PHY1033 | 2025/6 | University of Exeter

MODULE TITLE	Mathematics Skills	CREDIT VALUE	30
MODULE CODE	PHY1033	MODULE CONVENER	Dr Peter Petrov, Dr Gregory James Chaplain

DURATION: TERM	1	2	3
DURATION: WEEKS

Number of Students Taking Module (anticipated)

DESCRIPTION - summary of the module content

This module covers the fundamental mathematical skills Physics students need during their first year. It includes areas such as differential calculus with single and multiple variables, matrices, solutions of linear ordinary differential equations, vector calculus, complex numbers, vector spaces, eigenvalues and eigenvectors, along with Fourier Series and Transforms are covered, that are used daily by physicists. Emphasis is placed on the use of mathematical techniques rather than their rigorous proof. All those tools have wide applicability throughout physics. It emphasises problem solving with examples taken from physical sciences.

AIMS - intentions of the module

All physicists must possess a sound grasp of mathematical methods and a good level of 'fluency' in their application. The aim of this module is to provide a firm foundation in the mathematical techniques required in the modules Physics: Newtonian Mechanics, Classical Thermodynamics, Lagrangian Mechanics and Special Relativity, and The Structure of Our Universe as well as in the Stage 1 Laboratory.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

Module Specific Skills and Knowledge:

1 Make efficient use of the techniques and concepts of foundation-level mathematics: algebra, trigonometry and calculus

2 Perform basic operations on matrices and solve systems of simultaneous linear equations

3 Make series expansions of simple functions and determine their asymptotic behaviour

4 Evaluate single, double and triple integrals in straightforward cases

5 Solve simple first-order differential equations and second-order differential equations with constant coefficients;

6 Calculate and manipulate partial and total derivatives of functions of more than one variable

7 Evaluate single, double and triple integrals using commonly occurring coordinate systems

8 Apply differential operators to vector functions;

9 Apply Stokes's and Gauss's theorems;

10 Perform basic arithmetic and algebra with complex numbers;

11 Identify vector spaces and their properties, diagonalise matrices;

12 Construct and solve eigenvalue problems;

13 Calculate Fourier series and transforms, and use them to solve simple problems.

Discipline Specific Skills and Knowledge:

14 Tackle, with facility, mathematically formed problems and their solution

Personal and Key Transferable/ Employment Skills and Knowledge:

15 Work co-operatively and use peer group as a learning resource

16 Develop appropriate time-management strategies and meet deadlines for completion of work

SYLLABUS PLAN - summary of the structure and academic content of the module

1. Foundation Mathematics (Preliminary Self-Study and Self-Evaluation Pack):

1.1 Algebra

1.2 Trigonometric functions Binomial theorem

1.3 Scalars and Vectors

2. Matrices:

2.1 Matrix addition, subtraction, multiplication

2.2 Inversion of matrices

2.3 Applications to the solution of systems of homogeneous and inhomogeneous linear equations

2.4 Evaluating numerical determinant

2.5 Eigenvalues and Eigenvectors; applications to physics problems

3 Calculus with a Single Variable:

3.1 Advanced methods of Differentiation

3.2 Advanced methods of Integration

4 Solution of linear ordinary differential equations:

4.1 First-order separable, homogeneous, exact and integrating-factor types

4.2 Linear second-order equations with constant coefficients; damped harmonic motion

5 Calculus with Several Variables:

5.1 Partial differentiation, the differential, Reciprocal and Reciprocity Theorems, total derivatives of implicit functions, higher order partial derivatives

5.2 Coordinate systems in 2- and 3-dimensional geometries - Cartesian, plane-polar, cylindrical and spherical polar coordinate systems

5.3 Two-dimensional and three-dimensional integrals and their application to finding volumes and masses

5.4 Line integrals: parametrisation; work as a line integral

6 Sequences, Series and curve sketching

6.1 Sequences and their limits

6.2 Taylor and Maclaurin series

7 Vector Calculus

1.1 The grad operator and its interpretation as a slope

1.2 The divergence operator and its physical interpretation

1.3 The divergence theorem

1.4 The curl operator and its physical interpretation

1.5 Stokes's theorem

8 Multi-Variable Calculus

2.1 Green's Theorem in the plane

2.3 Surface integrals and their application to finding surface areas

9.3 Evaluation of multiple integrals in different coordinate systems and using parameterisation

9 Complex Numbers

3.1 Argand diagram, modulus-argument form, exponential form, de Moivre's theorem

3.2 Trigonometric and hyperbolic functions

10 Linear Algebra

4.1 Vector spaces: linear independence, basis, dimension, and span

4.2 Orthogonality, the Gram-Schmidt process

4.4 Diagonalisation

4.5 Eigenspaces and Eigenvalue problems

11 Generalised Functions, Fourier series, and Fourier transforms

5.1 The Dirac delta and the Heaviside functions

5.2 Real and complex Fourier Series, Fourier series as linear combinations

5.3 The Fourier Transform, duality between real and reciprocal spaces

5.4 The Convolution theorem

5.5 Using integral transforms to solve differential equations

LEARNING AND TEACHING

LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)

