## Module and Programme Catalogue

### 10 creditsClass Size: 50

Module manager: Dr Alex Rap
Email: a.rap@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2022/23

### Pre-requisites

Module replaces

SOEE2040 Mathematics for Geophysical and Environmental Sciences 3

This module is not approved as a discovery module

### Module summary

The topics covered in this module are essential mathematical tools for treating many physical phenomena. Matrices provide a powerful tool for storing, displaying and manipulating information about linear systems of algebraic and differential equations. They are, for example, used extensively in the analysis of vibrating systems such as those encountered in seismology.The operations of differentiating and integrating scalar and vector fields arise naturally in areas of geophysics such as fluid flow and heat transfer.

### Objectives

To provide the students with sufficient Mathematical background for understanding their studies in Geophysical and Atmospheric Sciences.

Learning outcomes
On completion of this module, students should be able to:
a) carry out basic manipulations involving determinants and matrices;
b) find eigenvalues and eigenvectors of given matrices;
c) calculate the gradient of a scalar field and the divergence and curl of a vector field;
d) evaluate line, surface and volume integrals using Cartesian and polar co-ordinates.

### Syllabus

- Determinants and Matrices: Determinants and solution of linear equations.
- Basic matrix algebra.
- Transpose and inverse of a matrix.
- Symmetric, orthogonal and Hermitian matrices.
- Eigenvalues and eigenvectors: rotation of co-ordinate axes.
- Diagonalisation of real symmetric matrices; quadratic forms.
- Vector Calculus: Gradient, divergence and curl.
- Second order derivatives; the Laplacian; vector identities.
- Expressions in spherical polar co-ordinates.
- Line, surface and volume integrals involving vector fields.
- Flux and the divergence theorem; Circulation and Stokes' theorem.
- Laplace's equation, diffusion equation.
- Solution by separation of variables.

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 20 1.00 20.00 Tutorial 10 1.00 10.00 Private study hours 70.00 Total Contact hours 30.00 Total hours (100hr per 10 credits) 100.00

### Private study

Completing of assignments for workshops and assessments.
Studying and revising of course material.

### Opportunities for Formative Feedback

Regular problem solving assignments

### Methods of assessment

Coursework
 Assessment type Notes % of formal assessment Assignment Problem Sheet - One Assignment 15.00 Total percentage (Assessment Coursework) 15.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) (S1) 2 hr 00 mins 85.00 Total percentage (Assessment Exams) 85.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated