Vector Calculus

2021

• Paper 3, Section I, B

  Vector Calculus | Part IA, 2021

  (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

  Vector Calculus | Part IA, 2021

  (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

  Vector Calculus | Part IA, 2021

  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

  Vector Calculus | Part IA, 2021

  (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

  Vector Calculus | Part IA, 2021

  (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 $1<y<\sqrt{3} x$.

  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

  Vector Calculus | Part IA, 2021

  (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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2020

• Paper 2, Section I, B

  Vector Calculus | Part IA, 2020

  (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 $b<a$ 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

  Vector Calculus | Part IA, 2020

  (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

  Vector Calculus | Part IA, 2020

  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

  $$z=5-x^{2}-y^{2}, \quad 1<z<4$$

  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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2019

• Paper 3, Section I, B

  Vector Calculus | Part IA, 2019

  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

  Vector Calculus | Part IA, 2019

  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

  Vector Calculus | Part IA, 2019

  (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 $|\mathbf{x}|<a$, 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 $\rho(r)=\left\{\begin{array}{ll}\rho_{0} & \text { if } 0 \leqslant r \leqslant b \\ 0 & \text { if } b<r \leqslant a\end{array}\right.$, 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 $b<r<a$.

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• Paper 3, Section II, B

  Vector Calculus | Part IA, 2019

  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

  Vector Calculus | Part IA, 2019

  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

  Vector Calculus | Part IA, 2019

  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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2018

• Paper 3, Section I, $4 \mathbf{C}$

  Vector Calculus | Part IA, 2018

  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

  Vector Calculus | Part IA, 2018

  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

  Vector Calculus | Part IA, 2018

  (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

  Vector Calculus | Part IA, 2018

  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 $r<a$ 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

  Vector Calculus | Part IA, 2018

  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

  Vector Calculus | Part IA, 2018

  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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2017

• Paper 3, Section $I$, B

  Vector Calculus | Part IA, 2017

  (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

  Vector Calculus | Part IA, 2017

  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

  Vector Calculus | Part IA, 2017

  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 $V$.

  comment

• Paper 3, Section II, B

  Vector Calculus | Part IA, 2017

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

  comment

• Paper 3, Section II, B

  Vector Calculus | Part IA, 2017

  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 $\phi(x, y, z)=z^{2}$.

  comment

• Paper 3, Section II, B

  Vector Calculus | Part IA, 2017

  (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

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

  comment

2016

• Paper 3, Section I, C

  Vector Calculus | Part IA, 2016

  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.

  comment

• Paper 3, Section I, C

  Vector Calculus | Part IA, 2016

  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 $u(-1,1)$.

  comment

• Paper 3, Section II, C

  Vector Calculus | Part IA, 2016

  (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 $\oint_{C} \mathbf{F} \cdot d \mathbf{x}$.

  comment

• Paper 3, Section II, C

  Vector Calculus | Part IA, 2016

  (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 $(*)$.]

  comment

• Paper 3, Section II, C

  Vector Calculus | Part IA, 2016

  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?

  comment

• Paper 3, Section II, C

  Vector Calculus | Part IA, 2016

  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. $F(x, y)=F(x+1, y)=F(x, y+1)$.

  comment

2015

• Paper 3, Section I, A

  Vector Calculus | Part IA, 2015

  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 $u$.

  comment

• Paper 3, Section I, A

  Vector Calculus | Part IA, 2015

  (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 $\mathbf{H}$.

  comment

• Paper 3, Section II, A

  Vector Calculus | Part IA, 2015

  (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.

  comment

• Paper 3, Section II, A

  Vector Calculus | Part IA, 2015

  (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 $m \leqslant \phi(\mathbf{0}) \leqslant M$.

  comment

• Paper 3, Section II, A

  Vector Calculus | Part IA, 2015

  State Stokes' theorem.

  Let $S$ be the surface in $\mathbb{R}^{3}$ given by $z^{2}=x^{2}+y^{2}+1-\lambda$, where $0 \leqslant z \leqslant 1$ and $\lambda$ is a positive constant. Sketch the surface $S$ for representative values of $\lambda$ and find the surface element $\mathbf{d} \mathbf{S}$ with respect to the Cartesian coordinates $x$ and $y$.

  Compute $\nabla \times \mathbf{F}$ for the vector field

  $$\mathbf{F}(\mathbf{x})=\left(\begin{array}{c} -y \\ x \\ z \end{array}\right)$$

  and verify Stokes' theorem for $\mathbf{F}$ on the surface $S$ for every value of $\lambda$.

  Now compute $\nabla \times \mathbf{G}$ for the vector field

  $$\mathbf{G}(\mathbf{x})=\frac{1}{x^{2}+y^{2}}\left(\begin{array}{c} -y \\ x \\ 0 \end{array}\right)$$

  and find the line integral $\int_{\partial S} \mathbf{G} \cdot \mathbf{d x}$ for the boundary $\partial S$ of the surface $S$. Is it possible to obtain this result using Stokes' theorem? Justify your answer.

  comment

• Paper 3, Section II, A

  Vector Calculus | Part IA, 2015

  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 $V$.

  comment

2014

• Paper 3, Section I, A

  Vector Calculus | Part IA, 2014

  Let $\mathbf{F}(\mathbf{x})$ be a vector field defined everywhere on the domain $G \subset \mathbb{R}^{3}$.

  (a) Suppose that $\mathbf{F}(\mathbf{x})$ has a potential $\phi(\mathbf{x})$ such that $\mathbf{F}(\mathbf{x})=\nabla \phi(\mathbf{x})$ for $\mathbf{x} \in G$. Show that

  $$\int_{\gamma} \mathbf{F} \cdot \mathbf{d} \mathbf{x}=\phi(\mathbf{b})-\phi(\mathbf{a})$$

  for any smooth path $\gamma$ from a to $\mathbf{b}$ in $G$. Show further that necessarily $\nabla \times \mathbf{F}=\mathbf{0}$ on $G$.

  (b) State a condition for $G$ which ensures that $\nabla \times \mathbf{F}=\mathbf{0}$ implies $\int_{\gamma} \mathbf{F} \cdot \mathbf{d x}$ is pathindependent.

  (c) Compute the line integral $\oint_{\gamma} \mathbf{F} \cdot \mathbf{d} \mathbf{x}$ for the vector field

  $$\mathbf{F}(\mathbf{x})=\left(\begin{array}{c} \frac{-y}{x^{2}+y^{2}} \\ \frac{x}{x^{2}+y^{2}} \\ 0 \end{array}\right)$$

  where $\gamma$ denotes the anti-clockwise path around the unit circle in the $(x, y)$-plane. Compute $\nabla \times \mathbf{F}$ and comment on your result in the light of (b).

  comment

• Paper 3, Section I, A

  Vector Calculus | Part IA, 2014

  (a) For $\mathbf{x} \in \mathbb{R}^{n}$ and $r=|\mathbf{x}|$, show that

  $$\frac{\partial r}{\partial x_{i}}=\frac{x_{i}}{r}$$

  (b) Use index notation and your result in (a), or otherwise, to compute

  (i) $\nabla \cdot(f(r) \mathbf{x})$, and

  (ii) $\nabla \times(f(r) \mathbf{x})$ for $n=3$.

  (c) Show that for each $n \in \mathbb{N}$ there is, up to an arbitrary constant, just one vector field $\mathbf{F}(\mathbf{x})$ of the form $f(r) \mathbf{x}$ such that $\nabla \cdot \mathbf{F}(\mathbf{x})=0$ everywhere on $\mathbb{R}^{n} \backslash\{\mathbf{0}\}$, and determine $\mathbf{F}$.

  comment

• Paper 3, Section II, 11A

  Vector Calculus | Part IA, 2014

  (i) Starting with Poisson's equation in $\mathbb{R}^{3}$,

  $$\nabla^{2} \phi(\mathbf{x})=f(\mathbf{x})$$

  derive Gauss' flux theorem

  $$\int_{V} f(\mathbf{x}) d V=\int_{\partial V} \mathbf{F}(\mathbf{x}) \cdot \mathbf{d} \mathbf{S}$$

  for $\mathbf{F}(\mathbf{x})=\nabla \phi(\mathbf{x})$ and for any volume $V \subseteq \mathbb{R}^{3}$.

  (ii) Let

  $$I=\int_{S} \frac{\mathbf{x} \cdot \mathbf{d} \mathbf{S}}{|\mathbf{x}|^{3}} .$$

  Show that $I=4 \pi$ if $S$ is the sphere $|\mathbf{x}|=R$, and that $I=0$ if $S$ bounds a volume that does not contain the origin.

  (iii) Show that the electric field defined by

  $$\mathbf{E}(\mathbf{x})=\frac{q}{4 \pi \epsilon_{0}} \frac{\mathbf{x}-\mathbf{a}}{|\mathbf{x}-\mathbf{a}|^{3}}, \quad \mathbf{x} \neq \mathbf{a}$$

  satisfies

  $$\int_{\partial V} \mathbf{E} \cdot \mathbf{d} \mathbf{S}= \begin{cases}0 & \text { if } \mathbf{a} \notin V \\ \frac{q}{\epsilon_{0}} & \text { if } \mathbf{a} \in V\end{cases}$$

  where $\partial V$ is a surface bounding a closed volume $V$ and $\mathbf{a} \notin \partial V$, and where the electric charge $q$ and permittivity of free space $\epsilon_{0}$ are constants. This is Gauss' law for a point electric charge.

  (iv) Assume that $f(\mathbf{x})$ is spherically symmetric around the origin, i.e., it is a function only of $|\mathbf{x}|$. Assume that $\mathbf{F}(\mathbf{x})$ is also spherically symmetric. Show that $\mathbf{F}(\mathbf{x})$ depends only on the values of $f$ inside the sphere with radius $|\mathbf{x}|$ but not on the values of $f$ outside this sphere.

  comment

• Paper 3, Section II, A

  Vector Calculus | Part IA, 2014

  (a) Show that any rank 2 tensor $t_{i j}$ can be written uniquely as a sum of two rank 2 tensors $s_{i j}$ and $a_{i j}$ where $s_{i j}$ is symmetric and $a_{i j}$ is antisymmetric.

  (b) Assume that the rank 2 tensor $t_{i j}$ is invariant under any rotation about the $z$-axis, as well as under a rotation of angle $\pi$ about any axis in the $(x, y)$-plane through the origin.

  (i) Show that there exist $\alpha, \beta \in \mathbb{R}$ such that $t_{i j}$ can be written as

  $$t_{i j}=\alpha \delta_{i j}+\beta \delta_{i 3} \delta_{j 3} .$$

  (ii) Is there some proper subgroup of the rotations specified above for which the result $(*)$ still holds if the invariance of $t_{i j}$ is restricted to this subgroup? If so, specify the smallest such subgroup.

  (c) The array of numbers $d_{i j k}$ is such that $d_{i j k} s_{i j}$ is a vector for any symmetric matrix $s_{i j}$.

  (i) By writing $d_{i j k}$ as a sum of $d_{i j k}^{s}$ and $d_{i j k}^{a}$ with $d_{i j k}^{s}=d_{j i k}^{s}$ and $d_{i j k}^{a}=-d_{j i k}^{a}$, show that $d_{i j k}^{s}$ is a rank 3 tensor. [You may assume without proof the Quotient Theorem for tensors.]

