# Course Pre-requisites

### Engineering/Physics Mathematics

To cope with the course material on the CDT training programme, you will need to have covered at least a 1st year undergraduate engineering/physics mathematics syllabus covering the material listed below. While some of this will be recapped - very briefly - during the course, it is often too fast to learn from first sight. Make sure you have completed one of the courses mentioned below if you are not confident in the material. (The number of lectures in brackets gives an approximate indication of the amount of time which you should have spent covering this at undergraduate level.)

Typical course work with which you should be confident:

**Vectors**(4 lectures)

Vectors. Scalars vs vectors. Directed line segments. Direction cosines. Adding, subtracting and scalar multiplication. Magnitude of a vector. Components: i,j, and k. Scalar product and projection. Concept of work done. Cross product and applications. Scalar and vector triple products, evaluation as a determinant and applications. Vector equations of lines, planes and spheres. Differentiation of a vector with respect to a single argument.**Complex Numbers**(4 lectures)

Complex Numbers. Motivation and notation. Real and imaginary parts. Argand diagram, polar form, modulus and argument. Complex conjugate. Solving quadratics. Addition, subtraction, multiplication and division of complex numbers. Euler formula, exponential form, logarithms. Hyperbolic and trig functions with complex arguments. De Moivre's formula with applications.**Matrix Algebra**(12 lectures)

Definition of a matrix. Column and row vectors, scalar product. Matrix addition and multiplication. Determinant of a matrix. Row operations for solving linear equations. Rank, linear dependence. Inverse matrix. Eigenvalues and eigenvectors.**Sequences, Series and Limits**(6 lectures)

Definition of a sequence, limit of a sequence. Definition of a series, convergence of series. Definition of a power series, convergence radius of a power series. Taylor series expansion. Functions of a real variable, inverse functions, standard functions. Continuity of a function.**Differentiation and Integration**(6 lectures)

Rate of change and interpretation of derivatives. Definition of differentiability. Techniques of differentiation. Taylor series, L'Hopital's rule. Interpretation of integral as accumulation over time, distance. Definite integral: relation between integral and derivative. Techniques of integration: substitution, integration by parts, partial fractions. Improper integrals.**Partial Differentiation**(6 lectures)

Definition; graph, contours. Partial derivatives: geometrical interpretation, directional derivative, gradient. Differentials - use for error analysis. Implicit differentiation; chain rule; differentiation of functions defined by integrals. Extremum problems in more than one variable, second partial derivatives.**Differential Equations**(9 lectures)

Classification of differential equations. Initial and boundary conditions. Solution of first-order equations: separable equations, integrating factor. Differential operators and linearity. Solution of linear constant-coefficient ODEs: characteristic equation, complementary function, particular integral. Sets of linear constant-coefficient equations. Discretising equations.**Probability**(10 lectures)

Data analysis; sample spaces and events; axioms and definitions; total and conditional probability; Bayes rule; independence. Random variables: continuous and discrete; distribution and density functions; mean and variance - definitions and properties. Special distributions: uniform; binomial; Poisson; Normal - definitions, properties and examples; Correlation: correlation coefficient, cross correlation function, autocorrelation, goodness of fit & R2; Statistics: t-test, Jarque.Bera test, rank sum test, chi squared test.**Fourier Analysis**(10 lectures)

Fourier series of a periodic function; A convergence theorem; .Derivatives, integrals, and uniform convergence; Fourier series on intervals; The Gibbs phenomenon; Orthogonal sets of functions; Inner products; Convergence and completeness; L2 space; Regular Sturm-Liouville problems; Boundary value problems; 1D heat flow and wave motion; The Dirichlet problem; Multiple Fourier series; The Fourier transform; Convolution; Applications of Fourier transforms; The Fourier transform of several variables; Autocorrelation function and power spectral density estimation.

If, after reading through that list you are not completely confident that you are ready, then please look at the following course from MIT's OpenCourseWare: Computational Science and Engineering I

It contains video lectures, course notes and presentations, and covers much useful material.

Also the following Engineering Mathematics course at Bristol university includes much of the above, (except Fourier).

### Computer Programming

You will also be expected to complete a significant amount of programming on the course. Although little programming experience is really needed, you should familiarise yourself with the computational environment & language (Matlab) by attempting the first two tutorials (or more) from here: http://www.mathworks.com/products/matlab/demos.html. If you are familiar with another language, this will mean this should be trivial for you, although you may want to watch out for transitions from procedural languages to Matlab's vectorised approach.

Most importantly, you will be expected to employ mathematical analysis in this computational environment. You will therefore be asked to complete pre-course material before arriving to begin the course.