## MAS152 Essential Mathematical Skills and Techniques

Both semesters, 2017/18 | 20 Credits | ||||

Lecturer: | Prof Koji Ohkitani | 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.

There are no prerequisites for this module.

The following modules have this module as a prerequisite:

MAS253 | Mathematics for Engineering Modelling |

MAS254 | Computational and Numerical Methods |

## 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 |
---|---|---|---|---|---|

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

B |
G. James and D. Burley | Modern Engineering Mathematics | 510.2462 (J) | Blackwells | Amazon |

B |
K. A. Stroud and D. J. Booth | Engineering Mathematics | 510.2462 (S) | Blackwells | Amazon |

(

**A**= essential,

**B**= recommended,

**C**= background.)

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