  (ii) Does $d_{i j k}^{a}$ necessarily have to be a tensor? Justify your answer.

  comment

• Paper 3, Section II, A

  Vector Calculus | Part IA, 2014

  (a) State Stokes' Theorem for a surface $S$ with boundary $\partial S$.

  (b) Let $S$ be the surface in $\mathbb{R}^{3}$ given by $z^{2}=1+x^{2}+y^{2}$ where $\sqrt{2} \leqslant z \leqslant \sqrt{5}$. Sketch the surface $S$ and find the surface element $\mathbf{d} \mathbf{S}$ with respect to the Cartesian coordinates $x$ and $y$.

  (c) Compute $\nabla \times \mathbf{F}$ for the vector field

  $$\mathbf{F}(\mathbf{x})=\left(\begin{array}{c} -y \\ x \\ x y(x+y) \end{array}\right)$$

  and verify Stokes' Theorem for $\mathbf{F}$ on the surface $S$.

  comment

• Paper 3, Section II, A

  Vector Calculus | Part IA, 2014

  The surface $C$ in $\mathbb{R}^{3}$ is given by $z^{2}=x^{2}+y^{2}$.

  (a) Show that the vector field

  $$\mathbf{F}(\mathbf{x})=\left(\begin{array}{l} x \\ y \\ z \end{array}\right)$$

  is tangent to the surface $C$ everywhere.

  (b) Show that the surface integral $\int_{S} \mathbf{F} \cdot \mathbf{d} \mathbf{S}$ is a constant independent of $S$ for any surface $S$ which is a subset of $C$, and determine this constant.

  (c) The volume $V$ in $\mathbb{R}^{3}$ is bounded by the surface $C$ and by the cylinder $x^{2}+y^{2}=1$. Sketch $V$ and compute the volume integral

  $$\int_{V} \nabla \cdot \mathbf{F} d V$$

  directly by integrating over $V$.

  (d) Use the Divergence Theorem to verify the result you obtained in part (b) for the integral $\int_{S} \mathbf{F} \cdot \mathbf{d} \mathbf{S}$, where $S$ is the portion of $C$ lying in $-1 \leqslant z \leqslant 1$.

  comment

2013

• Paper 3, Section $I$, C

  Vector Calculus | Part IA, 2013

  State a necessary and sufficient condition for a vector field $\mathbf{F}$ on $\mathbb{R}^{3}$ to be conservative.

  Check that the field

  $$\mathbf{F}=\left(2 x \cos y-2 z^{3}, 3+2 y e^{z}-x^{2} \sin y, y^{2} e^{z}-6 x z^{2}\right)$$

  is conservative and find a scalar potential for $\mathbf{F}$.

  comment

• Paper 3, Section I, C

  Vector Calculus | Part IA, 2013

  The curve $C$ is given by

  $$\mathbf{r}(t)=\left(\sqrt{2} e^{t},-e^{t} \sin t, e^{t} \cos t\right), \quad-\infty<t<\infty$$

  (i) Compute the arc length of $C$ between the points with $t=0$ and $t=1$.

  (ii) Derive an expression for the curvature of $C$ as a function of arc length $s$ measured from the point with $t=0$.

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2013

  (a) Prove that

  $$\nabla \times(\mathbf{F} \times \mathbf{G})=\mathbf{F}(\nabla \cdot \mathbf{G})-\mathbf{G}(\nabla \cdot \mathbf{F})+(\mathbf{G} \cdot \nabla) \mathbf{F}-(\mathbf{F} \cdot \nabla) \mathbf{G}$$

  (b) State the divergence theorem for a vector field $\mathbf{F}$ in a closed region $\Omega \subset \mathbb{R}^{3}$ bounded by $\partial \Omega$.

  For a smooth vector field $\mathbf{F}$ and a smooth scalar function $g$ prove that

  $$\int_{\Omega} \mathbf{F} \cdot \nabla g+g \nabla \cdot \mathbf{F} d V=\int_{\partial \Omega} g \mathbf{F} \cdot \mathbf{n} d S,$$

  where $\mathbf{n}$ is the outward unit normal on the surface $\partial \Omega$.

  Use this identity to prove that the solution $u$ to the Laplace equation $\nabla^{2} u=0$ in $\Omega$ with $u=f$ on $\partial \Omega$ is unique, provided it exists.

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2013

  If $\mathbf{E}$ and $\mathbf{B}$ are vectors in $\mathbb{R}^{3}$, show that

  $$T_{i j}=E_{i} E_{j}+B_{i} B_{j}-\frac{1}{2} \delta_{i j}\left(E_{k} E_{k}+B_{k} B_{k}\right)$$

  is a second rank tensor.

  Now assume that $\mathbf{E}(\mathbf{x}, t)$ and $\mathbf{B}(\mathbf{x}, t)$ obey Maxwell's equations, which in suitable units read

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

  where $\rho$ is the charge density and $\mathbf{J}$ the current density. Show that

  $$\frac{\partial}{\partial t}(\mathbf{E} \times \mathbf{B})=\mathbf{M}-\rho \mathbf{E}-\mathbf{J} \times \mathbf{B} \quad \text { where } \quad M_{i}=\frac{\partial T_{i j}}{\partial x_{j}}$$

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2013

  Consider the bounded surface $S$ that is the union of $x^{2}+y^{2}=4$ for $-2 \leqslant z \leqslant 2$ and $(4-z)^{2}=x^{2}+y^{2}$ for $2 \leqslant z \leqslant 4$. Sketch the surface.

  Using suitable parametrisations for the two parts of $S$, calculate the integral

  $$\int_{S}(\nabla \times \mathbf{F}) \cdot d \mathbf{S}$$

  for $\mathbf{F}=y z^{2} \mathbf{i}$.

  Check your result using Stokes's Theorem.

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2013

  Give an explicit formula for $\mathcal{J}$ which makes the following result hold:

  $$\int_{D} f d x d y d z=\int_{D^{\prime}} \phi|\mathcal{J}| d u d v d w$$

  where the region $D$, with coordinates $x, y, z$, and the region $D^{\prime}$, with coordinates $u, v, w$, are in one-to-one correspondence, and

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

  Explain, in outline, why this result holds.

  Let $D$ be the region in $\mathbb{R}^{3}$ defined by $4 \leqslant x^{2}+y^{2}+z^{2} \leqslant 9$ and $z \geqslant 0$. Sketch the region and employ a suitable transformation to evaluate the integral

  $$\int_{D}\left(x^{2}+y^{2}\right) d x d y d z$$

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2012

• Paper 3, Section I, C

  Vector Calculus | Part IA, 2012

  What does it mean for a second-rank tensor $T_{i j}$ to be isotropic? Show that $\delta_{i j}$ is isotropic. By considering rotations through $\pi / 2$ about the coordinate axes, or otherwise, show that the most general isotropic second-rank tensor in $\mathbb{R}^{3}$ has the form $T_{i j}=\lambda \delta_{i j}$, for some scalar $\lambda$.

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• Paper 3, Section I, C

  Vector Calculus | Part IA, 2012

  Define what it means for a differential $P d x+Q d y$ to be exact, and derive a necessary condition on $P(x, y)$ and $Q(x, y)$ for this to hold. Show that one of the following two differentials is exact and the other is not:

  $$\begin{aligned} &y^{2} d x+2 x y d y \\ &y^{2} d x+x y^{2} d y \end{aligned}$$

  Show that the differential which is not exact can be written in the form $g d f$ for functions $f(x, y)$ and $g(y)$, to be determined.

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2012

  (i) Let $V$ be a bounded region in $\mathbb{R}^{3}$ with smooth boundary $S=\partial V$. Show that Poisson's equation in $V$

  $$\nabla^{2} u=\rho$$

  has at most one solution satisfying $u=f$ on $S$, where $\rho$ and $f$ are given functions.

  Consider the alternative boundary condition $\partial u / \partial n=g$ on $S$, for some given function $g$, where $n$ is the outward pointing normal on $S$. Derive a necessary condition in terms of $\rho$ and $g$ for a solution $u$ of Poisson's equation to exist. Is such a solution unique?

  (ii) Find the most general spherically symmetric function $u(r)$ satisfying

  $$\nabla^{2} u=1$$

  in the region $r=|\mathbf{r}| \leqslant a$ for $a>0$. Hence in each of the following cases find all possible solutions satisfying the given boundary condition at $r=a$ : (a) $u=0$, (b) $\frac{\partial u}{\partial n}=0$.

  Compare these with your results in part (i).

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2012

  (a) Prove the identity

  $$\nabla(\mathbf{F} \cdot \mathbf{G})=(\mathbf{F} \cdot \nabla) \mathbf{G}+(\mathbf{G} \cdot \nabla) \mathbf{F}+\mathbf{F} \times(\nabla \times \mathbf{G})+\mathbf{G} \times(\nabla \times \mathbf{F})$$

  (b) If $\mathbf{E}$ is an irrotational vector field (i.e. $\nabla \times \mathbf{E}=\mathbf{0}$ everywhere), prove that there exists a scalar potential $\phi(\mathbf{x})$ such that $\mathbf{E}=-\nabla \phi$.

  Show that the vector field

  $$\left(x y^{2} z e^{-x^{2} z},-y e^{-x^{2} z}, \frac{1}{2} x^{2} y^{2} e^{-x^{2} z}\right)$$

  is irrotational, and determine the corresponding potential $\phi$.

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2012

  Consider the transformation of variables

  $$x=1-u, \quad y=\frac{1-v}{1-u v} .$$

  Show that the interior of the unit square in the $u v$ plane

  $$\{(u, v): 0<u<1,0<v<1\}$$

  is mapped to the interior of the unit square in the $x y$ plane,

  $$R=\{(x, y): 0<x<1,0<y<1\} .$$

  [Hint: Consider the relation between $v$ and $y$ when $u=\alpha$, for $0<\alpha<1$ constant.]

  Show that

  $$\frac{\partial(x, y)}{\partial(u, v)}=\frac{(1-(1-x) y)^{2}}{x}$$

  Now let

  $$u=\frac{1-t}{1-w t}, \quad v=1-w$$

  By calculating

  $$\frac{\partial(x, y)}{\partial(t, w)}=\frac{\partial(x, y)}{\partial(u, v)} \frac{\partial(u, v)}{\partial(t, w)}$$

  as a function of $x$ and $y$, or otherwise, show that

  $$\int_{R} \frac{x(1-y)}{(1-(1-x) y)\left(1-\left(1-x^{2}\right) y\right)^{2}} d x d y=1$$

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2012

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

  Consider the surface $S$ defined by

  $$z=x^{2}+y^{2}, \quad \frac{1}{9} \leqslant z \leqslant 1 .$$

  Sketch the surface and calculate the area element $d \mathbf{S}$ in terms of suitable coordinates or parameters. For the vector field

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

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

  Use Stokes' Theorem to express $I$ as an integral over $\partial S$ and verify that this gives the same result.

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2011

• Paper 3, Section I, C

  Vector Calculus | Part IA, 2011

  State the value of $\partial x_{i} / \partial x_{j}$ and find $\partial r / \partial x_{j}$, where $r=|\mathbf{x}|$.

