## Overview

To give students a solid grounding in mathematical methods and ideas that they will use later in their degree course.

#### Module Details

**Title:** Maths for Electrncs&Electrical**Code:** MATH2021

**Credits:** ECTS credits**Taught in:**

## Aims and Objectives

Mathematical methods of Laplace transform theory, elementary Fourier series, eigenvalues and eigenfunctions, complex variable theory and vector calculus.

## Syllabus

Laplace Transform Theory
The (one-sided) Laplace transform and its existence. Use of Laplace transforms in solving simple ODEs with constant coefficients and given boundary conditions. Step functions and their transforms. Laplace transforms of standard functions. Uniqueness of the inverse. Elementary properties - linearity, first and second shifting theorems, change of scale. Transforms of derivatives and integrals and of products with powers of t. Transforms of periodic functions.The limit of F(s) as s->infinity. The initial and final value theorems and their uses. Laplace transforms of some further special functions - the saw-tooth function, the dirac delta function. Theorems relating to inversion. The solution of slightly more complicated ordinary differential equations with given initial or boundary conditions - constant coefficient equations, simultaneous equations, some equations with non-constant coefficients, equations with discontinuous forcing terms. (About 8 lectures)
Fourier series:
Definition of Fourier series. Calculation of coefficients in easy cases. Examples of whole and half range series over various ranges. Elementary properties. (About 5 lectures)
Eigenvalues, eigenvectors and eigenfunctions:
Eigenvalues and eigenvectors of matrices. Simple harmonic equation. Eigenvalues and eigenfunctions of the simple-harmonic equation with various boundary conditions. Applications of eigenvalues, eigenvectors and eigenfunctions. (About 5 lectures)
Complex Variable Theory
Revision of complex numbers including the polar form, de Moivre's theorem, simple complex functions, loci in the argand diagram, differentiability and the Cauchy-Riemann equations. (About 4 lectures)
Vector Calculus
A survey of div, grad and curl and associated theorems - geometric interpretation.Divergence theorem and Stokes' theorem. The Laplacian in polar co-ordinates. (About 10 lectures)

## Learning and Teaching

- Lecture - 36 hours per semester
- Tutorial - 12 hours per semester

## Assessment

- 20% - Coursework assignaments. Frequency: 3
- 80% - Exam, 0 hour(s)

**Referral policy:** By examination

## Resources

**Other resource requirements:**

Resource type: Background textbook

Stephenson and Radmore Advanced Mathematical Methods for Engineering and Science Students Cambridge

Resource type: Background textbook

Jeffrey Mathematics for Engineers and Scientists Nelson, 2nd Edition

Resource type: Background textbook

Greenberg MD, Advanced Engineering Mathematics, Prentice Hall 1999.

Resource type: Background textbook

Kreyszig E, Advanced Engineering Mathematics Wiley, 1979

Resource type: Background textbook

Spiegel M R, Schaum's Outline of Theory and Problems of Vector Analysis Schaum

Resource type: Background textbook

Spiegel M R, Schaum's Outline of Theory and Problems of Laplace Transforms Schaum

Resource type: Background textbook

Many other examples (including old examination questions and solutions) are available via the web. Students are also recommended to attend the Mathematics Workshop.