# Prelims Mathematics 2019-20

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**Synopses**

The synopses give some additional detail and show how the material is split between the different lecture courses. They include details of recommended reading.

**Practical Work**

The requirement in the *Examination Regulations * to pursue an adequate course of practical work will be satisfied by following the Computational Mathematics course and submitting two Computational Mathematics projects. Details about submission of these projects will be given in the Computational Mathematics handbook.

The syllabus here is that referred to in the *Examination Regulations 2019* Special Regulations for the Preliminary Examination in Mathematics (https://www.admin.ox.ac.uk/examregs/) and has been approved by the Mathematics Teaching Committee for examination in Trinity Term 2020.

Examination Conventions can be found at: http://www.maths.ox.ac.uk/members/students/undergraduate-courses/examina...

**A**

The subject of the examination shall be mathematics and its applications. The syllabus and number of papers shall be prescribed by regulation from time to time by the Mathematical, Physical and Life Sciences Board.

**B**

1. Candidates shall take five written papers. The titles of the papers shall be:

Mathematics I,

Mathematics II,

Mathematics III,

Mathematics IV,

Mathematics V.

2. In addition to the five papers in cl 1, a candidate must also offer a practical work assessment.

3. Candidates shall be deemed to have passed the examination if they have satisfied the Moderators in all five papers and the practical assessment at a single examination or passed all five papers and the practical assessment in accordance with the proviso of cl 4.

4. A candidate who fails to satisfy the Moderators in one or two of papers I-V may offer those papers on one subsequent occasion; a candidate who fails to satisfy the Moderators in three or more of papers I-V may offer all five papers on one subsequent occasion; a candidate who fails to satisfy the Moderators in the practical work assessment may also offer the assessment on one subsequent occasion.

5. The Moderators may award a distinction to candidates of special merit who have passed all five written papers and the practical work assessment at a single examination.

6. The syllabus for each paper shall be published by the Mathematical Institute on the Mathematical Institute's website by the beginning of the Michaelmas Full Term in the academic year of the examination, after consultation with the Mathematics Teaching Committee. Each paper will contain questions of a straight forward character.

7. The Chair of Mathematics, or a deputy, shall make available to the Moderators evidence showing the extent to which each candidate has pursued an adequate course of practical work. In assessing a candidate's performance in the examination the Moderators shall take this evidence into account. Deadlines for handing in practical work will be published in a handbook for candidates by the beginning of Michaelmas Full Term in the academic year of the examination.

Candidates are usually required to submit such practical work electronically; details shall be given in the handbook for the practical course. Any candidate who is unable for some reason to submit work electronically must apply to the Academic Administrator, Mathematical Institute, for permission to submit the work in paper form. Such applications must reach the Academic Administrator two weeks before the deadline for submitting the practical work.

8. The use of hand held pocket calculators is generally not permitted but certain kinds may be permitted for some papers. Specifications of which papers and which types of calculator are permitted for those exceptional papers will be announced by the Moderators in the Hilary Term preceding the examination.

**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. Matrices and the beginnings of matrix algebra. Use of matrices to describe systems of linear equations.

Elementary Row Operations (EROs) on matrices. Reduction of matrices to echelon form. Application to the solution of systems of linear equations.

Inverse of a square matrix. Reduced row echelon (RRE) form and the use of EROs to compute inverses; computational efficiency of the method. Transpose of a matrix; orthogonal matrices.

Vector spaces: definition of a vector space over a field (such as $\mathbb{R}$, $\mathbb{Q}$, $\mathbb{C}$). Subspaces. Many explicit examples of vector spaces and subspaces.

Span of a set of vectors. Examples such as row space and column space of a matrix. Linear dependence and independence. Bases of vector spaces; examples. The Steinitz Exchange Lemma; dimension. Application to matrices: row space and column space, row rank and column rank. Coordinates associated with a basis of a vector space.

Use of EROs to find bases of subspaces. Sums and intersections of subspaces; the dimension formula. Direct sums of subspaces.

Linear transformations: definition and examples (including projections associated with direct-sum decompositions). Some algebra of linear transformations; inverses. Kernel and image, Rank-Nullity Theorem. Applications including algebraic characterisation of projections (as idempotent linear transformations).

Matrix of a linear transformation with respect to bases. Change of Bases Theorem. Applications including proof that row rank and column rank of a matrix are equal.

Bilinear forms; real inner product spaces; examples. Mention of complex inner product spaces. Cauchy--Schwarz inequality. Distance and angle. The importance of 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, examples. Groups of order 8 or less up to isomorphism (stated without proof). Homomorphisms of groups with motivating examples. Kernels. Images. Normal subgroups. Quotient groups; examples. 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. Examples.

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. 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 definitions 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'Hôpital'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.