## MATH1054 Mathematics for Engineering and the Environment

### Module Overview

To present, after some preparation and revision, the basic mathematical methods of differential and integral calculus, complex numbers, matrices, and ordinary differential equations.

### Module Details

**Title: **Mathematics for Engineering and the Environment**Code: **MATH1054**Semester: **1 and 2

**CATS points: **15** ECTS points: **7.5**Level: **Undergraduate**Co-ordinator(s): **Professor Carsten Gundlach,

### Pre-requisites and / or co-requisites

A-level Mathematics or equivalent

### Aims and objectives

**The aims of this module are to: **To present, after some preparation and revision, the basic mathematical methods of differential and integral calculus, complex numbers, matrices, and ordinary differential equations.

**Learning Outcomes:**

On successful completion of the module the students should be able to:

- Demonstrate knowledge and understanding of basic differential and integral calculus, complex numbers and differential equations, and be familiar with partial differentiation and some more advanced techniques of calculus.;
- Critically analyse and solve some mathematical problems.;
- Show logical thinking in problem solving.;
- Perform calculations in simple situations and work through some longer examples.;
- Work more effectively with self-study material.;
- Demonstrate organisational and time-management skills.

### Syllabus

A-Level Revision

Algebra: simplification of expressions and functions; indices; linear and quadratic equations; simultaneous linear equations; inequalities; partial fractions.

Trigonometry: solution of triangles; multiple angle formulae; trigonometric equations.

The following topics are studied and tested over two semesters:

- Differential calculus: standard rules; Newton's method for finding roots; simple partial differentiation.
- Integral calculus: standard integrals; integration by parts; numerical integration.
- Complex numbers I: algebra; Argand diagram; polar form; Euler's formula.
- Differential equations : classification; simple first and second order differential equations.
- Functions: inverse; trigonometric; exponential and hyperbolic.
- Differentiation: maxima, minima and points of inflection; curve sketching; implicit, parametric and logarithmic differentiation.
- Integration: substitution; applications to centroids, volumes of revolution etc.
- Integration: integration of rational functions; improper integrals.
- Integration: double integral; polar coordinates; triple integrals.
- Vectors I: basic properties, Cartesian components, scalar and vector products.
- Vectors II: triple products, differentiation and integration of vectors, vector equations of lines and planes.
- Complex numbers II: De Moivre's theorem; roots; logarithm of a complex number.
- Matrix algebra I: terminology; addition, subtraction and multiplication of matrices; determinants.
- Matrix algebra II: inverse of matrix using cofactors; sets of linear equations; solution of sets of linear equations using elimination method; inverse of matrix using elimination method.
- Matrix algebra III: rank; eigenvalues and eigenvectors.
- Ordinary differential equations: solution of first order equations (separable, homogenous, linear and exact).
- Ordinary differential equations: linear operators; second order linear inhomogeneous equations with constant coefficients; free and forced oscillations.

### Learning and teaching

### Study time allocation

**Contact hours: **18**Private study hours: **132**
Total study time:
150
hours
**

### Teaching and learning methods

**Teaching methods include:**

- Self-study notes produced for each weekly topic, including examples and specimen test (with solutions).

**Learning activities include:**

- Individual study of identified sections in course text.
- Working through examples and specimen test in each weekly set of self-study notes, with solutions provided.

**Feedback and student support during module study (formative assessment) **

- Examples and specimen test in each set of self-study notes, with solutions, provide self-assessment.
- Test at end of each topic is marked 1-1 with immediate feedback, at the rate of about two tests in three weeks.
- Twice weekly timetabled workshops available.
- Past examination papers and solutions available on website

### Resources and reading list

**Core Text**G James,

*Modern Engineering Mathematics*, Prentice Hall, 2007 ISBN 9780132391443, TA150 JAM (4 copies in short loan)

**Recommended Text**

(Very useful for weaker students) K A Stroud,

*Engineering Mathematics*, Palgrave 2007, ISBN 9781403942463, QA100STR (8 copies, 2 in short loan)

View the module web pages

### Assessment

### Assessment methods

2 hour unseen written examination, using Formula Sheet. 80%;

Coursework mark generated from tests at end of each topic. (17 items) 20%

The end of module examination is structured into two sections. The first contains 20 multiple choice questions which test basic knowledge of all weekly topics and whether simple calculations can be performed successfully. The second section consists of longer questions which test the depth of understanding of topics on the syllabus and the ability to carry out longer pieces of work.

The general skills elements are not explicitly assessed, but their development will reflect on the quality of the overall outcomes.

Feedback is through regular testing during the course and is given in person by markers and teaching assistants during the assessment sessions