The University of Southampton
Courses

# MATH2015 Mathematical Methods for Scientists

## Module Overview

Students will be introduced to a number of practical methods for solving linear matrix equations and linear ordinary and partial differential equations and for approximating functions with Fourier series.

### Aims and Objectives

#### Module Aims

To introduce a number of practical methods for solving linear matrix equations and linear ordinary and partial differential equations and for approximating functions with Fourier series. As well as taking a pragmatic approach to the solution of such problems, the module highlights the universality of the underlying concept of linearity; that is if f and g are two solutions to a given linear problem, then the linear combination a*f+b*g is also a solution.

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Perform matrix mathematics, including computing inverses, determinants, eigenvalues and eigenvectors
• Compute the Fourier series expansion of a given function
• Solve a range of first and second order ordinary differential equations, including both initial and boundary value problems, using exactness, integrating factors and methods of reduction of order and variation of parameters
• Solve a range of first and second order partial differential equations using methods of characteristic curves and separation of variables
• Solve some of the fundamental equations of mathematical physics using methods of separation of variables and Fourier series.

### Syllabus

Matrices: Properties of matrices, determinants, and inverses. Linear independence and orthogonality of vectors. Matrices and systems of ordinary differential equations. Solution of both homogeneous and inhomogeneous linear systems by Gauss elimination to Hermite form. Calculating eigenvalues and eigenvectors of a matrix. Matrix diagonalization. Fourier series: Full and half-range and complex Fourier series expansions. Orthogonality/orthonormality relations. Convergence and Gibbs phenomenon. Orthogonal functions. Introduction to Fourier transforms. Ordinary differential equations: Initial and boundary value problems. Exact integration. Integrating factors. Methods of reduction of order and variation of parameters. Eigenfunction expansion. Partial differential equations: Method of characteristic curves. Method of separation of variables solution of wave equation, diffusion equation and Laplace equation.

### Learning and Teaching

TypeHours
Teaching48
Independent Study102
Total study time150

Riley, K.F. Mathematical Methods for Physics and Engineering.

McQuarrie, D.A. Mathematical Methods for Scientists and Engineers.

Jordan, D.W and Smith, P. Mathematical Techniques.

Arfken, G.B. Mathematical Methods for Physicists.

### Assessment

#### Summative

MethodPercentage contribution
Closed book Examination  (120 minutes) 80%
In-class Test  (40 minutes) 20%

#### Referral

MethodPercentage contribution
Exam  (120 minutes) %

Prerequisites: MATH1006 (or MATH1008) and MATH1007 (or MATH1009 or MEDI2013 or MEDI2013)

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Course texts are provided by the library, and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.