MPhys Physics

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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 2D 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. employ matrix notation, carry out matrix algebra and use matrices to solve systems of linear equations. 
6. compute the properties of determinants, be able to evaluate them, and use them to test for unique solutions of linear equations. 
7. solve first and second order ordinary differential equations using a range of techniques.

1.  Functions and 2D coordinates

Properties of functions. 2D and 3D coordinate systems. Index notation, Sketching functions, logarithmic functions.          

2.  Complex numbers

Definition, modulus and argument; addition, multiplication, division; roots of quadratic equations; complex numbers in polar form; De Moivre's theorem; Hyperbolic functions.

3.  Differential Calculus

Review of differentiation, the differential; differentiation of products, functions of functions; maxima, minima and inflexions; partial differentiation; examples and applications from physics.                                                              

4.  Power Series

Series, limits of series; binomial expansion; Taylor's and Maclaurin's series expansions.

5.  Integral Calculus

Review of integration; integration by parts, substitution, standard integrals, partial fractions and completing the square; simple line integrals; physical applications.   

6.  Linear Algebra

Matrix algebra, inverse matrix. Definition and properties of determinants, scalar triple product, test of unique solution to linear equations. Eigenvalues and eigenvectors, eigenanalysis.

7.  Ordinary Differential Equations

Physical motivation. 1st order separable. 1st order homogeneous. 1st order linear: integrating factors. 2nd order with constant coefficients. Physical applications.  

* Other

10% Weekly online 

10% Tutorial Work/attendance 

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 is Mathematics for Physicists by Martin and Shaw (Manchester Physics Series). 

Supplementary texts.

Jordan, D. & Smith, P. Mathematical Techniques (OUP)

Tinker, M. & Lambourne, R. Further Mathematics for the Physical Sciences (Wiley) 

ASupplementary texts:

Lambourne, R. & Tinker, M. Basic Mathematics for the Physics Sciences (Wiley)