MPhys Physics with Astrophysics

Mathematics 1

To allow students to develop their mathematical competence with functions, calculus, complex numbers, power series, linear algebra and differential equations to a level where they can cope with the demands of the first year of the physics course and beyond.

On completion successful students will be able to:

1. Describe the properties of different types of functions and be able to sketch them in both Cartesian and polar coordinates 
2. Integrate and differentiate functions of one variable using a range of techniques and be able to apply integration and differentiation to a range of physical problems. 
3. Show how smooth functions can be expressed in terms of power series.
4. Explain the properties of complex numbers and construct some basic complex functions.
5. Solve first and second order ordinary differential equations using a range of techniques.
6. Calculate surface and volume integrals in various coordinate systems

1. Functions, 2D coordinates, and the basics of vectors

Properties of functions. 2D and 3D coordinate systems. Sketching functions. Exponential and logarithmic functions. Vectors.

2. Polar coordinates and differential calculus

Sketching and expressing functions in polar coordinates.  The differential; differentiation of products and functions of functions; maxima, minima and inflexion points. Partial differentiation; relationship between partial and total derivative; multivariate maxima, minima and saddle points; examples and applications from physics.

3. Complex numbers

Definition, modulus and argument; multiplication and division. Complex roots of quadratic equations. Complex numbers in polar and exponential form. Examples of applications from physics. De Moivre’s theorem.  Hyperbolic functions. 

4. Power series

Series. Limits of series. Binomial expansion. Taylor and Maclaurin series expansions. Taylor’s theorem for multivariate functions.

5. Integral calculus

Common and standard integral

Assessment - Exam

Workshop

Lecture

Tutorial

* Other

10% Online Test 

10% Tutorial Exercise 

Online quizzes will also be incorporated into the weekly learning material to give students instant feedback on their understanding and ability to apply their knowledge and skills.

Recommended texts:

Our recommended text: Mathematics for Physicists by Martin and Shaw (Manchester Physics Series)

Mathematics for Physicists - Martin, Brian R. Shaw, Graham – Wiley – 2015 - ISBN: 0470660228

Mathematical techniques: an introduction for the engineering, physical, and mathematical sciences - Jordan, D. W.; Smith, Peter - Oxford University Press – 2008 - ISBN: 0199282013

Online Resource Centre for Jordan & Smith: Mathematical Techniques (4th edition) - web site: www.oup.com - URL: Web site: www.oup.com…

Further mathematics for the physical sciences - Tinker, Michael; Lambourne, Robert - John Wiley – 2000 -ISBN: 0471866911

Mathematical methods in the physical sciences - Boas, Mary L. – Wiley – 2006 - ISBN: 0471198269

Basic mathematics for the physical sciences - Lambourne, Robert; Tinker, Michael - Wiley – 2000 - ISBN: 0471852066

Tuition fees for home students commencing their studies in September 2027 will be £10,050 per annum. Tuition fees for international students will be £40,300 per annum.

For general information please see the undergraduate finance pages.

All students should normally be able to complete their programme of study without incurring additional study costs over and above the tuition fee for that programme. Any unavoidable additional compulsory costs totalling more than 1% of the annual home undergraduate fee per annum, regardless of whether the programme in question is undergraduate or postgraduate taught, will be made clear to you at the point of application. Further information can be found in the University's Policy on additional costs incurred by students on undergraduate and postgraduate taught programmes (PDF document, 91KB).

The University of Manchester is committed to attracting and supporting the very best students. We have a focus on nurturing talent and ability and we want to make sure that you have the opportunity to study here, regardless of your financial circumstances.

For information about scholarships and bursaries please visit our undergraduate student finance pages and our Department funding pages .