  Vector fields $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^{3}$ are given by $\mathbf{u}=r^{\alpha} \mathbf{x}$ and $\mathbf{v}=\mathbf{k} \times \mathbf{u}$, where $\alpha$ is a constant and $\mathbf{k}$ is a constant vector. Calculate the second-rank tensor $d_{i j}=\partial u_{i} / \partial x_{j}$, and deduce that $\boldsymbol{\nabla} \times \mathbf{u}=\mathbf{0}$ and $\boldsymbol{\nabla} \cdot \mathbf{v}=0$. When $\alpha=-3$, show that $\boldsymbol{\nabla} \cdot \mathbf{u}=0$ and

  $$\nabla \times \mathbf{v}=\frac{3(\mathbf{k} \cdot \mathbf{x}) \mathbf{x}-\mathbf{k} r^{2}}{r^{5}}$$

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• Paper 3, Section I, C

  Vector Calculus | Part IA, 2011

  Cartesian coordinates $x, y, z$ and spherical polar coordinates $r, \theta, \phi$ are related by

  $$x=r \sin \theta \cos \phi, \quad y=r \sin \theta \sin \phi, \quad z=r \cos \theta$$

  Find scalars $h_{r}, h_{\theta}, h_{\phi}$ and unit vectors $\mathbf{e}_{r}, \mathbf{e}_{\theta}, \mathbf{e}_{\phi}$ such that

  $$\mathrm{d} \mathbf{x}=h_{r} \mathbf{e}_{r} \mathrm{~d} r+h_{\theta} \mathbf{e}_{\theta} \mathrm{d} \theta+h_{\phi} \mathbf{e}_{\phi} \mathrm{d} \phi$$

  Verify that the unit vectors are mutually orthogonal.

  Hence calculate the area of the open surface defined by $\theta=\alpha, 0 \leqslant r \leqslant R$, $0 \leqslant \phi \leqslant 2 \pi$, where $\alpha$ and $R$ are constants.

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2011

  The vector fields $\mathbf{A}(\mathbf{x}, t)$ and $\mathbf{B}(\mathbf{x}, t)$ obey the evolution equations

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

  where $\mathbf{u}$ is a given vector field and $\psi$ is a given scalar field. Use suffix notation to show that the scalar field $h=\mathbf{A} \cdot \mathbf{B}$ obeys an evolution equation of the form

  $$\frac{\partial h}{\partial t}=\mathbf{B} \cdot \nabla f-\mathbf{u} \cdot \nabla h$$

  where the scalar field $f$ should be identified.

  Suppose that $\boldsymbol{\nabla} \cdot \mathbf{B}=0$ and $\boldsymbol{\nabla} \cdot \mathbf{u}=0$. Show that, if $\mathbf{u} \cdot \mathbf{n}=\mathbf{B} \cdot \mathbf{n}=0$ on the surface $S$ of a fixed volume $V$ with outward normal $\mathbf{n}$, then

  $$\frac{\mathrm{d} H}{\mathrm{~d} t}=0, \quad \text { where } H=\int_{V} h \mathrm{~d} V .$$

  Suppose that $\mathbf{A}=a r^{2} \sin \theta \mathbf{e}_{\theta}+r\left(a^{2}-r^{2}\right) \sin \theta \mathbf{e}_{\phi}$ with respect to spherical polar coordinates, and that $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}$. Show that

  $$h=a r^{2}\left(a^{2}+r^{2}\right) \sin ^{2} \theta$$

  and calculate the value of $H$ when $V$ is the sphere $r \leqslant a$.

  $$\left[\text { In spherical polar coordinates } \nabla \times \mathbf{F}=\frac{1}{r^{2} \sin \theta}\left|\begin{array}{ccc} \mathbf{e}_{r} & r \mathbf{e}_{\theta} & r \sin \theta \mathbf{e}_{\phi} \\ \partial / \partial r & \partial / \partial \theta & \partial / \partial \phi \\ F_{r} & r F_{\theta} & r \sin \theta F_{\phi} \end{array}\right|\right.$$

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2011

  The electric field $\mathbf{E}(\mathbf{x})$ due to a static charge distribution with density $\rho(\mathbf{x})$ satisfies

  $$\mathbf{E}=-\boldsymbol{\nabla} \phi, \quad \boldsymbol{\nabla} \cdot \mathbf{E}=\frac{\rho}{\varepsilon_{0}},$$

  where $\phi(\mathbf{x})$ is the corresponding electrostatic potential and $\varepsilon_{0}$ is a constant.

  (a) Show that the total charge $Q$ contained within a closed surface $S$ is given by Gauss' Law

  $$Q=\varepsilon_{0} \int_{S} \mathbf{E} \cdot \mathrm{d} \mathbf{S} .$$

  Assuming spherical symmetry, deduce the electric field and potential due to a point charge $q$ at the origin i.e. for $\rho(\mathbf{x})=q \delta(\mathbf{x})$.

  (b) Let $\mathbf{E}_{1}$ and $\mathbf{E}_{2}$, with potentials $\phi_{1}$ and $\phi_{2}$ respectively, be the solutions to (1) arising from two different charge distributions with densities $\rho_{1}$ and $\rho_{2}$. Show that

  $$\frac{1}{\varepsilon_{0}} \int_{V} \phi_{1} \rho_{2} \mathrm{~d} V+\int_{\partial V} \phi_{1} \nabla \phi_{2} \cdot \mathrm{d} \mathbf{S}=\frac{1}{\varepsilon_{0}} \int_{V} \phi_{2} \rho_{1} \mathrm{~d} V+\int_{\partial V} \phi_{2} \nabla \phi_{1} \cdot \mathrm{d} \mathbf{S}$$

  for any region $V$ with boundary $\partial V$, where $\mathrm{d} \mathbf{S}$ points out of $V$.

  (c) Suppose that $\rho_{1}(\mathbf{x})=0$ for $|\mathbf{x}| \leqslant a$ and that $\phi_{1}(\mathbf{x})=\Phi$, a constant, on $|\mathbf{x}|=a$. Use the results of (a) and (b) to show that

  $$\Phi=\frac{1}{4 \pi \varepsilon_{0}} \int_{r>a} \frac{\rho_{1}(\mathbf{x})}{r} \mathrm{~d} V$$

  [You may assume that $\phi_{1} \rightarrow 0$ as $|\mathbf{x}| \rightarrow \infty$ sufficiently rapidly that any integrals over the 'sphere at infinity' in (2) are zero.]

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2011

  State the divergence theorem for a vector field $\mathbf{u}(\mathbf{x})$ in a region $V$ bounded by a piecewise smooth surface $S$ with outward normal $\mathbf{n}$.

  Show, by suitable choice of $\mathbf{u}$, that

  $$\int_{V} \nabla f \mathrm{~d} V=\int_{S} f \mathrm{~d} \mathbf{S}$$

  for a scalar field $f(\mathbf{x})$.

  Let $V$ be the paraboloidal region given by $z \geqslant 0$ and $x^{2}+y^{2}+c z \leqslant a^{2}$, where $a$ and $c$ are positive constants. Verify that $(*)$ holds for the scalar field $f=x z$.

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2011

  Write down the most general isotropic tensors of rank 2 and 3. Use the tensor transformation law to show that they are, indeed, isotropic.

  Let $V$ be the sphere $0 \leqslant r \leqslant a$. Explain briefly why

  $$T_{i_{1} \ldots i_{n}}=\int_{V} x_{i_{1}} \ldots x_{i_{n}} \mathrm{~d} V$$

  is an isotropic tensor for any $n$. Hence show that

  $$\int_{V} x_{i} x_{j} \mathrm{~d} V=\alpha \delta_{i j}, \quad \int_{V} x_{i} x_{j} x_{k} \mathrm{~d} V=0 \quad \text { and } \int_{V} x_{i} x_{j} x_{k} x_{l} \mathrm{~d} V=\beta\left(\delta_{i j} \delta_{k l}+\delta_{i k} \delta_{j l}+\delta_{i l} \delta_{j k}\right)$$

  for some scalars $\alpha$ and $\beta$, which should be determined using suitable contractions of the indices or otherwise. Deduce the value of

  $$\int_{V} \mathbf{x} \times(\boldsymbol{\Omega} \times \mathbf{x}) \mathrm{d} V$$

  where $\boldsymbol{\Omega}$ is a constant vector.

  [You may assume that the most general isotropic tensor of rank 4 is

  $$\lambda \delta_{i j} \delta_{k l}+\mu \delta_{i k} \delta_{j l}+\nu \delta_{i l} \delta_{j k}$$

  where $\lambda, \mu$ and $\nu$ are scalars.]

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2010

• Paper 3 , Section II, C

  Vector Calculus | Part IA, 2010

  State the divergence theorem (also known as Gauss' theorem) relating the surface and volume integrals of appropriate fields.

  The surface $S_{1}$ is defined by the equation $z=3-2 x^{2}-2 y^{2}$ for $1 \leqslant z \leqslant 3$; the surface $S_{2}$ is defined by the equation $x^{2}+y^{2}=1$ for $0 \leqslant z \leqslant 1$; the surface $S_{3}$ is defined by the equation $z=0$ for $x, y$ satisfying $x^{2}+y^{2} \leqslant 1$. The surface $S$ is defined to be the union of the surfaces $S_{1}, S_{2}$ and $S_{3}$. Sketch the surfaces $S_{1}, S_{2}, S_{3}$ and (hence) $S$.

  The vector field $\mathbf{F}$ is defined by

  $$\mathbf{F}(x, y, z)=\left(x y+x^{6},-\frac{1}{2} y^{2}+y^{8}, z\right)$$

  Evaluate the integral

  $$\oint_{S} \mathbf{F} \cdot \mathrm{d} \mathbf{S}$$

  where the surface element $\mathrm{d} \mathbf{S}$ points in the direction of the outward normal to $S$.

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• Paper 3, Section I, C

  Vector Calculus | Part IA, 2010

  A curve in two dimensions is defined by the parameterised Cartesian coordinates

  $$x(u)=a e^{b u} \cos u, \quad y(u)=a e^{b u} \sin u$$

  where the constants $a, b>0$. Sketch the curve segment corresponding to the range $0 \leqslant u \leqslant 3 \pi$. What is the length of the curve segment between the points $(x(0), y(0))$ and $(x(U), y(U))$, as a function of $U$ ?

  A geometrically sensitive ant walks along the curve with varying speed $\kappa(u)^{-1}$, where $\kappa(u)$ is the curvature at the point corresponding to parameter $u$. Find the time taken by the ant to walk from $(x(2 n \pi), y(2 n \pi))$ to $(x(2(n+1) \pi), y(2(n+1) \pi))$, where $n$ is a positive integer, and hence verify that this time is independent of $n$.

  [You may quote without proof the formula $\kappa(u)=\frac{\left|x^{\prime}(u) y^{\prime \prime}(u)-y^{\prime}(u) x^{\prime \prime}(u)\right|}{\left(\left(x^{\prime}(u)\right)^{2}+\left(y^{\prime}(u)\right)^{2}\right)^{3 / 2}} .$ ]

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• Paper 3, Section I, C

  Vector Calculus | Part IA, 2010

  Consider the vector field

  $$\mathbf{F}=\left(-y /\left(x^{2}+y^{2}\right), x /\left(x^{2}+y^{2}\right), 0\right)$$

  defined on all of $\mathbb{R}^{3}$ except the $z$ axis. Compute $\boldsymbol{\nabla} \times \mathbf{F}$ on the region where it is defined.

