ and $\mathbf{J}$ satisfy

$\nabla \times \mathbf{B}=\mathbf{J}$

Show that $\nabla \cdot \mathbf{J}=0$. In the case that $\mathbf{B}=\nabla \times \mathbf{A}$, with $\nabla \cdot \mathbf{A}=0$, show that

$\mathbf{A}(\mathbf{x})=\int_{\mathbb{R}^{3}} \frac{\mathbf{J}(\mathbf{y})}{4 \pi|\mathbf{x}-\mathbf{y}|} d V$

and hence that

$\mathbf{B}(\mathbf{x})=\int_{\mathbb{R}^{3}} \frac{\mathbf{J}(\mathbf{y}) \times(\mathbf{x}-\mathbf{y})}{4 \pi|\mathbf{x}-\mathbf{y}|^{3}} d V$

Verify that $\mathbf{A}$ given by $(*)$ does indeed satisfy $\nabla \cdot \mathbf{A}=0$. [It may be useful to make a change of variables in the right hand side of $(*)$.]

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• # Paper 3, Section II, C

Define the Jacobian $J[\mathbf{u}]$ of a smooth mapping $\mathbf{u}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$. Show that if $\mathbf{V}$ is the vector field with components

$V_{i}=\frac{1}{3 !} \epsilon_{i j k} \epsilon_{a b c} \frac{\partial u_{a}}{\partial x_{j}} \frac{\partial u_{b}}{\partial x_{k}} u_{c}$

then $J[\mathbf{u}]=\nabla \cdot \mathbf{V}$. If $\mathbf{v}$ is another such mapping, state the chain rule formula for the derivative of the composition $\mathbf{w}(\mathbf{x})=\mathbf{u}(\mathbf{v}(\mathbf{x}))$, and hence give $J[\mathbf{w}]$ in terms of $J[\mathbf{u}]$ and $J[\mathbf{v}]$.

Let $\mathbf{F}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ be a smooth vector field. Let there be given, for each $t \in \mathbb{R}$, a smooth mapping $\mathbf{u}_{t}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ such that $\mathbf{u}_{t}(\mathbf{x})=\mathbf{x}+t \mathbf{F}(\mathbf{x})+o(t)$ as $t \rightarrow 0$. Show that

$J\left[\mathbf{u}_{t}\right]=1+t Q(x)+o(t)$

for some $Q(x)$, and express $Q$ in terms of $\mathbf{F}$. Assuming now that $\mathbf{u}_{t+s}(\mathbf{x})=\mathbf{u}_{t}\left(\mathbf{u}_{s}(\mathbf{x})\right)$, deduce that if $\nabla \cdot \mathbf{F}=0$ then $J\left[\mathbf{u}_{t}\right]=1$ for all $t \in \mathbb{R}$. What geometric property of the mapping $\mathbf{x} \mapsto \mathbf{u}_{t}(\mathbf{x})$ does this correspond to?

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• # Paper 3, Section II, C

What is a conservative vector field on $\mathbb{R}^{n}$ ?

State Green's theorem in the plane $\mathbb{R}^{2}$.

(a) Consider a smooth vector field $\mathbf{V}=(P(x, y), Q(x, y))$ defined on all of $\mathbb{R}^{2}$ which satisfies

$\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}=0$

By considering

$F(x, y)=\int_{0}^{x} P\left(x^{\prime}, 0\right) d x^{\prime}+\int_{0}^{y} Q\left(x, y^{\prime}\right) d y^{\prime}$

or otherwise, show that $\mathbf{V}$ is conservative.

(b) Now let $\mathbf{V}=(1+\cos (2 \pi x+2 \pi y), 2+\cos (2 \pi x+2 \pi y))$. Show that there exists a smooth function $F(x, y)$ such that $\mathbf{V}=\nabla F$.

Calculate $\int_{C} \mathbf{V} \cdot d \mathbf{x}$, where $C$ is a smooth curve running from $(0,0)$ to $(m, n) \in \mathbb{Z}^{2}$. Deduce that there does not exist a smooth function $F(x, y)$ which satisfies $\mathbf{V}=\nabla F$ and which is, in addition, periodic with period 1 in each coordinate direction, i.e. $F(x, y)=F(x+1, y)=F(x, y+1)$.

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• # Paper 3, Section I, A

The smooth curve $\mathcal{C}$ in $\mathbb{R}^{3}$ is given in parametrised form by the function $\mathbf{x}(u)$. Let $s$ denote arc length measured along the curve.

(a) Express the tangent $\mathbf{t}$ in terms of the derivative $\mathbf{x}^{\prime}=d \mathbf{x} / d u$, and show that $d u / d s=\left|\mathbf{x}^{\prime}\right|^{-1}$.

