Preliminary calculus (8 hours, including problems classes)

Differentiation [Limits - continuity, Differentiation from first principles; product, chain, implicit and inverse, logarithmic differentiation; Higher derivatives - Leibnitz' theorem; radius of curvature]

Integration [Integration from first principles; the inverse of differentiation; infinite and improper integrals; parts, tan-half-angle, reduction formulae, integral inequalities, improper integrals;]

Probability and distributions (6 hours, including problems classes)

Definition of probability, Permutations and combinations

Discrete distributions [mean and variance; expectation values; Binomial and Poisson distributions]

Continuous distributions [expectation values, Gaussian distribution including as an approximation for Binomial and Poisson; simple applications, e.g. velocity distributions.]

Central limit theorem

Complex numbers and hyperbolic functions (5 hours, including problems classes)

The need for complex numbers Manipulation of complex numbers [Addition and subtraction; modulus and argument; multiplication; complex conjugate; division]

Polar representation of complex numbers [Multiplication and division in polar form] de Moivre's theorem [Trigonometric identities; finding the nth roots of unity; solving polynomial equations]

Complex logarithms and complex powers Applications to differentiation and integration

Hyperbolic functions [Definitions; hyperbolic-trigonometric analogies; identities of hyperbolic functions; solving hyperbolic equations; inverses of hyperbolic functions; calculus of hyperbolic functions]

Vector algebra (4 hours, including problems classes)

Multiplication of vectors [vector product; scalar triple product; vector triple product]

Equations of lines, planes and spheres

Using vectors to find distances [Point to line; point to plane; line to line; line to plane; plane polar coordinates]

Reciprocal vectors

Series and limits (4 hours, including problems classes)

Series, Operations with series

Power series [Convergence of power series; operations with power series]

Partial differentiation (6 hours, including problems classes)

Definition of the partial derivative, The total differential and total derivative

Exact and inexact differentials, The chain rule, Taylor's theorem for many-variable functions, Stationary values of many-variable functions, Least squares fits

Multiple integrals (7 hours, including problems classes)

Double integrals, Triple integrals

Applications of multiple integrals

[Areas and volumes; masses, centres of mass and centroids; Pappus' theorems; moments of inertia; mean values of functions]

Change of variables in multiple integrals

[Change of variables in double integrals; change of variables in triple integrals]

First-order ordinary differential equations (7 hours, including problems classes)

General form of solution

First-degree first-order equations

[Separable-variable equations; exact equations; inexact equations, integrating factors; linear equations; homogeneous equations; Bernoulli's equation; miscellaneous equations]

Higher-degree first-order equations

[Equations soluble for p; for x; for y;]

Higher-order ordinary differential equations (4 hours, including problems classes)

Linear equations with constant coefficients

[Finding the complementary function yc(x); finding the particular integral yp(x); constructing the general solution yc(x) + yp(x); linear recurrence relations;

Matrices and determinants (5 hours, including problems classes)

Linear operators, Matrices, Basic matrix algebra[Matrix addition; multiplication by a scalar; matrix multiplication]

Functions of matrices, The transpose of a matrix

The complex and Hermitian conjugates of a matrix

The trace of a matrix

The determinant of a matrix [Properties of determinants]

The inverse of a matrix

The rank of a matrix

Special types of square matrix [Diagonal; triangular; symmetric and antisymmetric; orthogonal; Hermitian and anti-Hermitian; unitary; normal]

Linear Algebra (5 hours, including problems classes)

Eigenvectors and eigenvalues

[Of a normal matrix; of Hermitian and anti-Hermitian matrices; of a unitary matrix; of a general square matrix]

Determination of eigenvalues and eigenvectors

[Degenerate eigenvalues]

Typical oscillatory systems

Change of basis and similarity transformations

Diagonalization of matrices

Quadratic and Hermitian forms

[Stationary properties of the eigenvectors; quadratic surfaces]

Simultaneous linear equations

[Range; null space; N simultaneous linear equations in N unknowns; singular value decomposition

Modelling (5 hours, including problems classes)

Mathematisation of systems described by text. Test elements and applying physical laws. e.g. conservation of number, Newton's third law, Rates of change.