  Let $\gamma_{1}$ be the closed curve defined by the circle in the $x y$-plane with centre $(2,2,0)$ and radius 1 , and $\gamma_{2}$ be the closed curve defined by the circle in the $x y$-plane with centre $(0,0,0)$ and radius 1 .

  By using your earlier result, or otherwise, evaluate the line integral $\oint_{\gamma_{1}} \mathbf{F} \cdot \mathrm{d} \mathbf{x}$.

  By explicit computation, evaluate the line integral $\oint_{\gamma_{2}} \mathbf{F} \cdot \mathrm{d} \mathbf{x}$. Is your result consistent with Stokes' theorem? Explain your answer briefly.

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2010

  Given a spherically symmetric mass distribution with density $\rho$, explain how to obtain the gravitational field $\mathbf{g}=-\nabla \phi$, where the potential $\phi$ satisfies Poisson's equation

  $$\nabla^{2} \phi=4 \pi G \rho$$

  The remarkable planet Geometria has radius 1 and is composed of an infinite number of stratified spherical shells $S_{n}$ labelled by integers $n \geqslant 1$. The shell $S_{n}$ has uniform density $2^{n-1} \rho_{0}$, where $\rho_{0}$ is a constant, and occupies the volume between radius $2^{-n+1}$ and $2^{-n}$.

  Obtain a closed form expression for the mass of Geometria.

  Obtain a closed form expression for the gravitational field $\mathbf{g}$ due to Geometria at a distance $r=2^{-N}$ from its centre of mass, for each positive integer $N \geqslant 1$. What is the potential $\phi(r)$ due to Geometria for $r>1$ ?

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2010

  Let $f(x, y)$ be a function of two variables, and $R$ a region in the $x y$-plane. State the rule for evaluating $\int_{R} f(x, y) \mathrm{d} x \mathrm{~d} y$ as an integral with respect to new variables $u(x, y)$ and $v(x, y)$.

  Sketch the region $R$ in the $x y$-plane defined by

  $$R=\left\{(x, y): x^{2}+y^{2} \leqslant 2, x^{2}-y^{2} \geqslant 1, x \geqslant 0, y \geqslant 0\right\}$$

  Sketch the corresponding region in the $u v$-plane, where

  $$u=x^{2}+y^{2}, \quad v=x^{2}-y^{2}$$

  Express the integral

  $$I=\int_{R}\left(x^{5} y-x y^{5}\right) \exp \left(4 x^{2} y^{2}\right) \mathrm{d} x \mathrm{~d} y$$

  as an integral with respect to $u$ and $v$. Hence, or otherwise, calculate $I$.

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• Paper 3, Section II, C

  Vector Calculus | Part IA, 2010

  (a) Define a rank two tensor and show that if two rank two tensors $A_{i j}$ and $B_{i j}$ are the same in one Cartesian coordinate system, then they are the same in all Cartesian coordinate systems.

  The quantity $C_{i j}$ has the property that, for every rank two tensor $A_{i j}$, the quantity $C_{i j} A_{i j}$ is a scalar. Is $C_{i j}$ necessarily a rank two tensor? Justify your answer with a proof from first principles, or give a counterexample.

  (b) Show that, if a tensor $T_{i j}$ is invariant under rotations about the $x_{3}$-axis, then it has the form

  $$\left(\begin{array}{ccc} \alpha & \omega & 0 \\ -\omega & \alpha & 0 \\ 0 & 0 & \beta \end{array}\right)$$

  (c) The inertia tensor about the origin of a rigid body occupying volume $V$ and with variable mass density $\rho(\mathbf{x})$ is defined to be

  $$I_{i j}=\int_{V} \rho(\mathbf{x})\left(x_{k} x_{k} \delta_{i j}-x_{i} x_{j}\right) \mathrm{d} V$$

  The rigid body $B$ has uniform density $\rho$ and occupies the cylinder

  $$\left\{\left(x_{1}, x_{2}, x_{3}\right):-2 \leqslant x_{3} \leqslant 2, x_{1}^{2}+x_{2}^{2} \leqslant 1\right\}$$

  Show that the inertia tensor of $B$ about the origin is diagonal in the $\left(x_{1}, x_{2}, x_{3}\right)$ coordinate system, and calculate its diagonal elements.

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2009

• Paper 3, Section $I$, B

  Vector Calculus | Part IA, 2009

  State the value of $\partial x_{i} / \partial x_{j}$ and find $\partial r / \partial x_{j}$, where $r=|\mathbf{x}|$.

  A vector field $\mathbf{u}$ is given by

  $$\mathbf{u}=\frac{\mathbf{k}}{r}+\frac{(\mathbf{k} \cdot \mathbf{x}) \mathbf{x}}{r^{3}}$$

  where $\mathbf{k}$ is a constant vector. Calculate the second-rank tensor $d_{i j}=\partial u_{i} / \partial x_{j}$ using suffix notation, and show that $d_{i j}$ splits naturally into symmetric and antisymmetric parts. Deduce that $\boldsymbol{\nabla} \cdot \mathbf{u}=0$ and that

  $$\boldsymbol{\nabla} \times \mathbf{u}=\frac{2 \mathbf{k} \times \mathbf{x}}{r^{3}}$$

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• Paper 3, Section I, B

  Vector Calculus | Part IA, 2009

  What does it mean for a vector field $\mathbf{F}$ to be irrotational ?

  The field $\mathbf{F}$ is irrotational and $\mathbf{x}_{0}$ is a given point. Write down a scalar potential $V(\mathbf{x})$ with $\mathbf{F}=-\nabla V$ and $V\left(\mathbf{x}_{0}\right)=0$. Show that this potential is well defined.

  For what value of $m$ is the field $\frac{\cos \theta \cos \phi}{r} \mathbf{e}_{\theta}+\frac{m \sin \phi}{r} \mathbf{e}_{\phi}$ irrotational, where $(r, \theta, \phi)$ are spherical polar coordinates? What is the corresponding potential $V(\mathbf{x})$ when $\mathbf{x}_{0}$ is the point $r=1, \theta=0$ ?

  $$\left[\text { In spherical polar coordinates } \nabla \times \mathbf{F}=\frac{1}{r^{2} \sin \theta} \mid \begin{array}{ccc} \mathbf{e}_{r} & r \mathbf{e}_{\theta} & r \sin \theta \mathbf{e}_{\phi} \\ \partial / \partial r & \partial / \partial \theta & \partial / \partial \phi \\ F_{r} & r F_{\theta} & r \sin \theta F_{\phi} \end{array}\right] .$$

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• Paper 3, Section II, B

  Vector Calculus | Part IA, 2009

  State the divergence theorem for a vector field $\mathbf{u}(\mathbf{x})$ in a region $V$ of $\mathbb{R}^{3}$ bounded by a smooth surface $S$.

  Let $f(x, y, z)$ be a homogeneous function of degree $n$, that is, $f(k x, k y, k z)=$ $k^{n} f(x, y, z)$ for any real number $k$. By differentiating with respect to $k$, show that

  $$\mathbf{x} \cdot \nabla f=n f$$

  Deduce that

  $$\tag{†} \int_{V} f \mathrm{~d} V=\frac{1}{n+3} \int_{S} f \mathbf{x} \cdot \mathbf{d} \mathbf{A}$$

  Let $V$ be the cone $0 \leqslant z \leqslant \alpha, \alpha \sqrt{x^{2}+y^{2}} \leqslant z$, where $\alpha$ is a positive constant. Verify that $(†)$ holds for the case $f=z^{4}+\alpha^{4}\left(x^{2}+y^{2}\right)^{2}$.

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• Paper 3, Section II, B

  Vector Calculus | Part IA, 2009

  A second-rank tensor $T(\mathbf{y})$ is defined by

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

  where $\mathbf{y}$ is a fixed vector with $|\mathbf{y}|=a, n>-1$, and the integration is over all points $\mathbf{x}$ lying on the surface $S$ of the sphere of radius $a$, centred on the origin. Explain briefly why $T$ might be expected to have the form

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

  where $\alpha$ and $\beta$ are scalar constants.

  Show that $\mathbf{y} \cdot(\mathbf{y}-\mathbf{x})=a^{2}(1-\cos \theta)$, where $\theta$ is the angle between $\mathbf{y}$ and $\mathbf{x}$, and find a similar expression for $|\mathbf{y}-\mathbf{x}|^{2}$. Using suitably chosen spherical polar coordinates, show that

  $$y_{i} T_{i j} y_{j}=\frac{\pi a^{2}(2 a)^{2 n+2}}{n+2}$$

  Hence, by evaluating another scalar integral, determine $\alpha$ and $\beta$, and find the value of $n$ for which $T$ is isotropic.

  comment

• Paper 3, Section II, B

  Vector Calculus | Part IA, 2009

  Give a necessary condition for a given vector field $\mathbf{J}$ to be the curl of another vector field $\mathbf{B}$. Is the vector field $\mathbf{B}$ unique? If not, explain why not.

  State Stokes' theorem and use it to evaluate the area integral

  $$\int_{S}\left(y^{2}, z^{2}, x^{2}\right) \cdot \mathbf{d} \mathbf{A}$$

  where $S$ is the half of the ellipsoid

  $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$$

  that lies in $z \geqslant 0$, and the area element dA points out of the ellipsoid.

  comment

• Paper 3, Section II, B

  Vector Calculus | Part IA, 2009

  Let $S$ be a bounded region of $\mathbb{R}^{2}$ and $\partial S$ be its boundary. Let $u$ be the unique solution to Laplace's equation in $S$, subject to the boundary condition $u=f$ on $\partial S$, where $f$ is a specified function. Let $w$ be any smooth function with $w=f$ on $\partial S$. By writing $w=u+\delta$, or otherwise, show that

  $$\int_{S}|\nabla w|^{2} \mathrm{~d} A \geqslant \int_{S}|\nabla u|^{2} \mathrm{~d} A$$

  Let $S$ be the unit disc in $\mathbb{R}^{2}$. By considering functions of the form $g(r) \cos \theta$ on both sides of $(*)$, where $r$ and $\theta$ are polar coordinates, deduce that

  $$\int_{0}^{1}\left(r\left(\frac{\mathrm{d} g}{\mathrm{~d} r}\right)^{2}+\frac{g^{2}}{r}\right) \mathrm{d} r \geqslant 1$$

  for any differentiable function $g(r)$ satisfying $g(1)=1$ and for which the integral converges at $r=0$.

  $$\left[\nabla f(r, \theta)=\left(\frac{\partial f}{\partial r}, \frac{1}{r} \frac{\partial f}{\partial \theta}\right), \quad \nabla^{2} f(r, \theta)=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial f}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} f}{\partial \theta^{2}} \cdot\right]$$

  comment

2008

• 3.I.3C

  Vector Calculus | Part IA, 2008

  A curve is given in terms of a parameter $t$ by

  $$\mathbf{x}(t)=\left(t-\frac{1}{3} t^{3}, t^{2}, t+\frac{1}{3} t^{3}\right)$$

  (i) Find the arc length of the curve between the points with $t=0$ and $t=1$.

