Prelims Syllabus

The syllabus here is that referred to in the Examination Regulations 2016 - Special Regulations for the Preliminary Examination in Mathematics and has been approved by the Mathematics Teaching Committee for examination in Trinity Term 2017.

Mathematics I

Sets: examples including the natural numbers, the integers, the rational numbers, the real numbers; inclusion, union, intersection, power set, ordered pairs and cartesian product of sets. Relations. Definition of an equivalence relation.

The well-ordering property of the natural numbers. Induction as a method of proof, including a proof of the binomial theorem with non-negative integral coefficients.

Maps: composition, restriction, injective (one-to-one), surjective (onto) and invertible maps, images and preimages.

Systems of linear equations. Expression as an augmented matrix (just understood as an array at this point). Elementary Row Operations (EROs). Solutions by row reduction.

Abstract vector spaces: Definition of a vector space over a field (expected examples $\mathbb{R}$, $\mathbb{Q}$, $\mathbb{C}\,$). Examples of vector spaces: solution space of homogeneous system of equations and differential equations; function spaces; polynomials; $\mathbb{C}$ as an $\mathbb{R}$-vector space; sequence spaces. Subspaces, spanning sets and spans.

Linear independence, definition of a basis, examples. Steinitz exchange lemma, and definition of dimension. Coordinates associated with a
basis. Algorithms involving finding a basis of a subspace with EROs. Sums, intersections and direct sums of subspaces. Dimension formula.

Linear transformations: definition and examples including projections. Kernel and image, rank nullity formula.

Algebra of linear transformations. Inverses. Matrix of a linear transformation with respect to a basis. Algebra of matrices. Transformation
of a matrix under change of basis. Determining an inverse with EROs. Column space, column rank.

Bilinear forms. Positive definite symmetric bilinear forms. Inner Product Spaces. Examples: $\mathbb{R}^{n}$ with dot product, function spaces. Comment on (positive definite) Hermitian forms. Cauchy-Schwarz inequality. Distance and angle. Transpose of
a matrix. Orthogonal matrices.

Introduction to determinant of a square matrix: existence and uniqueness and relation to volume. Proof of existence by induction. Basic properties, computation by row operations.

Determinants and linear transformations: multiplicativity of the determinant, definition of the determinant of a linear transformation.
Invertibility and the determinant. Permutation matrices and explicit formula for the determinant deduced from properties of determinant.

Eigenvectors and eigenvalues, the characteristic polynomial. Trace. Proof that eigenspaces form a direct sum. Examples. Discussion of diagonalisation. Geometric and algebraic multiplicity of eigenvalues.

Gram-Schmidt procedure.

Spectral theorem for real symmetric matrices. Matrix realisation of bilinear maps given a basis and application to orthogonal transformation of quadrics into normal form. Statement of classification of orthogonal transformations.

Axioms for a group and for an Abelian group. Examples including geometric symmetry groups, matrix groups ($GL_{n}$, $SL_{n}$, $O_{n}$, $% SO_{n}$, $U_{n}$), cyclic groups. Products of groups.

Permutations of a finite set under composition. Cycles and cycle notation. Order. Transpositions; every permutation may be expressed as a
product of transpositions. The parity of a permutation is well-defined via determinants. Conjugacy in permutation groups.

Subgroups; examples. Intersections. The subgroup generated by a subset of a group. A subgroup of a cyclic group is cyclic. Connection with hcf and lcm. Bezout's Lemma.

Recap on equivalence relations including congruence mod n and conjugacy in a group. Proof that equivalence classes partition a set. Cosets and Lagrange's Theorem; examples. The order of an element. Fermat's Little Theorem.

Isomorphisms. Groups up to isomorphism of order 8 (stated without proof). Homomorphisms of groups. Kernels. Images. Normal subgroups. Quotient groups. First Isomorphism Theorem. Simple examples determining all homomorphisms between groups.

Group actions; examples. Definition of orbits and stabilizers. Transitivity. Orbits partition the set. Stabilizers are subgroups.

Orbit-stabilizer Theorem. Examples and applications including Cauchy's Theorem and to conjugacy classes. Orbit-counting formula.

The representation $G\rightarrow \mathrm{Sym}(S)$ associated with an action of $G$ on $S$. Cayley's Theorem. Symmetry groups of the
tetrahedron and cube.

Mathematics II

Real numbers: arithmetic, ordering, suprema, infima; real numbers as a complete ordered field. Countable sets. The rational numbers are countable. The real numbers are uncountable.

The complex number system. The Argand diagram; modulus and argument. De Moivre's Theorem, polar form, the triangle inequality. Statement of the Fundamental Theorem of Algebra. Roots of unity. De Moivre's Theorem. Simple transformations in the complex plane. Polar form, with applications.

Sequences of (real or complex) numbers. Limits of sequences of numbers; the algebra of limits. Order notation.

Subsequences; every subsequence of a convergent sequence converges to the same limit. Bounded monotone sequences converge. Bolzano--Weierstrass Theorem. Cauchy's convergence criterion. Limit point of a subset of the line or plane.

