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The University of Southampton

MATH1054 Mathematics for Engineering and the Environment

Module Overview

This course lays the mathematical foundation for all engineering degrees. Its structure allows students with different levels of previous knowledge to work at their own pace. Pre-requisite for MATH2048 One of the pre-requisites for MATH3081 and MATH3082

Aims and Objectives

Module Aims

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

Learning Outcomes

Knowledge and Understanding

Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:

  • Demonstrate knowledge and understanding of basic differential and integral calculus, complex numbers, vectors and matrices, statistics and differential equations, and be familiar with partial differentiation and some more advanced techniques of calculus
  • Show logical thinking in problem solving
  • Work effectively with self-study material
Transferable and Generic Skills

Having successfully completed this module you will be able to:

  • Demonstrate organisational and time-management skills
Subject Specific Practical Skills

Having successfully completed this module you will be able to:

  • Critically analyse and solve some mathematical problems
  • Perform calculations in simple situations and work through some longer examples


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: 1. Differentiation: basic rules; differentiation of standard functions; Newton's method for finding roots; partial differentiation. 2. Integration: definition of integral; standard integrals; substitution; integration by parts; numerical integration. 3. Complex numbers: graphical representation; algebra; polar form; Euler's formula and exponential form. 4. Differential equations : classification; simple first and second order differential equations. 5. Functions: inverse; trigonometric; exponential, logarithmic and hyperbolic. 6. Differentiation: maxima, minima and points of inflection; curve sketching; parametric, implicit and logarithmic differentiation; Taylor and Maclaurin series. 7. Integration: substitution; applications to centroids, volumes of revolution, etc. 8. Integration: rational functions; improper integrals. 9. Integration: double integrals; polar coordinates; triple integrals. 10. Differential equations: solution of first order ODEs (separable, homogenous, linear and exact). 11. Differential equations: linear operators; linear inhomogeneous second order ODEs; free and forced oscillators. 12. Vectors: basic properties; Cartesian components, scalar and vector products. 13. Vectors: triple products; differentiation and integration of vectors; vector equations of lines and planes. 14. Matrix algebra: terminology; addition, subtraction and multiplication of matrices; determinants. 15. Matrix algebra: inverse of a matrix using cofactors; systems of linear equations; inverse of a matrix using the elimination method. 16. Matrix algebra: rank; eigenvalues and eigenvectors. 17. Further calculus: sequences and series; Rolle's and mean value theorems; Taylor's and Maclaurin's theorems 18. Complex numbers: trigonometric and hyperbolic functions; logarithm of a complex number; De Moivre's theorem; roots; simple loci in the complex plane. 19. Statistics: probability; conditional probability; combinations and permutations; discrete and continuous random variables. 20. Statistics: mean and standard error of sample data; normal distribution; sampling; confidence intervals; hypothesis testing.

Special Features

This is a self-study course: there are no lectures except an introductory lecture on the first session in Week 1. The other sessions are testing and marking sessions, which the student attends at their own pace.

Learning and Teaching

Teaching and learning methods

Teaching methods include • Self-study notes for each topic. • Examples and specimen test in each set of self-study notes, with solutions, for self-assessment. • Test at end of each topic is marked one-to-one with immediate feedback, at the rate of about two tests in three weeks. • Twice weekly timetabled workshops available. • Past examination papers and solutions. Learning activities include • Individual study of identified sections in course textbook. • Working through examples and specimen test for each topic in the set of self-study notes, with solutions provided.

Independent Study149
Total study time150

Resources & Reading list

K A Stroud (2007). Engineering Mathematics. 

Glyn James (2015). Modern Engineering Mathematics. 


Assessment Strategy

• The end of module examination is structured into two parts. The first part contains 20 multiple choice questions which test basic knowledge of all topics and whether simple calculations can be performed successfully. The second part 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.


MethodPercentage contribution
Exam  (2 hours) 100%


MethodPercentage contribution
Exam 100%

Repeat Information

Repeat type: Internal & External


Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

Books and Stationery equipment

The core textbook (James, 5th edition) is provided by the Faculty. Theare are copies of the alternative core textbook (Stroud, 6th edition) in the Hartley library (8 copies, 2 in short loan, library code QA100 STR). Older versions of the core textbooks (James 4th or 3rd edition, Stroud 5th edition) can also be used, and the library holds several copies

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at

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