Part IA, 2003, Paper 3

Algebra and Geometry

• 3.I.1A

  Algebra and Geometry | Part IA, 2003

  The mapping $\alpha$ of $\mathbb{R}^{3}$ into itself is a reflection in the plane $x_{2}=x_{3}$. Find the matrix $A$ of $\alpha$ with respect to any basis of your choice, which should be specified.

  The mapping $\beta$ of $\mathbb{R}^{3}$ into itself is a rotation about the line $x_{1}=x_{2}=x_{3}$ through $2 \pi / 3$, followed by a dilatation by a factor of 2 . Find the matrix $B$ of $\beta$ with respect to a choice of basis that should again be specified.

  Show explicitly that

  $$B^{3}=8 A^{2}$$

  and explain why this must hold, irrespective of your choices of bases.

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• 3.I.2B

  Algebra and Geometry | Part IA, 2003

  Show that if a group $G$ contains a normal subgroup of order 3, and a normal subgroup of order 5 , then $G$ contains an element of order 15 .

  Give an example of a group of order 10 with no element of order $10 .$

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• 3.II.5E

  Algebra and Geometry | Part IA, 2003

  (a) Show, using vector methods, that the distances from the centroid of a tetrahedron to the centres of opposite pairs of edges are equal. If the three distances are $u, v, w$ and if $a, b, c, d$ are the distances from the centroid to the vertices, show that

  $$u^{2}+v^{2}+w^{2}=\frac{1}{4}\left(a^{2}+b^{2}+c^{2}+d^{2}\right) .$$

  [The centroid of $k$ points in $\mathbb{R}^{3}$ with position vectors $\mathbf{x}_{i}$ is the point with position vector $\left.\frac{1}{k} \sum \mathbf{x}_{i} .\right]$

  (b) Show that

  $$|\mathbf{x}-\mathbf{a}|^{2} \cos ^{2} \alpha=[(\mathbf{x}-\mathbf{a}) \cdot \mathbf{n}]^{2},$$

  with $\mathbf{n}^{2}=1$, is the equation of a right circular double cone whose vertex has position vector a, axis of symmetry $\mathbf{n}$ and opening angle $\alpha$.

  Two such double cones, with vertices $\mathbf{a}_{1}$ and $\mathbf{a}_{2}$, have parallel axes and the same opening angle. Show that if $\mathbf{b}=\mathbf{a}_{1}-\mathbf{a}_{2} \neq \mathbf{0}$, then the intersection of the cones lies in a plane with unit normal

  $$\mathbf{N}=\frac{\mathbf{b} \cos ^{2} \alpha-\mathbf{n}(\mathbf{n} \cdot \mathbf{b})}{\sqrt{\mathbf{b}^{2} \cos ^{4} \alpha+(\mathbf{b} \cdot \mathbf{n})^{2}\left(1-2 \cos ^{2} \alpha\right)}}$$

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• 3.II.6E

  Algebra and Geometry | Part IA, 2003

  Derive an expression for the triple scalar product $\left(\mathbf{e}_{1} \times \mathbf{e}_{2}\right) \cdot \mathbf{e}_{3}$ in terms of the determinant of the matrix $E$ whose rows are given by the components of the three vectors $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$.

  Use the geometrical interpretation of the cross product to show that $\mathbf{e}_{a}, a=1,2,3$, will be a not necessarily orthogonal basis for $\mathbb{R}^{3}$ as long as $\operatorname{det} E \neq 0$.

  The rows of another matrix $\hat{E}$ are given by the components of three other vectors $\hat{\mathbf{e}}_{b}, b=1,2,3$. By considering the matrix $E \hat{E}^{\mathrm{T}}$, where ${ }^{\mathrm{T}}$ denotes the transpose, show that there is a unique choice of $\hat{E}$ such that $\hat{\mathbf{e}}_{b}$ is also a basis and

  $$\mathbf{e}_{a} \cdot \hat{\mathbf{e}}_{b}=\delta_{a b}$$

  Show that the new basis is given by

  $$\hat{\mathbf{e}}_{1}=\frac{\mathbf{e}_{2} \times \mathbf{e}_{3}}{\left(\mathbf{e}_{1} \times \mathbf{e}_{2}\right) \cdot \mathbf{e}_{3}} \quad \text { etc. }$$

  Show that if either $\mathbf{e}_{a}$ or $\hat{\mathbf{e}}_{b}$ is an orthonormal basis then $E$ is a rotation matrix.

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• 3.II.7B

  Algebra and Geometry | Part IA, 2003

  Let $G$ be the group of Möbius transformations of $\mathbb{C} \cup\{\infty\}$ and let $X=\{\alpha, \beta, \gamma\}$ be a set of three distinct points in $\mathbb{C} \cup\{\infty\}$.

  (i) Show that there exists a $g \in G$ sending $\alpha$ to $0, \beta$ to 1 , and $\gamma$ to $\infty$.

  (ii) Hence show that if $H=\{g \in G \mid g X=X\}$, then $H$ is isomorphic to $S_{3}$, the symmetric group on 3 letters.

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• 3.II.8B

  Algebra and Geometry | Part IA, 2003

  (a) Determine the characteristic polynomial and the eigenvectors of the matrix

  $$\left(\begin{array}{ccc} 0 & 1 & 0 \\ -1 & 2 & 0 \\ 1 & -1 & 2 \end{array}\right)$$

  Is it diagonalizable?

  (b) Show that an $n \times n$ matrix $A$ with characteristic polynomial $f(t)=(t-\mu)^{n}$ is diagonalizable if and only if $A=\mu I$.

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Vector Calculus

• 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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