  (ii) Find the unit tangent vector at the point with parameter $t$, and show that the principal normal is orthogonal to the $z$ direction at each point on the curve.

  comment

• 3.I.4C

  Vector Calculus | Part IA, 2008

  What does it mean to say that $T_{i j}$ transforms as a second rank tensor?

  If $T_{i j}$ transforms as a second rank tensor, show that $\frac{\partial T_{i j}}{\partial x_{j}}$ transforms as a vector.

  comment

• 3.II.10C

  Vector Calculus | Part IA, 2008

  Find the effect of a rotation by $\pi / 2$ about the $z$-axis on the tensor

  $$\left(\begin{array}{lll} S_{11} & S_{12} & S_{13} \\ S_{21} & S_{22} & S_{23} \\ S_{31} & S_{32} & S_{33} \end{array}\right)$$

  Hence show that the most general isotropic tensor of rank 2 is $\lambda \delta_{i j}$, where $\lambda$ is an arbitrary scalar.

  Prove that there is no non-zero isotropic vector, and write down without proof the most general isotropic tensor of rank 3 .

  Deduce that if $T_{i j k l}$ is an isotropic tensor then the following results hold, for some scalars $\mu$ and $\nu$ : (i) $\epsilon_{i j k} T_{i j k l}=0$; (ii) $\delta_{i j} T_{i j k l}=\mu \delta_{k l}$; (iii) $\epsilon_{i j m} T_{i j k l}=\nu \epsilon_{k l m}$.

  Verify these three results in the case $T_{i j k l}=\alpha \delta_{i j} \delta_{k l}+\beta \delta_{i k} \delta_{j l}+\gamma \delta_{i l} \delta_{j k}$, expressing $\mu$ and $\nu$ in terms of $\alpha, \beta$ and $\gamma$.

  comment

• 3.II.11C

  Vector Calculus | Part IA, 2008

  Let $V$ be a volume in $\mathbb{R}^{3}$ bounded by a closed surface $S$.

  (a) Let $f$ and $g$ be twice differentiable scalar fields such that $f=1$ on $S$ and $\nabla^{2} g=0$ in $V$. Show that

  $$\int_{V} \nabla f \cdot \nabla g d V=0$$

  (b) Let $V$ be the sphere $|\mathbf{x}| \leqslant a$. Evaluate the integral

  $$\int_{V} \nabla u \cdot \nabla v d V$$

  in the cases where $u$ and $v$ are given in spherical polar coordinates by: (i) $u=r, \quad v=r \cos \theta$; (ii) $u=r / a, \quad v=r^{2} \cos ^{2} \theta$; (iii) $u=r / a, \quad v=1 / r$.

  Comment on your results in the light of part (a).

  comment

• 3.II.12C

  Vector Calculus | Part IA, 2008

  Let $A$ be the closed planar region given by

  $$y \leqslant x \leqslant 2 y, \quad \frac{1}{y} \leqslant x \leqslant \frac{2}{y} .$$

  (i) Evaluate by means of a suitable change of variables the integral

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

  (ii) Let $C$ be the boundary of $A$. Evaluate the line integral

  $$\oint_{C} \frac{x^{2}}{2 y} d y-d x$$

  by integrating along each section of the boundary.

  (iii) Comment on your results.

  comment

• 3.II.9C

  Vector Calculus | Part IA, 2008

  Let $\mathbf{F}=\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{x})$, where $\mathbf{x}$ is the position vector and $\boldsymbol{\omega}$ is a uniform vector field.

  (i) Use the divergence theorem to evaluate the surface integral $\int_{S} \mathbf{F} \cdot d \mathbf{S}$, where $S$ is the closed surface of the cube with vertices $(\pm 1, \pm 1, \pm 1)$.

  (ii) Show that $\boldsymbol{\nabla} \times \mathbf{F}=0$. Show further that the scalar field $\phi$ given by

  $$\phi=\frac{1}{2}(\boldsymbol{\omega} \cdot \mathbf{x})^{2}-\frac{1}{2}(\boldsymbol{\omega} \cdot \boldsymbol{\omega})(\mathbf{x} \cdot \mathbf{x})$$

  satisfies $\mathbf{F}=\boldsymbol{\nabla} \phi$. Describe geometrically the surfaces of constant $\phi$.

  comment

2007

• Paper 3 , Section I, A

  Vector Calculus | Part IA, 2007

  (i) Give definitions for the unit tangent vector $\hat{\mathbf{T}}$ and the curvature $\kappa$ of a parametrised curve $\mathbf{x}(t)$ in $\mathbb{R}^{3}$. Calculate $\hat{\mathbf{T}}$ and $\kappa$ for the circular helix

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

  where $a$ and $b$ are constants.

  (ii) Find the normal vector and the equation of the tangent plane to the surface $S$ in $\mathbb{R}^{3}$ given by

  $$z=x^{2} y^{3}-y+1$$

  at the point $x=1, y=1, z=1$.

  comment

• Paper 3, Section I, A

  Vector Calculus | Part IA, 2007

  By using suffix notation, prove the following identities for the vector fields $\mathbf{A}$ and B in $\mathbb{R}^{3}$ :

  $$\begin{gathered} \nabla \cdot(\mathbf{A} \times \mathbf{B})=\mathbf{B} \cdot(\nabla \times \mathbf{A})-\mathbf{A} \cdot(\nabla \times \mathbf{B}) \\ \nabla \times(\mathbf{A} \times \mathbf{B})=(\mathbf{B} \cdot \nabla) \mathbf{A}-\mathbf{B}(\nabla \cdot \mathbf{A})-(\mathbf{A} \cdot \nabla) \mathbf{B}+\mathbf{A}(\nabla \cdot \mathbf{B}) \end{gathered}$$

  comment

• Paper 3, Section II, A

  Vector Calculus | Part IA, 2007

  Show that any second rank Cartesian tensor $P_{i j}$ in $\mathbb{R}^{3}$ can be written as a sum of a symmetric tensor and an antisymmetric tensor. Further, show that $P_{i j}$ can be decomposed into the following terms

  $$\tag{†} P_{i j}=P \delta_{i j}+S_{i j}+\epsilon_{i j k} A_{k},$$

  where $S_{i j}$ is symmetric and traceless. Give expressions for $P, S_{i j}$ and $A_{k}$ explicitly in terms of $P_{i j}$.

  For an isotropic material, the stress $P_{i j}$ can be related to the strain $T_{i j}$ through the stress-strain relation, $P_{i j}=c_{i j k l} T_{k l}$, where the elasticity tensor is given by

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

  and $\alpha, \beta$ and $\gamma$ are scalars. As in $(†)$, the strain $T_{i j}$ can be decomposed into its trace $T$, a symmetric traceless tensor $W_{i j}$ and a vector $V_{k}$. Use the stress-strain relation to express each of $T, W_{i j}$ and $V_{k}$ in terms of $P, S_{i j}$ and $A_{k}$.

  Hence, or otherwise, show that if $T_{i j}$ is symmetric then so is $P_{i j}$. Show also that the stress-strain relation can be written in the form

  $$P_{i j}=\lambda \delta_{i j} T_{k k}+\mu T_{i j}$$

  where $\mu$ and $\lambda$ are scalars.

  comment

• Paper 3, Section II, A

  Vector Calculus | Part IA, 2007

  The function $\phi(x, y, z)$ satisfies $\nabla^{2} \phi=0$ in $V$ and $\phi=0$ on $S$, where $V$ is a region of $\mathbb{R}^{3}$ which is bounded by the surface $S$. Prove that $\phi=0$ everywhere in $V$.

  Deduce that there is at most one function $\psi(x, y, z)$ satisfying $\nabla^{2} \psi=\rho$ in $V$ and $\psi=f$ on $S$, where $\rho(x, y, z)$ and $f(x, y, z)$ are given functions.

  Given that the function $\psi=\psi(r)$ depends only on the radial coordinate $r=|\mathbf{x}|$, use Cartesian coordinates to show that

  $$\nabla \psi=\frac{1}{r} \frac{d \psi}{d r} \mathbf{x}, \quad \nabla^{2} \psi=\frac{1}{r} \frac{d^{2}(r \psi)}{d r^{2}}$$

  Find the general solution in this radial case for $\nabla^{2} \psi=c$ where $c$ is a constant.

  Find solutions $\psi(r)$ for a solid sphere of radius $r=2$ with a central cavity of radius $r=1$ in the following three regions:

  (i) $0 \leqslant r \leqslant 1$ where $\nabla^{2} \psi=0$ and $\psi(1)=1$ and $\psi$ bounded as $r \rightarrow 0$;

  (ii) $1 \leqslant r \leqslant 2$ where $\nabla^{2} \psi=1$ and $\psi(1)=\psi(2)=1$;

  (iii) $r \geqslant 2$ where $\nabla^{2} \psi=0$ and $\psi(2)=1$ and $\psi \rightarrow 0$ as $r \rightarrow \infty$.

  comment

• Paper 3, Section II, A

  Vector Calculus | Part IA, 2007

  For a given charge distribution $\rho(x, y, z)$ and divergence-free current distribution $\mathbf{J}(x, y, z)$ (i.e. $\nabla \cdot \mathbf{J}=0)$ in $\mathbb{R}^{3}$, the electric and magnetic fields $\mathbf{E}(x, y, z)$ and $\mathbf{B}(x, y, z)$ satisfy the equations

  $$\nabla \times \mathbf{E}=0, \quad \nabla \cdot \mathbf{B}=0, \quad \nabla \cdot \mathbf{E}=\rho, \quad \nabla \times \mathbf{B}=\mathbf{J}$$

  The radiation flux vector $\mathbf{P}$ is defined by $\mathbf{P}=\mathbf{E} \times \mathbf{B}$. For a closed surface $S$ around a region $V$, show using Gauss' theorem that the flux of the vector $\mathbf{P}$ through $S$ can be expressed as

  $$\iint_{S} \mathbf{P} \cdot \mathbf{d} \mathbf{S}=-\iiint_{V} \mathbf{E} \cdot \mathbf{J} d V$$

  For electric and magnetic fields given by

  $$\mathbf{E}(x, y, z)=(z, 0, x), \quad \mathbf{B}(x, y, z)=(0,-x y, x z)$$

  find the radiation flux through the quadrant of the unit spherical shell given by

  $$x^{2}+y^{2}+z^{2}=1, \quad \text { with } \quad 0 \leqslant x \leqslant 1, \quad 0 \leqslant y \leqslant 1, \quad-1 \leqslant z \leqslant 1$$

  [If you use (*), note that an open surface has been specified.]

  comment

• Paper 3, Section II, A

  Vector Calculus | Part IA, 2007

  (i) Define what is meant by a conservative vector field. Given a vector field $\mathbf{A}=\left(A_{1}(x, y), A_{2}(x, y)\right)$ and a function $\psi(x, y)$ defined in $\mathbb{R}^{2}$, show that, if $\psi \mathbf{A}$ is a conservative vector field, then

  $$\psi\left(\frac{\partial A_{1}}{\partial y}-\frac{\partial A_{2}}{\partial x}\right)=A_{2} \frac{\partial \psi}{\partial x}-A_{1} \frac{\partial \psi}{\partial y}$$

  (ii) Given two functions $P(x, y)$ and $Q(x, y)$ defined in $\mathbb{R}^{2}$, prove Green's theorem,

  $$\oint_{C}(P d x+Q d y)=\iint_{R}\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right) d x d y$$

  where $C$ is a simple closed curve bounding a region $R$ in $\mathbb{R}^{2}$.