Series of (real or complex) numbers.  Convergence of series. Simple examples to include geometric progressions and power series.  Alternating series test, absolute convergence, comparison test, ratio test, integral test.

Power series, radius of convergence, important examples to include defintions of relationships between exponential, trigonometric functions and hyperbolic functions.

Continuous functions of a single real or complex variable. Definition of continuity of real valued functions of several variables.

The algebra of continuous functions. A continuous real-valued function on a closed bounded interval is bounded, achieves its bounds and is uniformly continuous.  Intermediate Value Theorem. Inverse Function Theorem for continuous strictly monotonic functions.

Sequences and series of functions.  The uniform limit of a sequence of continuous functions is continuous.  Weierstrass's M-test.
Continuity of functions defined by power series.

Definition of derivative of a function of a single real variable. The algebra of differentiable functions. Rolle's Theorem.  Mean Value Theorem. Cauchy's (Generalized) Mean Value Theorem. L'H\^{o}pital's Formula.  Taylor's expansion with remainder in Lagrange's form. Binomial theorem with arbitrary index.

Step functions and their integrals. The integral of a continuous function on a closed bounded interval. Properties of the integral including linearity and the interchange of integral and limit for a uniform limit of continuous functions on a bounded interval. The Mean Value Theorem for Integrals.  The Fundamental Theorem of Calculus; integration by parts and substitution.

Term-by-term differentiation of a (real) power series (interchanging limit and derivative for a series of functions where the derivatives converge uniformly).

Mathematics III

General linear homogeneous ODEs: integrating factor for first order linear ODEs, second solution when one solution is known for second order linear ODEs. First and second order linear ODEs with constant coefficients. General solution of linear inhomogeneous ODE as particular
solution plus solution of homogeneous equation. Simple examples of finding particular integrals by guesswork.

Partial derivatives. Second order derivatives and statement of condition for equality of mixed partial derivatives. Chain rule, change of variable, including planar polar coordinates. Solving some simple partial differential equations (e.g. $f_{xy} = 0$, $f_x = f_y$).

Parametric representation of curves, tangents. Arc length. Line integrals.

Jacobians with examples including plane polar coordinates. Some simple double integrals calculating area and also $\int_{\mathbb{R}^2} e^{-(x^2+y^2)} dA$.

Simple examples of surfaces, especially as level sets. Gradient vector; normal to surface; directional derivative; $\int^B_A \nabla \phi \cdot d\mathbf{r} = \phi(B)-\phi(A)$.

Taylor's Theorem for a function of two variables (statement only). Critical points and classification using directional derivatives and Taylor's theorem. Informal (geometrical) treatment of Lagrange multipliers.

Sample space, algebra of events, probability measure. Permutations and combinations, sampling with or without replacement. Conditional probability, partitions of the sample space, theorem of total probability, Bayes' Theorem. Independence.

Discrete random variables, probability mass functions, examples: Bernoulli, binomial, Poisson, geometric. Expectation: mean and variance. Joint distributions of several discrete random variables. Marginal and conditional distributions. Independence. Conditional expectation, theorem of total probability for expectations. Expectations of functions of more than one discrete random variable, covariance, variance of a sum of dependent discrete random variables.

Solution of first and second order linear difference equations. Random walks (finite state space only).

Probability generating functions, use in calculating expectations. Random sample, sums of independent random variables, random sums. Chebyshev's inequality, Weak Law of Large Numbers.

Continuous random variables, cumulative distribution functions, probability density functions, examples: uniform, exponential, gamma, normal. Expectation: mean and variance. Functions of a single continuous random variable. Joint probability density functions of several continuous random variables (rectangular regions only). Marginal distributions. Independence. Expectations of functions of jointly continuous random variables, covariance, variance of a sum of dependent jointly continuous random variables.

Random samples, concept of a statistic and its distribution, sample mean as a measure of location and sample variance as a measure of
spread.

Concept of likelihood; examples of likelihood for simple distributions. Estimation for a single unknown parameter by maximising
likelihood. Examples drawn from: Bernoulli, binomial, geometric, Poisson, exponential (parametrized by mean), normal (mean only, variance known). Data to include simple surveys, opinion polls, archaeological studies, etc. Properties of estimators---unbiasedness, Mean Squared Error = (bias$^{2}$ + variance). Statement of Central Limit Theorem (excluding proof). Confidence intervals using CLT. Simple straight line fit, $Y_{t}=a+bx_{t}+\varepsilon _{t}$, with $\varepsilon _{t}$ normal independent errors of zero mean and common known variance. Estimators for $a$, $b$ by maximising likelihood using partial differentiation, unbiasedness and calculation of variance as linear sums of $Y_{t}$. (No confidence intervals). Examples (use scatter plots to show suitability of linear regression).

Linear regression with 2 regressors. Special case of quadratic regression $Y_t = a + bx_t + cx^2_t + \epsilon_t$. Model diagnostics and outlier detection. Residual plots. Heteroscedasticity. Outliers and studentized residuals. High-leverage points and leverage statistics.

