Relations and functions between sets: domain, co-domain, range, composition and inverse, equivalence relations.

Complex numbers: algebraic manipulation, geometric viewpoint, DeMoivre’s theorem and simple applications.

Mathematical Induction: applications including summation of series, divisibility properties, recurrence relations, infinite sequences and series, introduction to limits and convergence.

Real functions: including polynomials (Remainder Theorem, Rational Root Theorem), other elementary functions, rational functions and more exotic examples.

Curve sketching and connection with transformations of the cartesian plane.

Conic sections: classification via translation and rotation of coordinate axes.

Vectors: scalar product of vectors, its properties and applications; vector product and its properties, applications; triple products.

Matrix algebra, affine transformations; eigenvalues and eigenvectors.

Groups: definitions and examples, including symmetry, permutation and transformation groups; subgroups, Lagrange’s theorem.

The completeness of the real number system; limits of sequences and functions.

The Riemann integral as the limit of a summation; the fundamental theorem of calculus.

Convergence and divergence of infinite series; tests for convergence; power series, radius of convergence.

Techniques and application of differentiation and integration, including extrema problems, rates of change, areas, volumes and arc lengths.

Solving systems of linear equations, matrix inversion, determinants, Cramer’s rule.

Systems of linear equations, linear dependence, solution set; matrix inversion, determinants.

Real vector spaces, axioms and examples; subspaces; linear independence, basis, dimension; coordinates, change of basis.

Linear transformations, definitions, examples, kernel and image space; rank and nullity; composites and inverses; matrix representation, diagonalisation, eigenvectors.

Inner-product spaces, axioms, properties including Cauchy-Schwarz inequality; length, distance, angle; orthonormal bases, the Gram-Schmidt process.

Continuity of functions.

Differentiability of functions.

Partial derivatives, directional derivatives; maxima and minima, Lagrange multipliers.

Grad f, generalised chain rule.

Limits and continuity of complex functions; differentiability, Cauchy-Riemann equations, analytic functions; harmonic functions, and harmonic conjugates.

Taylor and Laurent series.

Complex integration, Cauchy’s integral formulae; poles and other singularities, residues; Cauchy’s residue theorem.