Part IA, 2002, Paper 3

Algebra and Geometry

• 3.I.1A

  Algebra and Geometry | Part IA, 2002

  Given two real non-zero $2 \times 2$ matrices $A$ and $B$, with $A B=0$, show that $A$ maps $\mathbb{R}^{2}$ onto a line. Is it always true that $B A=0 ?$ Show that there is always a non-zero matrix $C$ with $C A=0=A C$. Justify your answers.

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• 3.I.2B

  Algebra and Geometry | Part IA, 2002

  (a) What does it mean for a group to be cyclic? Give an example of a finite abelian group that is not cyclic, and justify your assertion.

  (b) Suppose that $G$ is a finite group of rotations of $\mathbb{R}^{2}$ about the origin. Is $G$ necessarily cyclic? Justify your answer.

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• 3.II.5E

  Algebra and Geometry | Part IA, 2002

  Prove, using the standard formula connecting $\delta_{i j}$ and $\epsilon_{i j k}$, that

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

  Define, in terms of the dot and cross product, the triple scalar product [a, b, c $]$ of three vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ in $\mathbb{R}^{3}$ and show that it is invariant under cyclic permutation of the vectors.

  Let $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ be a not necessarily orthonormal basis for $\mathbb{R}^{3}$, and define

  $$\hat{\mathbf{e}}_{1}=\frac{\mathbf{e}_{2} \times \mathbf{e}_{3}}{\left[\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right]}, \quad \hat{\mathbf{e}}_{2}=\frac{\mathbf{e}_{3} \times \mathbf{e}_{1}}{\left[\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right]}, \quad \hat{\mathbf{e}}_{3}=\frac{\mathbf{e}_{1} \times \mathbf{e}_{2}}{\left[\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right]} .$$

  By calculating $\left[\hat{\mathbf{e}}_{1}, \hat{\mathbf{e}}_{2}, \hat{\mathbf{e}}_{3}\right]$, show that $\hat{\mathbf{e}}_{1}, \hat{\mathbf{e}}_{2}, \hat{\mathbf{e}}_{3}$ is also a basis for $\mathbb{R}^{3}$.

  The vectors $\hat{\mathbf{e}}_{1}, \hat{\mathbf{e}}_{2}, \hat{\mathbf{e}}_{3}$ are constructed from $\hat{\mathbf{e}}_{1}, \hat{\mathbf{e}}_{2}, \hat{\mathbf{e}}_{3}$ in the same way that $\hat{\mathbf{e}}_{1}, \hat{\mathbf{e}}_{2}, \hat{\mathbf{e}}_{3}$ are constructed from $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$. Show that

  $$\hat{\mathbf{e}}_{1}=\mathbf{e}_{1}, \hat{\hat{\mathbf{e}}}_{2}=\mathbf{e}_{2}, \hat{\mathbf{e}}_{3}=\mathbf{e}_{3},$$

  Show that a vector $\mathbf{V}$ has components $\mathbf{V} \cdot \hat{\mathbf{e}}_{1}, \mathbf{V} \cdot \hat{\mathbf{e}}_{2}, \mathbf{V} \cdot \hat{\mathbf{e}}_{3}$ with respect to the basis $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$. What are the components of the vector $\mathbf{V}$ with respect to the basis $\hat{\mathbf{e}}_{1}, \hat{\mathbf{e}}_{2}, \hat{\mathbf{e}}_{3}$ ?

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• 3.II.6E

  Algebra and Geometry | Part IA, 2002

  (a) Give the general solution for $\mathbf{x}$ and $\mathbf{y}$ of the equations

  $$\mathbf{x}+\mathbf{y}=2 \mathbf{a}, \quad \mathbf{x} \cdot \mathbf{y}=c \quad(c<\mathbf{a} \cdot \mathbf{a})$$

  Show in particular that $\mathbf{x}$ and $\mathbf{y}$ must lie at opposite ends of a diameter of a sphere whose centre and radius should be specified.

  (b) If two pairs of opposite edges of a tetrahedron are perpendicular, show that the third pair are also perpendicular to each other. Show also that the sum of the lengths squared of two opposite edges is the same for each pair.

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• 3.II.7A

  Algebra and Geometry | Part IA, 2002

  Explain why the number of solutions $\mathbf{x} \in \mathbb{R}^{3}$ of the simultaneous linear equations $A \mathbf{x}=\mathbf{b}$ is 0,1 or infinite, where $A$ is a real $3 \times 3$ matrix and $\mathbf{b} \in \mathbb{R}^{3}$. Let $\alpha$ be the mapping which $A$ represents. State necessary and sufficient conditions on $\mathbf{b}$ and $\alpha$ for each of these possibilities to hold.

  Let $A$ and $B$ be $3 \times 3$ matrices representing linear mappings $\alpha$ and $\beta$. Give necessary and sufficient conditions on $\alpha$ and $\beta$ for the existence of a $3 \times 3$ matrix $X$ with $A X=B$. When is $X$ unique?

  Find $X$ when

  $$A=\left(\begin{array}{lll} 4 & 1 & 1 \\ 1 & 2 & 1 \\ 0 & 3 & 1 \end{array}\right), \quad B=\left(\begin{array}{lll} 1 & 1 & 1 \\ 0 & 1 & 0 \\ 3 & 1 & 2 \end{array}\right)$$

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• 3.II.8B

  Algebra and Geometry | Part IA, 2002

  Suppose that a,b, $\mathbf{c}, \mathbf{d}$ are the vertices of a regular tetrahedron $T$ in $\mathbb{R}^{3}$ and that $\mathbf{a}=(1,1,1), \mathbf{b}=(-1,-1,1), \mathbf{c}=(-1,1,-1), \mathbf{d}=(1, x, y)$.

  (a) Find $x$ and $y$.

  (b) Find a matrix $M$ that is a rotation leaving $T$ invariant such that $M \mathbf{a}=\mathbf{b}$ and $M \mathbf{b}=\mathbf{a} .$

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Vector Calculus

• 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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