1.  Introduction:  This Autumn Semester unit is designed for students of geophysical science who have already passed the first year applied mathematics pre-requisites Mathematical Methods I and Mathematical Methods II. This unit is the pre-requisite for Mathematics for Geophysical Science II, studied in the Spring. Mathematics for Geophysical Science I is taught by lectures and seminars, and computer laboratory sessions. The unit gives students some of the essential methods of applied mathematics. This unit covers matrix algebra for the solution of systems of linear equations and Markov chain modelling of competing populations. A variety of numerical methods are introduced and practiced in computer sessions using the algebra package Maple. Programming skills required to implement the methods in Maple are taught. The calculus of vector fields is also treated. Students are also introduced to the theory of complex functions.

2.  Timetable Hours, Credits, Assessments:  This is a 20 UCU unit of 28 lectures with 6 seminars and 5 computer lab sessions. There are 4 modules: matrix algebra and numerical methods (12 lectures + 3 seminars), complex variable (4 lectures and 1 seminar), and vector calculus (8 lectures + 2 seminars), Programming in Maple (4 lectures + 5 labs). Assessment is by 40% coursework and 60% examination.

3.  Overview:  Among other tools matrix algebra is essential for solving real physical problem numerically. For example, changes in the competing animal populations can be described by a set of recurrence relations, which may be naturally expressed as a matrix system. Mathematics allows us to predict the long-term fate of the different species. Gauss elimination provides an elegant tool for solving large matrix systems of linear equations. In practice, however, more devious techniques may be required in order to obtain accurate solutions on a computer. We will discuss a variety of computational numerical methods available to solve such problems. We will study methods to integrate differential equations on a computer. We will use them to solve the Lorenz equations, which have been applied in weather prediction, and plot chaotic trajectories in space. The numerical methods will be implemented using Maple in the computer lab sessions.

The theory of functions of a complex variable is useful for evaluating difficult integrals and has many applications in applied mathematics, e.g. aerofoil theory and electromagnetism.

The vector calculus module is essential for modelling quantities or processes which vary in two or three dimensions. The vector operators div, grad and curl are defined, and various vector identities are explored. The divergence theorem and Stokes' theorem are introduced. These theorems are particularly useful in the study of geophysical fluid dynamics, dynamical meteorology and solid mechanics.

A structured approach to Maple programming is introduced and used to implement the numerical methods discussed in the Matrix Algebra lectures. This includes the use of functions and subroutines. The basic language elements are introduced, loops, conditional statements, arrays and input/output.

E Kreyszig "Advanced Engineering Mathematics" (Wiley)

W. Cheney & D. Kincaid "Numerical Mathematics and Computing"

Introduction to matrices. Simple application (predator/prey population model). Systems of linear equations (introduced via two equations in two unknowns case). Solving systems of linear equations using elementary operations. Matrices and systems of linear equations. Gaussian elimination, row echelon form and back-substitution. Reduced row echelon form; use of rank to distinguish cases of no solution, unique solution, infinitely many solutions. (4 lectures)

Matrix inversion. Determinants and how to calculate them by expansion. Determinants and solutions to systems of linear equations (including homogeneous systems). (2 lectures)

Linear independence, calculation of eigenvalues and eigenvectors. Diagonalisation of matrices with linearly independent eigenvectors. Application of diagonalisation to population model. Long-term fate of populations. (2 lectures)

Numerical methods for solving matrix systems. The Thomas algorithm for tridiagonal systems. Iterative methods including Jacobi iteration and Gauss-Seidel iteration. Convergence criteria. Numerical integration of ODEs. (4 lectures)

Notion of a complex number. Argand diagrams. Analytic functions and the Cauchy-Riemann equations. (4 lectures)

Scalar fields, gradient, directional derivative, level surfaces. (2 lectures)

Vector fields, divergence, Laplacian, curl, divergence theorem, Stokes's theorem, solenoidal vector fields, irrotational vector fields. (6 lectures)

Introduction to Maple. Defining and manipulating variables. Arrays. Data input and graphics output. Functions. If/Then statements and Do loops. (4 lectures)