  Through an appropriate choice for $P$ and $Q$, find an expression for the area of the region $R$, and apply this to evaluate the area of the ellipse bounded by the curve

  $$x=a \cos \theta, \quad y=b \sin \theta, \quad 0 \leqslant \theta \leqslant 2 \pi$$

  comment

2006

• 3.I.3A

  Vector Calculus | Part IA, 2006

  Consider the vector field $\mathbf{F}(\mathbf{x})=\left(\left(3 x^{3}-x^{2}\right) y,\left(y^{3}-2 y^{2}+y\right) x, z^{2}-1\right)$ and let $S$ be the surface of a unit cube with one corner at $(0,0,0)$, another corner at $(1,1,1)$ and aligned with edges along the $x$-, $y$ - and $z$-axes. Use the divergence theorem to evaluate

  $$I=\int_{S} \mathbf{F} \cdot d \mathbf{S}$$

  Verify your result by calculating the integral directly.

  comment

• 3.I.4A

  Vector Calculus | Part IA, 2006

  Use suffix notation in Cartesian coordinates to establish the following two identities for the vector field $\mathbf{v}$ :

  $$\nabla \cdot(\nabla \times \mathbf{v})=0, \quad(\mathbf{v} \cdot \nabla) \mathbf{v}=\nabla\left(\frac{1}{2}|\mathbf{v}|^{2}\right)-\mathbf{v} \times(\nabla \times \mathbf{v})$$

  comment

• 3.II.10A

  Vector Calculus | Part IA, 2006

  State Stokes' theorem for a vector field $\mathbf{A}$.

  By applying Stokes' theorem to the vector field $\mathbf{A}=\phi \mathbf{k}$, where $\mathbf{k}$ is an arbitrary constant vector in $\mathbb{R}^{3}$ and $\phi$ is a scalar field defined on a surface $S$ bounded by a curve $\partial S$, show that

  $$\int_{S} d \mathbf{S} \times \nabla \phi=\int_{\partial S} \phi d \mathbf{x}$$

  For the vector field $\mathbf{A}=x^{2} y^{4}(1,1,1)$ in Cartesian coordinates, evaluate the line integral

  $$I=\int \mathbf{A} \cdot d \mathbf{x}$$

  around the boundary of the quadrant of the unit circle lying between the $x$ - and $y$ axes, that is, along the straight line from $(0,0,0)$ to $(1,0,0)$, then the circular arc $x^{2}+y^{2}=1, z=0$ from $(1,0,0)$ to $(0,1,0)$ and finally the straight line from $(0,1,0)$ back to $(0,0,0)$.

  comment

• 3.II.11A

  Vector Calculus | Part IA, 2006

  In a region $R$ of $\mathbb{R}^{3}$ bounded by a closed surface $S$, suppose that $\phi_{1}$ and $\phi_{2}$ are both solutions of $\nabla^{2} \phi=0$, satisfying boundary conditions on $S$ given by $\phi=f$ on $S$, where $f$ is a given function. Prove that $\phi_{1}=\phi_{2}$.

  In $\mathbb{R}^{2}$ show that

  $$\phi(x, y)=\left(a_{1} \cosh \lambda x+a_{2} \sinh \lambda x\right)\left(b_{1} \cos \lambda y+b_{2} \sin \lambda y\right)$$

  is a solution of $\nabla^{2} \phi=0$, for any constants $a_{1}, a_{2}, b_{1}, b_{2}$ and $\lambda$. Hence, or otherwise, find a solution $\phi(x, y)$ in the region $x \geqslant 0$ and $0 \leqslant y \leqslant a$ which satisfies:

  $$\begin{aligned} &\phi(x, 0)=0, \quad \phi(x, a)=0, \quad x \geqslant 0 \\ &\phi(0, y)=\sin \frac{n \pi y}{a}, \quad \phi(x, y) \rightarrow 0 \quad \text { as } \quad x \rightarrow \infty, \quad 0 \leqslant y \leqslant a \end{aligned}$$

  where $a$ is a real constant and $n$ is an integer.

  comment

• 3.II.12A

  Vector Calculus | Part IA, 2006

  Define what is meant by an isotropic tensor. By considering a rotation of a second rank isotropic tensor $B_{i j}$ by $90^{\circ}$ about the $z$-axis, show that its components must satisfy $B_{11}=B_{22}$ and $B_{13}=B_{31}=B_{23}=B_{32}=0$. Now consider a second and different rotation to show that $B_{i j}$ must be a multiple of the Kronecker delta, $\delta_{i j}$.

  Suppose that a homogeneous but anisotropic crystal has the conductivity tensor

  $$\sigma_{i j}=\alpha \delta_{i j}+\gamma n_{i} n_{j}$$

  where $\alpha, \gamma$ are real constants and the $n_{i}$ are the components of a constant unit vector $\mathbf{n}$ $(\mathbf{n} \cdot \mathbf{n}=1)$. The electric current density $\mathbf{J}$ is then given in components by

  $$J_{i}=\sigma_{i j} E_{j}$$

  where $E_{j}$ are the components of the electric field $\mathbf{E}$. Show that

  (i) if $\alpha \neq 0$ and $\gamma \neq 0$, then there is a plane such that if $\mathbf{E}$ lies in this plane, then $\mathbf{E}$ and $\mathbf{J}$ must be parallel, and

  (ii) if $\gamma \neq-\alpha$ and $\alpha \neq 0$, then $\mathbf{E} \neq 0$ implies $\mathbf{J} \neq 0$.

  If $D_{i j}=\epsilon_{i j k} n_{k}$, find the value of $\gamma$ such that

  $$\sigma_{i j} D_{j k} D_{k m}=-\sigma_{i m}$$

  comment

• 3.II.9A

  Vector Calculus | Part IA, 2006

  Evaluate the line integral

  $$\int \alpha\left(x^{2}+x y\right) d x+\beta\left(x^{2}+y^{2}\right) d y$$

  with $\alpha$ and $\beta$ constants, along each of the following paths between the points $A=(1,0)$ and $B=(0,1)$ :

  (i) the straight line between $A$ and $B$;

  (ii) the $x$-axis from $A$ to the origin $(0,0)$ followed by the $y$-axis to $B$;

  (iii) anti-clockwise from $A$ to $B$ around the circular path centred at the origin $(0,0)$.

  You should obtain the same answer for the three paths when $\alpha=2 \beta$. Show that when $\alpha=2 \beta$, the integral takes the same value along any path between $A$ and $B$.

  comment

2005

• 3.I.3A

  Vector Calculus | Part IA, 2005

  Let $\mathbf{A}(t, \mathbf{x})$ and $\mathbf{B}(t, \mathbf{x})$ be time-dependent, continuously differentiable vector fields on $\mathbb{R}^{3}$ satisfying

  $$\frac{\partial \mathbf{A}}{\partial t}=\nabla \times \mathbf{B} \quad \text { and } \quad \frac{\partial \mathbf{B}}{\partial t}=-\nabla \times \mathbf{A}$$

  Show that for any bounded region $V$,

  $$\frac{d}{d t}\left[\frac{1}{2} \int_{V}\left(\mathbf{A}^{2}+\mathbf{B}^{2}\right) d V\right]=-\int_{S}(\mathbf{A} \times \mathbf{B}) \cdot d \mathbf{S}$$

  where $S$ is the boundary of $V$.

  comment

• 3.I.4A

  Vector Calculus | Part IA, 2005

  Given a curve $\gamma(s)$ in $\mathbb{R}^{3}$, parameterised such that $\left\|\gamma^{\prime}(s)\right\|=1$ and with $\gamma^{\prime \prime}(s) \neq 0$, define the tangent $\mathbf{t}(s)$, the principal normal $\mathbf{p}(s)$, the curvature $\kappa(s)$ and the binormal $\mathbf{b}(s)$.

  The torsion $\tau(s)$ is defined by

  $$\tau=-\mathbf{b}^{\prime} \cdot \mathbf{p}$$

  Sketch a circular helix showing $\mathbf{t}, \mathbf{p}, \mathbf{b}$ and $\mathbf{b}^{\prime}$ at a chosen point. What is the sign of the torsion for your helix? Sketch a second helix with torsion of the opposite sign.

  comment

• 3.II.11A

  Vector Calculus | Part IA, 2005

  Explain, with justification, the significance of the eigenvalues of the Hessian in classifying the critical points of a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$. In what circumstances are the eigenvalues inconclusive in establishing the character of a critical point?

  Consider the function on $\mathbb{R}^{2}$,

  $$f(x, y)=x y e^{-\alpha\left(x^{2}+y^{2}\right)}$$

  Find and classify all of its critical points, for all real $\alpha$. How do the locations of the critical points change as $\alpha \rightarrow 0$ ?

  comment

• 3.II.12A

  Vector Calculus | Part IA, 2005

  Express the integral

  $$I=\int_{0}^{\infty} d x \int_{0}^{1} d y \int_{0}^{x} d z x e^{-A x / y-B x y-C y z}$$

  in terms of the new variables $\alpha=x / y, \beta=x y$, and $\gamma=y z$. Hence show that

  $$I=\frac{1}{2 A(A+B)(A+B+C)}$$

  You may assume $A, B$ and $C$ are positive. [Hint: Remember to calculate the limits of the integral.]

  comment

• 3.II.9A

  Vector Calculus | Part IA, 2005

  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.

  comment

2004

• 3.I.3C

  Vector Calculus | Part IA, 2004

  If $\mathbf{F}$ and $\mathbf{G}$ are differentiable vector fields, show that

  (i) $\boldsymbol{\nabla} \times(\mathbf{F} \times \mathbf{G})=\mathbf{F}(\boldsymbol{\nabla} \cdot \mathbf{G})-\mathbf{G}(\boldsymbol{\nabla} \cdot \mathbf{F})+(\mathbf{G} \cdot \boldsymbol{\nabla}) \mathbf{F}-(\mathbf{F} \cdot \boldsymbol{\nabla}) \mathbf{G}$,

  (ii) $\boldsymbol{\nabla}(\mathbf{F} \cdot \mathbf{G})=(\mathbf{F} \cdot \boldsymbol{\nabla}) \mathbf{G}+(\mathbf{G} \cdot \boldsymbol{\nabla}) \mathbf{F}+\mathbf{F} \times(\boldsymbol{\nabla} \times \mathbf{G})+\mathbf{G} \times(\boldsymbol{\nabla} \times \mathbf{F})$.

  comment

• 3.I.4C

  Vector Calculus | Part IA, 2004

  Define the curvature, $\kappa$, of a curve in $\mathbb{R}^{3}$.

  The curve $C$ is parametrised by

  $$\mathbf{x}(t)=\left(\frac{1}{2} e^{t} \cos t, \frac{1}{2} e^{t} \sin t, \frac{1}{\sqrt{2}} e^{t}\right) \quad \text { for }-\infty<t<\infty$$

  Obtain a parametrisation of the curve in terms of its arc length, $s$, measured from the origin. Hence obtain its curvature, $\kappa(s)$, as a function of $s$.

  comment

• 3.II.10C

  Vector Calculus | Part IA, 2004

  Explain what is meant by an exact differential. The three-dimensional vector field $\mathbf{F}$ is defined by

  $$\mathbf{F}=\left(e^{x} z^{3}+3 x^{2}\left(e^{y}-e^{z}\right), e^{y}\left(x^{3}-z^{3}\right), 3 z^{2}\left(e^{x}-e^{y}\right)-e^{z} x^{3}\right)$$

  Find the most general function that has $\mathbf{F} \cdot \mathbf{d} \mathbf{x}$ as its differential.

