**Module Description**

This module will introduce engineering mathematics to students, including a number of special functions: trigonometric, lorgarithmic and exponential. It will explain the operations of differentiation and integration. Complex number operations are demonstrated along with complex routes of polynomials. Students will also learn to perform basic operations with vectors and matrices and find the Fourier series of aa given period function, and explain the concept of a spectrum.

**Learning Outcomes**

After completing this module, students will be expected to be able to:

1.Recognise,
and perform routine calculations with a number of special functions,
including trigonometric, logarithmic and exponential functions.

2.
Define and explain the operations of differentiation and integration,
and use appropriate rules to find derivatives and integrals.

3.
Execute basic operations with vectors (including products of vectors)
and matrices (+, -, x, determinants, inverse), and solve systems of
linear equations via matrices.

4. Execute basic operations with
complex numbers (+, -, x, /, *), and convert between Cartesian and
polar forms. Find roots of polynomials (real and complex).

5. Find the Fourier series of a given periodic function, and explain the concept of the spectrum of a periodic function.

**Outline Syllabus**

1. Special Functions

Polynomial, trigonometric, logarithmic and exponential functions

Powers and Logatithms; The inverse operations; Rules of logarithms

Right-angle triangles,;The quadratic function; Symmetry of functions and their graphs

Visualisation with MATLAB

2. Differentiation

The derivative function

Some comon derivativesFinding the derivative of combinations of functionsApplications of differentiation

Visualisation with MATLAB

3. Integration

The integral as the area under a graph

Finding integralsApplications of integrationnumerical methods of integrationNumerical integration with MATLAB

4. Vectors and Matrices

Vectors and vector quantities;

Basic vectors; Products of vectors;

Vector calculation (addition, substraction and multiplication)

Matrix calculations (addition, subtraction and multiplication)

The matrix form of a set of linear equations

Determinants and inversion of 2x2 matrices

Visualisation with MATLAB

5. Complex Numbers

The square-root of negative numbers, and the number j

Cartesian representation of complex numbers: addition, subtraction, multiplication and division;

Complex numbers and operations

Polar representation: multiplication and division, conversion between forms

Applications to A.C. Linear circuits

Complex arithmetic with MATLAB

6. Fourier Series

Periodic functions obtained by adding sinusoids

Sine and cosine seriesThe Fourier series of symmetric periodic
functionsAmplitude and phase representation of a Fourier
seriesComputation and visualisation of Fourier series with MATLAB

- Module Supervisor: Eirina Bourtsoulatze