Prerequisites

Whatever your background, we expect that you will be familiar with all the material listed below. We also expect that you will have experience in more advanced areas such as differential equations, fluid mechanics, numerical analysis, statistics, etc. The following gives details of the minimum prerequisite material.

Linear algebra

Definitions of a subspace, basis, dimension, span, linear dependence and independence.
Definitions of inner product spaces and norms; the Cauchy Schwarz inequality.
Matrix multiplication, transpose, determinant, trace, inverse. Identities such as (AB)^T=B^TA^T.
Systems of linear equations and their interpretation via matrices.
Elementary row operations, Gaussian elimination, and LU decomposition for the solution of linear systems.
Definition and concepts of eigenvalues and eigenvectors. Finding them by hand for up to 3x3 matrices.
Rotation of coordinates, orthogonal matrices.
Diagonalisation. Possibility of non-diagonalisable matrices.
Properties of real symmetric matrices.
Properties of singular matrices.

Background reading
Jordan & Smith chapters 7, 8, 12, 13; Kreyszig chapters 7, 8.

Real analysis

Concepts and practice of differentiation and integration.
Taylor's theorem.
Limits and L'Hôpital's rule.
The Mean Value Theorem and the Intermediate Value Theorem.

Background reading
Collins chapter 0; Jordan & Smith chapters 1-5, 14-17.

Calculus of several variables

Concept and practice of partial differentiation.
Change of coordinates, chain rule.
Line, surface and volume integrals. Change of variables, Jacobian.
div, grad, curl and the Laplacian in Cartesian and other coordinate systems.
Simple manipulation rules: $\nabla \cdot (\phi$ u) = $\nabla \phi \cdot$ u + $\phi \nabla \cdot$ u , and so forth.
Classification of stationary points: local minima, maxima and saddle points. Lagrange multipliers.
Divergence theorem and Stokes' theorem.

Background reading
Jordan & Smith chapters 28-34; Kreyszig chapters 9, 10.

Complex analysis

Basic manipulation of complex numbers and complex variables.
Properties of complex functions: zⁿ, e^z, log z, z^$\alpha$, Euler's formula.
Analytic functions and power series. Convergence, divergence and Cauchy sequences. Holomorphic functions. Cauchy-Riemann equations.
Analysis and classification of isolated singularities and branch points.
Contour integration, Cauchy's theorem, deformation of a contour.

Background reading
Kreyszig chapters 13-16; Priestley.

Ordinary differential equations

Picard's theorem.
Solution of simple ODEs: first-order separable, integrating factors, linear constant-coefficient ODEs, complementary function and particular integral. Stürm-Liouville problem for second-order linear ODE.
Solution of second order ODEs via Green's functions. Delta functions.
Elements of phase plane analysis: critical points and their classification.
Fourier series and eigenfunction expansions.

Background reading
Collins chapters 2, 3, 4, 15; Jordan & Smith chapters 18, 19, 23; Kreyszig chapters 1-4.

Partial differential equations

Solution of first order quasilinear PDEs using the method of characteristics.
Classification of second order semi-linear PDEs. Canonical form.
Basic treatment of Laplace, heat and wave equations. Their solution via separation of variables.
D'Alembert's solution of the wave equation.

Background reading
Collins chapters 5-7; Kreyszig chapter 12.

References

P. J. Collins, Differential and Integral Equations (2006). Oxford University Press.
D. W. Jordan and P. Smith, Mathematical Techniques, 4th Edition (2008) (or 3rd Edition (2002)). Oxford University Press.
E. Kreyszig, Advanced Engineering Mathematics, 10th Edition (2011). Wiley.
H. A. Priestley, Introduction to Complex Analysis, revised edition (1990). Oxford University Press.