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 is a minimum prequisite:

• Geometry
• Linear algebra
• Real analysis
• Calculus of several variables
• Partial differential equations
• Complex analysis
• References

Geometry

• Basic vector manipulation. Ideas of position vector, velocity, acceleration. Scalar and vector products.
• Parametrisation of a curve; tangent vector, arc length.
• Parametrisation of a surface; normal to a surface.

Background reading and exercises
Jordan & Smith chapters 9-11; Kreyszig sections 8.1-8.6.

Linear algebra

• Systems of linear equations and their interpretation via matrices.
• Elementary row operations, Gauss and Gauss-Jordan elimination, linear dependence and independence.
• Matrix multiplication, transpose, determinant, trace, inverse. Identities such as (AB)^T=B^TA^T.
• 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.

Background reading and exercises
Jordan & Smith chapters 7, 8, 12, 13; Kreyszig chapters 6, 7.

Real analysis

• Concepts and practice of differentiation and integration. Taylor'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.
• Elements of phase plane analysis: critical points and their classification.
• Standard sequences and series.
• Fourier series and eigenfunction expansions.

Background reading and exercises
Jordan & Smith chapters 1-5, 14-19, 23, 26; Kreyszig sections 1.1-1.6, 2.1-2.3, 2.7-2.10, chapters 3, 4, sections 10.1-10.4, A3.3.

Calculus of several variables

• Concept and practice of partial differentiation.
• Change of coordinates, chain rule.
• div, grad and curl. Simple manipulation rules: ∇·(φu) = ∇φ·u + φ∇·u , and so forth.
• Line, surface and volume integrals. Change of variables, Jacobian.
• Classification of stationary points: local minima, maxima and saddle points. Lagrange multipliers.
• Divergence theorem and Stokes' theorem.

Background reading and exercises
Jordan & Smith chapters 28-34; Kreyszig sections A3.2, 8.8-8.11, chapter 9.

Partial differential equations

• Basic treatment of Laplace, heat and wave equations. Their solution via separation of variables.
• Use of Fourier series, Fourier transform and Laplace transform to solve linear constant-coefficient ODEs and PDEs.

Background reading and exercises
Kreyszig chapter 11.

Complex analysis

• Basic manipulation of complex numbers and complex variables.
• Properties of complex functions: zⁿ, e^z, log z, z^α
• Analytic functions and power series. Convergence, divergence and Cauchy sequences. Cauchy-Riemann equations.
• Analysis and classification of isolated singularities and branch points.
• Contour integration and residue calculus.
• Conformal mapping.

Background reading and exercises
Jordan & Smith chapter 6; Kreyszig chapters 12-15; Priestley.

References

1. D. W. JORDAN & P. SMITH, Mathematical Techniques, 3rd Edition (2002). Oxford University Press.
2. ERWIN KREYSZIG, Advanced Engineering Mathematics, 8th Edition (1999). Wiley.
3. H. A. PRIESTLEY, Introduction to Complex Analysis, revised edition (1990). Oxford University Press.