Introduction to unsupervised learning with real world examples. Principal components analysis (PCA). Proof that PCs maximize directions of maximum variance and are orthogonal using Lagrange multipliers. PCA as eigendecomposition of covariance matrix. Eigenvalues as variances. Choosing number of PCs. The multivariate normal distribution pdf. Examples of PCA on multivariate normal data and clustered data. Clustering techniques; K-means clustering. Minimization of within-cluster variance. K-means algorithm and proof that it will decrease objective function. Local versus global optima and use of random initializations. Hierarchical clustering techniques. Agglomerative clustering using complete, average and single linkage.

Mathematics IV

Euclidean geometry in two and three dimensions approached by vectors and coordinates. Vector addition and scalar multiplication. The scalar product, equations of planes, lines and circles. Conics (normal form only), focus and directrix.

The vector product in three dimensions. Use of $\mathbf{a}, \mathbf{b}, \mathbf{a} \land \mathbf{b}$ as a basis. $\mathbf{r} \land \mathbf{a} = \mathbf{b}$ represents a line. Scalar triple products and vector triple products, vector algebra.

Orthogonal matrices. $2\times 2$ orthogonal matrices and the maps they represent. Orthonormal bases in $\mathbb{R}^3$. Orthogonal change of variable; $A\mathbf{u} \cdot A\mathbf{v} = \mathbf{u \cdot v}$ and $A\mathbf{u} \land A\mathbf{v} = \pm \mathbf{u} \land \mathbf{v}$. Showing the locus $Ax^2 + Bxy + Cy^2 = 1$ can be put in normal form via a rotation matrix; statement that a
real symmetric matrix can be orthogonally diagonalized. Simple examples identifying some conics and quadrics not in normal form.

$3 \times 3$ orthogonal matrices; $SO(3)$ and rotations; conditions for being a reflection. Isometries of $\mathbb{R}^3$.

Rotating frames in $2$ and $3$ dimensions. Angular velocity. $\mathbf{v} = \omega \land \mathbf{r}$.

Parametrised surfaces, including spheres, cones. Examples of coordinate systems including parabolic, spherical and cylindrical polars. Calculating normal as $\mathbf{r}_u \land \mathbf{r}_v$. Surface area.

Newton's laws, inertial frames, Galilean relativity. Dimensional analysis.

Forces, examples including gravity, electromagnetism, fluid drag. Conservative forces and the Newtonian gravitational potential. Energy and momentum.

Equilibria and the harmonic oscillator. Stability and instability via linearized equations, normal modes. Simple examples of equilibria in two variables via matrices.

Central forces, angular momentum and torque. Planar motion in polar coordinates, the effective potential, Kepler's laws and planetary motion.

Many particle systems, centre of mass motion. Rigid bodies, rotating frames and Coriolis force, inertia tensor, rigid body motion.

The Division Algorithm on Integers, Euclid's Algorithm including proof of termination with highest common factor. The solution of linear Diophantine equations.

Division and Euclid's algorithm for real polynomials. Examples.

Root finding for real polynomials. Fixed point iterations, examples. Convergence. Existence of fixed points and convergence of fixed point iterations by the contraction mapping theorem (using the mean value theorem).

Newton iteration. Quadratic convergence. Horner's Rule.

Mathematics V

Multiple integrals: Two dimensions. Informal definition and evaluation by repeated integration; example over a rectangle; properties. General domains. Change of variables. Examples.

Volume integrals: Jacobians for cylindrical and spherical polars, examples.

Recap on surface integrals. Flux integrals.

Scalar and vector fields. Vector differential operators: divergence and curl; physical interpretation. Calculation. Identities.

Divergence theorem. Example. Consequences: Greens 1st and second theorems. $\int_V \nabla \phi dV = \int_{\delta V} \phi dS$. Uniqueness of solutions of Poisson's equation. Derivation of heat equation. Divergence theorem in plane. Informal proof for plane.

Stokes's theorem. Examples. Consequences. The existence of potential for a conservative force.

Gauss' Flux Theorem. Examples. Equivalence with Poisson's equation.

Fourier series: Periodic, odd and even functions. Calculation of sine and cosine series. Simple applications concentrating on imparting familiarity with the calculation of Fourier coefficients and the use of Fourier series. The issue of convergence is discussed informally with examples. The link between convergence and smoothness is mentioned, together with its consequences for approximation purposes.

Partial differential equations: Introduction in descriptive mode on partial differential equations and how they arise. Derivation of (i) the wave equation of a string, (ii) the heat equation in one dimension (box argument only). Examples of solutions and their interpretation. D'Alembert's solution of the wave equation and applications. Characteristic diagrams (excluding reflection and transmission). Uniqueness of solutions of wave and heat equations.

PDEs with Boundary conditions. Solution by separation of variables. Use of Fourier series to solve the wave equation, Laplace's equation and the heat equation (all with two independent variables). (Laplace's equation in Cartesian and in plane polar coordinates). Applications.