(b) Find an expression for $d \mathbf{t} / d s$ in terms of derivatives of $\mathbf{x}$ with respect to $u$, and show that the curvature $\kappa$ is given by

$\kappa=\frac{\left|\mathbf{x}^{\prime} \times \mathbf{x}^{\prime \prime}\right|}{\left|\mathbf{x}^{\prime}\right|^{3}}$

[Hint: You may find the identity $\left(\mathbf{x}^{\prime} \cdot \mathbf{x}^{\prime \prime}\right) \mathbf{x}^{\prime}-\left(\mathbf{x}^{\prime} \cdot \mathbf{x}^{\prime}\right) \mathbf{x}^{\prime \prime}=\mathbf{x}^{\prime} \times\left(\mathbf{x}^{\prime} \times \mathbf{x}^{\prime \prime}\right)$ helpful.]

(c) For the curve

$\mathbf{x}(u)=\left(\begin{array}{c} u \cos u \\ u \sin u \\ 0 \end{array}\right)$

with $u \geqslant 0$, find the curvature as a function of $u$.

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• # Paper 3, Section I, A

(i) For $r=|\mathbf{x}|$ with $\mathbf{x} \in \mathbb{R}^{3} \backslash\{\mathbf{0}\}$, show that

$\frac{\partial r}{\partial x_{i}}=\frac{x_{i}}{r} \quad(i=1,2,3) .$

(ii) Consider the vector fields $\mathbf{F}(\mathbf{x})=r^{2} \mathbf{x}, \mathbf{G}(\mathbf{x})=(\mathbf{a} \cdot \mathbf{x}) \mathbf{x}$ and $\mathbf{H}(\mathbf{x})=\mathbf{a} \times \hat{\mathbf{x}}$, where $\mathbf{a}$ is a constant vector in $\mathbb{R}^{3}$ and $\hat{\mathbf{x}}$ is the unit vector in the direction of $\mathbf{x}$. Using suffix notation, or otherwise, find the divergence and the curl of each of $\mathbf{F}, \mathbf{G}$ and $\mathbf{H}$.

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• # Paper 3, Section II, A

(a) Let $t_{i j}$ be a rank 2 tensor whose components are invariant under rotations through an angle $\pi$ about each of the three coordinate axes. Show that $t_{i j}$ is diagonal.

(b) An array of numbers $a_{i j}$ is given in one orthonormal basis as $\delta_{i j}+\epsilon_{1 i j}$ and in another rotated basis as $\delta_{i j}$. By using the invariance of the determinant of any rank 2 tensor, or otherwise, prove that $a_{i j}$ is not a tensor.

(c) Let $a_{i j}$ be an array of numbers and $b_{i j}$ a tensor. Determine whether the following statements are true or false. Justify your answers.

(i) If $a_{i j} b_{i j}$ is a scalar for any rank 2 tensor $b_{i j}$, then $a_{i j}$ is a rank 2 tensor.

(ii) If $a_{i j} b_{i j}$ is a scalar for any symmetric rank 2 tensor $b_{i j}$, then $a_{i j}$ is a rank 2 tensor.

(iii) If $a_{i j}$ is antisymmetric and $a_{i j} b_{i j}$ is a scalar for any symmetric rank 2 tensor $b_{i j}$, then $a_{i j}$ is an antisymmetric rank 2 tensor.

(iv) If $a_{i j}$ is antisymmetric and $a_{i j} b_{i j}$ is a scalar for any antisymmetric rank 2 tensor $b_{i j}$, then $a_{i j}$ is an antisymmetric rank 2 tensor.

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• # Paper 3, Section II, A

(i) Starting with the divergence theorem, derive Green's first theorem

$\int_{V}\left(\psi \nabla^{2} \phi+\nabla \psi \cdot \nabla \phi\right) d V=\int_{\partial V} \psi \frac{\partial \phi}{\partial n} d S$

(ii) The function $\phi(\mathbf{x})$ satisfies Laplace's equation $\nabla^{2} \phi=0$ in the volume $V$ with given boundary conditions $\phi(\mathbf{x})=g(\mathbf{x})$ for all $\mathbf{x} \in \partial V$. Show that $\phi(\mathbf{x})$ is the only such function. Deduce that if $\phi(\mathbf{x})$ is constant on $\partial V$ then it is constant in the whole volume $V$.

(iii) Suppose that $\phi(\mathbf{x})$ satisfies Laplace's equation in the volume $V$. Let $V_{r}$ be the sphere of radius $r$ centred at the origin and contained in $V$. The function $f(r)$ is defined by

$f(r)=\frac{1}{4 \pi r^{2}} \int_{\partial V_{r}} \phi(\mathbf{x}) d S$

By considering the derivative $d f / d r$, and by introducing the Jacobian in spherical polar coordinates and using the divergence theorem, or otherwise, show that $f(r)$ is constant and that $f(r)=\phi(\mathbf{0})$.

(iv) Let $M$ denote the maximum of $\phi$ on $\partial V_{r}$ and $m$ the minimum of $\phi$ on $\partial V_{r}$. By using the result from (iii), or otherwise, show that $m \leqslant \phi(\mathbf{0}) \leqslant M$.

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• # Paper 3, Section II, A

State Stokes' theorem.

Let $S$ be the surface in