  Hence show that the line integral

  $$\int_{P_{1}}^{P_{2}} \mathbf{F} \cdot \mathbf{d} \mathbf{x}$$

  along any path in $\mathbb{R}^{3}$ between points $P_{1}=(0, a, 0)$ and $P_{2}=(b, b, b)$ vanishes for any values of $a$ and $b$.

  The two-dimensional vector field $\mathbf{G}$ is defined at all points in $\mathbb{R}^{2}$ except $(0,0)$ by

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

  $(\mathbf{G}$ is not defined at $(0,0)$.) Show that

  $$\oint_{C} \mathbf{G} \cdot \mathbf{d} \mathbf{x}=2 \pi$$

  for any closed curve $C$ in $\mathbb{R}^{2}$ that goes around $(0,0)$ anticlockwise precisely once without passing through $(0,0)$.

  comment

• 3.II.11C

  Vector Calculus | Part IA, 2004

  Let $S_{1}$ be the 3 -dimensional sphere of radius 1 centred at $(0,0,0), S_{2}$ be the sphere of radius $\frac{1}{2}$ centred at $\left(\frac{1}{2}, 0,0\right)$ and $S_{3}$ be the sphere of radius $\frac{1}{4}$ centred at $\left(\frac{-1}{4}, 0,0\right)$. The eccentrically shaped planet Zog is composed of rock of uniform density $\rho$ occupying the region within $S_{1}$ and outside $S_{2}$ and $S_{3}$. The regions inside $S_{2}$ and $S_{3}$ are empty. Give an expression for Zog's gravitational potential at a general coordinate $\mathbf{x}$ that is outside $S_{1}$. Is there a point in the interior of $S_{3}$ where a test particle would remain stably at rest? Justify your answer.

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• 3.II.12C

  Vector Calculus | Part IA, 2004

  State (without proof) the divergence theorem for a vector field $\mathbf{F}$ with continuous first-order partial derivatives throughout a volume $V$ enclosed by a bounded oriented piecewise-smooth non-self-intersecting surface $S$.

  By calculating the relevant volume and surface integrals explicitly, verify the divergence theorem for the vector field

  $$\mathbf{F}=\left(x^{3}+2 x y^{2}, y^{3}+2 y z^{2}, z^{3}+2 z x^{2}\right)$$

  defined within a sphere of radius $R$ centred at the origin.

  Suppose that functions $\phi, \psi$ are continuous and that their first and second partial derivatives are all also continuous in a region $V$ bounded by a smooth surface $S$.

  Show that

  $$\begin{aligned} \int_{V}\left(\phi \nabla^{2} \psi+\nabla \phi \cdot \nabla \psi\right) d \tau &=\int_{S} \phi \boldsymbol{\nabla} \psi \cdot \mathbf{d} \mathbf{S} \\ \int_{V}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d \tau &=\int_{S} \phi \boldsymbol{\nabla} \psi \cdot \mathbf{d} \mathbf{S}-\int_{S} \psi \boldsymbol{\nabla} \phi \cdot \mathbf{d} \mathbf{S} \end{aligned}$$

  Hence show that if $\rho(\mathbf{x})$ is a continuous function on $V$ and $g(\mathbf{x})$ a continuous function on $S$ and $\phi_{1}$ and $\phi_{2}$ are two continuous functions such that

  $$\begin{aligned} \nabla^{2} \phi_{1}(\mathbf{x}) &=\nabla^{2} \phi_{2}(\mathbf{x})=\rho(\mathbf{x}) \quad \text { for all } \mathbf{x} \text { in } V, \text { and } \\ \phi_{1}(\mathbf{x}) &=\phi_{2}(\mathbf{x})=g(\mathbf{x}) \quad \text { for all } \mathbf{x} \text { on } S \end{aligned}$$

  then $\phi_{1}(\mathbf{x})=\phi_{2}(\mathbf{x})$ for all $\mathbf{x}$ in $V$.

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• 3.II.9C

  Vector Calculus | Part IA, 2004

  For a function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ state if the following implications are true or false. (No justification is required.)

  (i) $f$ is differentiable $\Rightarrow f$ is continuous.

  (ii) $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exist $\Rightarrow f$ is continuous.

  (iii) directional derivatives $\frac{\partial f}{\partial \mathbf{n}}$ exist for all unit vectors $\mathbf{n} \in \mathbb{R}^{2} \Rightarrow f$ is differentiable.

  (iv) $f$ is differentiable $\Rightarrow \frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are continuous.

  (v) all second order partial derivatives of $f$ exist $\Rightarrow \frac{\partial^{2} f}{\partial x \partial y}=\frac{\partial^{2} f}{\partial y \partial x}$.

  Now let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be defined by

  $$f(x, y)= \begin{cases}\frac{x y\left(x^{2}-y^{2}\right)}{\left(x^{2}+y^{2}\right)} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}$$

  Show that $f$ is continuous at $(0,0)$ and find the partial derivatives $\frac{\partial f}{\partial x}(0, y)$ and $\frac{\partial f}{\partial y}(x, 0)$. Then show that $f$ is differentiable at $(0,0)$ and find its derivative. Investigate whether the second order partial derivatives $\frac{\partial^{2} f}{\partial x \partial y}(0,0)$ and $\frac{\partial^{2} f}{\partial y \partial x}(0,0)$ are the same. Are the second order partial derivatives of $f$ at $(0,0)$ continuous? Justify your answer.

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2003

• 3.I.3A

  Vector Calculus | Part IA, 2003

  Sketch the curve $y^{2}=x^{2}+1$. By finding a parametric representation, or otherwise, determine the points on the curve where the radius of curvature is least, and compute its value there.

  [Hint: you may use the fact that the radius of curvature of a parametrized curve $(x(t), y(t))$ is $\left(\dot{x}^{2}+\dot{y}^{2}\right)^{3 / 2} /|\dot{x} \ddot{y}-\ddot{x} \dot{y}|$.]

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• 3.I.4A

  Vector Calculus | Part IA, 2003

  Suppose $V$ is a region in $\mathbb{R}^{3}$, bounded by a piecewise smooth closed surface $S$, and $\phi(\mathbf{x})$ is a scalar field satisfying

  $$\begin{aligned} \nabla^{2} \phi &=0 \text { in } V, \\ \text { and } \phi &=f(\mathbf{x}) \text { on } S . \end{aligned}$$

  Prove that $\phi$ is determined uniquely in $V$.

  How does the situation change if the normal derivative of $\phi$ rather than $\phi$ itself is specified on $S$ ?

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• 3.II.10A

  Vector Calculus | Part IA, 2003

  Write down an expression for the Jacobian $J$ of a transformation

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

  Use it to show that

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

  where $D$ is mapped one-to-one onto $\Delta$, and

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

  Find a transformation that maps the ellipsoid $D$,

  $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} \leqslant 1$$

  onto a sphere. Hence evaluate

  $$\int_{D} x^{2} d x d y d z$$

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• 3.II.11A

  Vector Calculus | Part IA, 2003

  (a) Prove the identity

  $$\boldsymbol{\nabla}(\mathbf{F} \cdot \mathbf{G})=(\mathbf{F} \cdot \boldsymbol{\nabla}) \mathbf{G}+(\mathbf{G} \cdot \boldsymbol{\nabla}) \mathbf{F}+\mathbf{F} \times(\boldsymbol{\nabla} \times \mathbf{G})+\mathbf{G} \times(\boldsymbol{\nabla} \times \mathbf{F})$$

  (b) If $\mathbf{E}$ is an irrotational vector field $(\boldsymbol{\nabla} \times \mathbf{E}=\mathbf{0}$ everywhere $)$, prove that there exists a scalar potential $\phi(\mathbf{x})$ such that $\mathbf{E}=-\boldsymbol{\nabla} \phi$.

  Show that

  $$\left(2 x y^{2} z e^{-x^{2} z},-2 y e^{-x^{2} z}, x^{2} y^{2} e^{-x^{2} z}\right)$$

  is irrotational, and determine the corresponding potential $\phi$.

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• 3.II.12A

  Vector Calculus | Part IA, 2003

  State the divergence theorem. By applying this to $f(\mathbf{x}) \mathbf{k}$, where $f(\mathbf{x})$ is a scalar field in a closed region $V$ in $\mathbb{R}^{3}$ bounded by a piecewise smooth surface $S$, and $\mathbf{k}$ an arbitrary constant vector, show that

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

  A vector field $\mathbf{G}$ satisfies

  $$\begin{gathered} \boldsymbol{\nabla} \cdot \mathbf{G}=\rho(\mathbf{x}) \\ \text { with } \quad \rho(\mathbf{x})= \begin{cases}\rho_{0} & |\mathbf{x}| \leqslant a \\ 0 & |\mathbf{x}|>a\end{cases} \end{gathered}$$

  By applying the divergence theorem to $\int_{V} \nabla \cdot \mathbf{G} d V$, prove Gauss's law

  $$\int_{S} \mathbf{G} \cdot d \mathbf{S}=\int_{V} \rho(\mathbf{x}) d V$$

  where $S$ is the piecewise smooth surface bounding the volume $V$.

  Consider the spherically symmetric solution

  $$\mathbf{G}(\mathbf{x})=G(r) \frac{\mathbf{x}}{r}$$

  where $r=|\mathbf{x}|$. By using Gauss's law with $S$ a sphere of radius $r$, centre $\mathbf{0}$, in the two cases $0<r \leqslant a$ and $r>a$, show that

  $$\mathbf{G}(\mathbf{x})= \begin{cases}\frac{\rho_{0}}{3} \mathbf{x} & r \leqslant a \\ \frac{\rho_{0}}{3}\left(\frac{a}{r}\right)^{3} \mathbf{x} & r>a\end{cases}$$

  The scalar field $f(\mathbf{x})$ satisfies $\mathbf{G}=\nabla f$. Assuming that $f \rightarrow 0$ as $r \rightarrow \infty$, and that $f$ is continuous at $r=a$, find $f$ everywhere.

  By using a symmetry argument, explain why $(*)$ is clearly satisfied for this $f$ if $S$ is any sphere centred at the origin.

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• 3.II.9A

  Vector Calculus | Part IA, 2003

  Let $C$ be the closed curve that is the boundary of the triangle $T$ with vertices at the points $(1,0,0),(0,1,0)$ and $(0,0,1)$.

  Specify a direction along $C$ and consider the integral

  $$\oint_{C} \mathbf{A} \cdot d \mathbf{x}$$

  where $\mathbf{A}=\left(z^{2}-y^{2}, x^{2}-z^{2}, y^{2}-x^{2}\right)$. Explain why the contribution to the integral is the same from each edge of $C$, and evaluate the integral.

  State Stokes's theorem and use it to evaluate the surface integral

  $$\int_{T}(\boldsymbol{\nabla} \times \mathbf{A}) \cdot d \mathbf{S},$$

  the components of the normal to $T$ being positive.

