## MAS156 Mathematics (Electrical and Aerospace)

Both semesters, 2019/20 | 20 Credits | ||||

Lecturer: | Prof Elizabeth Winstanley | Home page (also MOLE) | Timetable | Reading List | |

Aims | Outcomes | Teaching Methods | Assessment | Full Syllabus |

This module aims to reinforce students' previous knowledge and to develop new basic mathematical techniques needed to support the engineering subjects taken at Levels 1 and 2. It also provides a foundation for the Level 2 mathematics courses in the appropriate engineering department. The module is delivered via online lectures, reinforced with 2 weekly interactive problem classes.

**Prerequisites:** B Tech or A-level mathematics or equivalent

The following modules have this module as a prerequisite:

AER201 | Mathematics for Aerospace Engineers |

MAS241 | Mathematics II (Electrical) |

## Aims

- To reinforce previous mathematical knowledge.
- To develop new mathematical concepts needed to support engineering at Level 1.
- To provide a foundation for the Level 2 mathematics courses for engineers.

## Learning outcomes

**Semester 1:**

- Ability to sketch functions and evaluate simple limits using algebraic techniques.
- Ability to differentiate, find maxima and minima and apply this technique to curve sketching.
- Ability to find 1st and 2nd order partial derivatives.
- An understanding of hyperbolic functions.
- Ability to apply l'Hôpital's rule.
- Ability to manipulate complex numbers.
- Knowledge of the basic properties of vectors.

**Semester 2:**

- Ability to evaluate indefinite and definite integrals using the techniques of substitution and integration by parts.
- Ability to manipulate matrices, evaluate determinants and find the inverse of a non-singular square matrix.
- Ability to apply matrix methods to the solution of systems of simultaneous linear equations.
- Ability to find eigenvalues and corresponding eigenvectors of a square matrix
- Ability to solve first order ordinary differential equations which are (i) variables separable, (ii) linear,
- Ability to solve second order linear homogeneous ordinary differential equations with constant coefficients.
- Ability to solve second order linear inhomogeneous ordinary differential equations with constant coefficients, using a trial technique for the particular integral.
- Ability to apply Laplace Transforms and use them to solve linear differential equations.

## Teaching methods

Online video lectures, online tests, problem classes, problem solving.

5 lectures, 40 tutorials

## Assessment

One three-hour exam at the end of the year (85%). Online tests (15%).

## Full syllabus

**Semester 1:**

**1. Functions of a real variable and limits:**

The concept
of a function and simple limits, continuity.

**2. Differentiation:**

Basic rules of differentiation:
maxima, minima and curve sketching. Inverse functions.

**3. Partial differentiation:**

1st and 2nd derivatives,
geometrical interpretation.

**4. Hyperbolic functions:**

Definitions and derivatives
of hyperbolic functions and their inverses.

**5. Series:**

Taylor and Maclaurin series, L'Hôpital's rule.

**6. Complex numbers:**

basic manipulation,
Argand diagram, de Moivre's theorem, Euler's relation.

**7. Vectors:**

Vector algebra, dot and cross
products, differentiation.

**Semester 2:**

**1. Integration:**

Indefinite integrals of simple functions. Simple substitutions. Standard forms involving inverse trigonometric and inverse hyperbolic functions. Examples using completing the square and partial fractions. Integration by parts. Definite integrals: properties, evaluation, application to area.

**2. Matrices and linear equations:**

Definition of an m ×n matrix. Special matrices (identity, zero, square etc.). Matrix algebra. Transpose. Symmetric and skew-symmetric matrices, and the decomposition of square matrices. Determinants. Inverse of a non-singular matrix. Use of matrices to solve systems of linear equations (homogeneous and nonhomogeneous). Gaussian elimination. Eigenvalues and eigenvectors.

**3. Ordinary differential equations:**

First order differential equations: variables separable, linear with integrating factor, general solution, solution satisfying given initial conditions. Second order linear differential equations with constant coefficients: auxiliary equation, complementary function. Particular integral for polynomials, exponentials, trigonometric functions and products of polynomials and exponential/trigonometric functions on right-hand side. Laplace Transforms and their use in the solution of linear differential equations.

## Reading list

Type | Author(s) | Title | Library | Blackwells | Amazon |
---|---|---|---|---|---|

A |
A. Croft, R. Davison and M. Hargreaves | Engineering Mathematics: a foundation for electronic, electrical, communications and systems engineer | 510.2462 (C) | Blackwells | Amazon |

A |
C. W. Evans | Engineering Mathematics | 510.2462 (E) | Blackwells | Amazon |

A |
G. James and D. Burley | Modern Engineering Mathematic | 510.2462 (J) | Blackwells | Amazon |

A |
K. A. Stroud and D.J. Booth | Engineering Mathematics |

(

**A**= essential,

**B**= recommended,

**C**= background.)

Most books on reading lists should also be available from the Blackwells shop at Jessop West.

## Timetable (semester 2)

Mon | 12:00 - 12:50 | lecture | Diamond Building LT1 | ||||

Tue | 09:00 - 09:50 | tutorial | (group EE1) | Hicks Seminar Room F20 | |||

Tue | 09:00 - 09:50 | tutorial | (group EE2) | Hicks Seminar Room F24 | |||

Tue | 15:00 - 15:50 | tutorial | (group AE1) | Hicks Lecture Theatre 4 | |||

Tue | 15:00 - 15:50 | tutorial | (group AE2) | Hicks Lecture Theatre 10 | |||

Tue | 15:00 - 15:50 | tutorial | (group AE3) | Hicks Seminar Room F28 | |||

Tue | 15:00 - 15:50 | tutorial | (group AE4) | Hicks Seminar Room F38 | |||

Thu | 11:00 - 11:50 | lab session | (group ee1) | Hicks Lecture Theatre D | |||

Thu | 11:00 - 11:50 | lab session | (group ee2) | Hicks Seminar Room F38 | |||

Thu | 12:00 - 12:50 | lab session | (group ae1) | Hicks Lecture Theatre 10 | |||

Thu | 12:00 - 12:50 | lab session | (group ae2) | Hicks Lecture Theatre D | |||

Thu | 12:00 - 12:50 | lab session | (group ae3) | Hicks Seminar Room F38 | |||

Thu | 12:00 - 12:50 | lab session | (group ae4) | Hicks Seminar Room F28 |