Part IA, 2001, Paper 3

Algebra and Geometry

• 3.I.1F

  Algebra and Geometry | Part IA, 2001

  For a $2 \times 2$ matrix $A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$, prove that $A^{2}=0$ if and only if $a=-d$ and $b c=-a^{2}$. Prove that $A^{3}=0$ if and only if $A^{2}=0$.

  [Hint: it is easy to check that $\left.A^{2}-(a+d) A+(a d-b c) I=0 .\right]$

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• 3.I.2D

  Algebra and Geometry | Part IA, 2001

  Show that the set of Möbius transformations of the extended complex plane $\mathbb{C} \cup\{\infty\}$ form a group. Show further that an arbitrary Möbius transformation can be expressed as the composition of maps of the form

  $$f(z)=z+a, \quad g(z)=k z \text { and } h(z)=1 / z$$

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• 3.II.5F

  Algebra and Geometry | Part IA, 2001

  Let $A, B, C$ be $2 \times 2$ matrices, real or complex. Define the trace $\operatorname{tr} C$ to be the sum of diagonal entries $C_{11}+C_{22}$. Define the commutator $[A, B]$ to be the difference $A B-B A$. Give the definition of the eigenvalues of a $2 \times 2$ matrix and prove that it can have at most two distinct eigenvalues. Prove that a) $\operatorname{tr}[A, B]=0$, b) $\operatorname{tr} C$ equals the sum of the eigenvalues of $C$, c) if all eigenvalues of $C$ are equal to 0 then $C^{2}=0$, d) either $[A, B]$ is a diagonalisable matrix or the square $[A, B]^{2}=0$, e) $[A, B]^{2}=\alpha I$ where $\alpha \in \mathbb{C}$ and $I$ is the unit matrix.

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• 3.II.7E

  Algebra and Geometry | Part IA, 2001

  State Lagrange's theorem. Use it to describe all groups of order $p$, where $p$ is a fixed prime number.

  Find all the subgroups of a fixed cyclic group $\langle x\rangle$ of order $n$.

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• 3.II.8D

  Algebra and Geometry | Part IA, 2001

  (i) Let $A_{4}$ denote the alternating group of even permutations of four symbols. Let $X$ be the 3-cycle $(123)$ and $P, Q$ be the pairs of transpositions $(12)(34)$ and $(13)(24)$. Find $X^{3}, P^{2}, Q^{2}, X^{-1} P X, X^{-1} Q X$, and show that $A_{4}$ is generated by $X, P$ and $Q$.

  (ii) Let $G$ and $H$ be groups and let

  $$G \times H=\{(g, h): g \in G, h \in H\}$$

  Show how to make $G \times H$ into a group in such a way that $G \times H$ contains subgroups isomorphic to $G$ and $H$.

  If $D_{n}$ is the dihedral group of order $n$ and $C_{2}$ is the cyclic group of order 2 , show that $D_{12}$ is isomorphic to $D_{6} \times C_{2}$. Is the group $D_{12}$ isomorphic to $A_{4}$ ?

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Vector Calculus

• 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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