  Show that $d \mathbf{S}$ in the above surface integral can be written in the form $(1,1,1) d y d z$.

  Use this to verify your result by a direct calculation of the surface integral.

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2002

• 3.I.3A

  Vector Calculus | Part IA, 2002

  Determine whether each of the following is the exact differential of a function, and if so, find such a function: (a) $(\cosh \theta+\sinh \theta \cos \phi) d \theta+(\cosh \theta \sin \phi+\cos \phi) d \phi$, (b) $3 x^{2}\left(y^{2}+1\right) d x+2\left(y x^{3}-z^{2}\right) d y-4 y z d z$.

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• 3.I.4A

  Vector Calculus | Part IA, 2002

  State the divergence theorem.

  Consider the integral

  $$I=\int_{S} r^{n} \mathbf{r} \cdot d \mathbf{S}$$

  where $n>0$ and $S$ is the sphere of radius $R$ centred at the origin. Evaluate $I$ directly, and by means of the divergence theorem.

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• 3.II.10A

  Vector Calculus | Part IA, 2002

  The domain $S$ in the $(x, y)$ plane is bounded by $y=x, y=a x(0 \leqslant a \leqslant 1)$ and $x y^{2}=1(x, y \geqslant 0)$. Find a transformation

  $$u=f(x, y), \quad v=g(x, y)$$

  such that $S$ is transformed into a rectangle in the $(u, v)$ plane.

  Evaluate

  $$\int_{D} \frac{y^{2} z^{2}}{x} d x d y d z$$

  where $D$ is the region bounded by

  $$y=x, \quad y=z x, \quad x y^{2}=1 \quad(x, y \geqslant 0)$$

  and the planes

  $$z=0, \quad z=1$$

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• 3.II.11A

  Vector Calculus | Part IA, 2002

  Prove that

  $$\nabla \times(\mathbf{a} \times \mathbf{b})=\mathbf{a} \nabla \cdot \mathbf{b}-\mathbf{b} \nabla \cdot \mathbf{a}+(\mathbf{b} \cdot \nabla) \mathbf{a}-(\mathbf{a} \cdot \nabla) \mathbf{b}$$

  $S$ is an open orientable surface in $\mathbb{R}^{3}$ with unit normal $\mathbf{n}$, and $\mathbf{v}(\mathbf{x})$ is any continuously differentiable vector field such that $\mathbf{n} \cdot \mathbf{v}=0$ on $S$. Let $\mathbf{m}$ be a continuously differentiable unit vector field which coincides with $\mathbf{n}$ on $S$. By applying Stokes' theorem to $\mathbf{m} \times \mathbf{v}$, show that

  $$\int_{S}\left(\delta_{i j}-n_{i} n_{j}\right) \frac{\partial v_{i}}{\partial x_{j}} d S=\oint_{C} \mathbf{u} \cdot \mathbf{v} d s$$

  where $s$ denotes arc-length along the boundary $C$ of $S$, and $\mathbf{u}$ is such that $\mathbf{u} d s=d \mathbf{s} \times \mathbf{n}$. Verify this result by taking $\mathbf{v}=\mathbf{r}$, and $S$ to be the disc $|\mathbf{r}| \leqslant R$ in the $z=0$ plane.

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• 3.II.12A

  Vector Calculus | Part IA, 2002

  (a) Show, using Cartesian coordinates, that $\psi=1 / r$ satisfies Laplace's equation, $\nabla^{2} \psi=0$, on $\mathbb{R}^{3} \backslash\{0\} .$

  (b) $\phi$ and $\psi$ are smooth functions defined in a 3-dimensional domain $V$ bounded by a smooth surface $S$. Show that

  $$\int_{V}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d V=\int_{S}(\phi \nabla \psi-\psi \nabla \phi) \cdot d \mathbf{S}$$

  (c) Let $\psi=1 /\left|\mathbf{r}-\mathbf{r}_{0}\right|$, and let $V_{\varepsilon}$ be a domain bounded by a smooth outer surface $S$ and an inner surface $S_{\varepsilon}$, where $S_{\varepsilon}$ is a sphere of radius $\varepsilon$, centre $\mathbf{r}_{0}$. The function $\phi$ satisfies

  $$\nabla^{2} \phi=-\rho(\mathbf{r}) .$$

  Use parts (a) and (b) to show, taking the limit $\varepsilon \rightarrow 0$, that $\phi$ at $\mathbf{r}_{0}$ is given by

  $$4 \pi \phi\left(\mathbf{r}_{0}\right)=\int_{V} \frac{\rho(\mathbf{r})}{\left|\mathbf{r}-\mathbf{r}_{0}\right|} d V+\int_{S}\left(\frac{1}{\left|\mathbf{r}-\mathbf{r}_{0}\right|} \frac{\partial \phi}{\partial n}-\phi(\mathbf{r}) \frac{\partial}{\partial n} \frac{1}{\left|\mathbf{r}-\mathbf{r}_{0}\right|}\right) d S,$$

  where $V$ is the domain bounded by $S$.

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• 3.II.9A

  Vector Calculus | Part IA, 2002

  Two independent variables $x_{1}$ and $x_{2}$ are related to a third variable $t$ by

  $$x_{1}=a+\alpha t, \quad x_{2}=b+\beta t,$$

  where $a, b, \alpha$ and $\beta$ are constants. Let $f$ be a smooth function of $x_{1}$ and $x_{2}$, and let $F(t)=f\left(x_{1}, x_{2}\right)$. Show, by using the Taylor series for $F(t)$ about $t=0$, that

  $$\begin{gathered} f\left(x_{1}, x_{2}\right)=f(a, b)+\left(x_{1}-a\right) \frac{\partial f}{\partial x_{1}}+\left(x_{2}-b\right) \frac{\partial f}{\partial x_{2}} \\ +\frac{1}{2}\left(\left(x_{1}-a\right)^{2} \frac{\partial^{2} f}{\partial x_{1}^{2}}+2\left(x_{1}-a\right)\left(x_{2}-b\right) \frac{\partial^{2} f}{\partial x_{1} \partial x_{2}}+\left(x_{2}-b\right)^{2} \frac{\partial^{2} f}{\partial x_{2}^{2}}\right)+\ldots \end{gathered}$$

  where all derivatives are evaluated at $x_{1}=a, x_{2}=b$.

  Hence show that a stationary point $(a, b)$ of $f\left(x_{1}, x_{2}\right)$ is a local minimum if

  $$H_{11}>0, \quad \operatorname{det} H_{i j}>0$$

  where $H_{i j}=\frac{\partial^{2} f}{\partial x_{i} \partial x_{j}}$ is the Hessian matrix evaluated at $(a, b)$.

  Find two local minima of

  $$f\left(x_{1}, x_{2}\right)=x_{1}^{4}-x_{1}^{2}+2 x_{1} x_{2}+x_{2}^{2}$$

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2001

• 3.I.3C

  Vector Calculus | Part IA, 2001

  For a real function $f(x, y)$ with $x=x(t)$ and $y=y(t)$ state the chain rule for the derivative $\frac{d}{d t} f(x(t), y(t))$.

  By changing variables to $u$ and $v$, where $u=\alpha(x) y$ and $v=y / x$ with a suitable function $\alpha(x)$ to be determined, find the general solution of the equation

  $$x \frac{\partial f}{\partial x}-2 y \frac{\partial f}{\partial y}=6 f$$

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• 3.I.4A

  Vector Calculus | Part IA, 2001

  Suppose that

  $$u=y^{2} \sin (x z)+x y^{2} z \cos (x z), \quad v=2 x y \sin (x z), \quad w=x^{2} y^{2} \cos (x z)$$

  Show that $u d x+v d y+w d z$ is an exact differential.

  Show that

  $$\int_{(0,0,0)}^{(\pi / 2,1,1)} u d x+v d y+w d z=\frac{\pi}{2}$$

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• 3.II.10A

  Vector Calculus | Part IA, 2001

  State the rule for changing variables in a double integral.

  Let $D$ be the region defined by

  $$\begin{cases}1 / x \leq y \leq 4 x & \text { when } \frac{1}{2} \leq x \leq 1 \\ x \leq y \leq 4 / x & \text { when } 1 \leq x \leq 2\end{cases}$$

  Using the transformation $u=y / x$ and $v=x y$, show that

  $$\int_{D} \frac{4 x y^{3}}{x^{2}+y^{2}} d x d y=\frac{15}{2} \ln \frac{17}{2}$$

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• 3.II.11B

  Vector Calculus | Part IA, 2001

  State the divergence theorem for a vector field $\mathbf{u}(\mathbf{r})$ in a closed region $V$ bounded by a smooth surface $S$.

  Let $\Omega(\mathbf{r})$ be a scalar field. By choosing $\mathbf{u}=\mathbf{c} \Omega$ for arbitrary constant vector $\mathbf{c}$, show that

  $$\int_{V} \nabla \Omega d v=\int_{S} \Omega d \mathbf{S}$$

  Let $V$ be the bounded region enclosed by the surface $S$ which consists of the cone $(x, y, z)=(r \cos \theta, r \sin \theta, r / \sqrt{3})$ with $0 \leq r \leq \sqrt{3}$ and the plane $z=1$, where $r, \theta, z$ are cylindrical polar coordinates. Verify that $(*)$ holds for the scalar field $\Omega=(a-z)$ where $a$ is a constant.

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• 3.II.12B

  Vector Calculus | Part IA, 2001

  In $\mathbb{R}^{3}$ show that, within a closed surface $S$, there is at most one solution of Poisson's equation, $\nabla^{2} \phi=\rho$, satisfying the boundary condition on $S$

  $$\alpha \frac{\partial \phi}{\partial n}+\phi=\gamma \text {, }$$

  where $\alpha$ and $\gamma$ are functions of position on $S$, and $\alpha$ is everywhere non-negative.

  Show that

  $$\phi(x, y)=e^{\pm l x} \sin l y$$

  are solutions of Laplace's equation $\nabla^{2} \phi=0$ on $\mathbb{R}^{2}$.

  Find a solution $\phi(x, y)$ of Laplace's equation in the region $0<x<\pi, 0<y<\pi$ that satisfies the boundary conditions

  $$\begin{array}{cccc} \phi=0 & \text { on } & 0<x<\pi & y=0 \\ \phi=0 & \text { on } & 0<x<\pi & y=\pi \\ \phi+\partial \phi / \partial n=0 & \text { on } & x=0 & 0<y<\pi \\ \phi=\sin (k y) & \text { on } & x=\pi & 0<y<\pi \end{array}$$

  where $k$ is a positive integer. Is your solution the only possible solution?

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• 3.II.9C

  Vector Calculus | Part IA, 2001

  Explain, with justification, how the nature of a critical (stationary) point of a function $f(\mathbf{x})$ can be determined by consideration of the eigenvalues of the Hessian matrix $H$ of $f(\mathbf{x})$ if $H$ is non-singular. What happens if $H$ is singular?

  Let $f(x, y)=\left(y-x^{2}\right)\left(y-2 x^{2}\right)+\alpha x^{2}$. Find the critical points of $f$ and determine their nature in the different cases that arise according to the values of the parameter $\alpha \in \mathbb{R}$.

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