Scheduled Learning & Teaching Activities	60	Guided Independent Study	240	Placement / Study Abroad	0

DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS

Category	Hours of study time	Description
Scheduled Learning and Teaching activities	40	20×1-hour lectures
Scheduled Learning and Teaching activities	20	10 x 1-hour problem classes
Guided independent study	30	Self-study packages
Guided independent study	40	Problem sets
Guided independent study	170	Reading, private study, and revision

ASSESSMENT

FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade

Form of Assessment	Size of Assessment (e.g. duration/length)	ILOs Assessed	Feedback Method
Guided self-study	Self-Study packages (fortnightly)	1-13	Discussion in class
Problems sets	4 hours per set (fortnightly)	1-16	Solutions discussed in classes

SUMMATIVE ASSESSMENT (% of credit)

Coursework	24	Written Exams	76	Practical Exams

DETAILS OF SUMMATIVE ASSESSMENT

Form of Assessment	% of Credit	Size of Assessment (e.g. duration/length)	ILOs Assessed	Feedback Method
End of Term examination 1	28	2 hours	1-13	Written, collective feedback via ELE and solutions
End of Term examination 2	28	2 hours	1-13	Written, collective feedback via ELE and solutions
Mid-Term test 1	10	1 hour	1-10	Written, collective feedback via ELE and solutions
Mid-Term test 2	10	1 hour	1-10	Written, collective feedback via ELE and solutions
Problem sets 1	12	Electronic problem sets	1-16	Solutions via ELE and interaction with demonstrators
Problem sets 2	12	Electronic problem sets	1-16	Solutions via ELE and interaction with demonstrators

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)

Original Form of Assessment	Form of Re-assessment	ILOs Re-assessed	Time Scale for Re-assessment
End of term examination 1, mid-Term test 1	Written exam (2 hours, 38%)	1-13	Referral/deferral period
End of term examination 2, mid-Term test 2	Written exam (2 hours, 38%)	1-13	Referral/deferral period
Problem sets 1 and 2	Problem sets (24%)	1-16	Referral/deferral period

RE-ASSESSMENT NOTES

Re-assessment is not available except when required by referral or deferral. Since mid-terms and end of Term examinations test the same ILOs, a student failing the entirety of the module will sit two written exams over Ref/Def period, each worth 38% of the module, where the content of each term will be assessed. Mid-terms will, though, be deferrable into the ref/def period.

RESOURCES

INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

Core text:

Stroud K.A. and Booth D.J. (2013), Engineering Mathematics (7^th edition), Palgrave MacMillan, ISBN 978-1-137-03120-4 (UL: 510.2462 STR)
K.F. Riley, Hobson M.P. and Bence S.J. (2006), Mathematical Methods for Physics and Engineering: A Comprehensive Guide (3^rd edition), Cambridge University Press, ISBN 978-0521679718 (UL: eBook)
Poole D. (2015) Linear Algebra: A Modern Introduction (4^th edition), Brooks/Cole, ISBN: 978-8-13-153024-5

Supplementary texts:

Alcock L. (2014), How to think about analysis, Oxford University Press, ISBN 9780191035371 (UL: eBook)
Arfken G.B. and Weber H.J. (2001), Mathematical methods for physicists (5^th edition), Academic Press, ISBN 0-120-59826-4 (UL: 510 ARF)
K.F. Riley and Hobson M.P. (2011), Foundation Mathematics for the Physical Sciences, Cambridge University Press, ISBN 978-0-521-19273-6 (UL: 500 RIL)
Spiegel M.R. (1971), Advanced Mathematics for Engineers and Scientists, Schaum Outline Series, McGraw-Hill, ISBN 0-070-60216-6 (UL: 510 SPI)
Stroud K.A. and Booth D.J. (2011), Advanced Engineering Mathematics (5^th edition), Paulgrave, ISBN 978-0-23-027548-5 (UL: 510.2462 STR)

Reading list for this module:

There are currently no reading list entries found for this module.

CREDIT VALUE	30	ECTS VALUE	15

PRE-REQUISITE MODULES	None
CO-REQUISITE MODULES	None

NQF LEVEL (FHEQ)	4	AVAILABLE AS DISTANCE LEARNING	No
ORIGIN DATE	Thursday 16th May 2024	LAST REVISION DATE	Tuesday 2nd September 2025

KEY WORDS SEARCH	Physics, Algebra, Calculus, Complex numbers, Differentiation, Equations, Functions, Integration, Matrices, Series, Derivatives; Differential equations; ls; Linear algebra; Operators; Theorems.

Mathematics Skills - 2025 entry