MPhys Physics with Theoretical Physics / Course details

Mathematics 2

To acquire the mathematical skills in linear algebra and vector calculus needed for applications in Dynamics, Electromagnetism, Fluid Mechanics, Statistical Mechanics, Waves and Fields, Relativity and Quantum Mechanics.

On successful completion of PHYS10372, a student will be able to:

• Describe the properties and uses of vectors, determinants and matrices and the algebraic operations that can be performed with vectors, determinants and matrices.
• Be able to solve linear equations and coupled ordinary differential equations using eigenvectors/values, matrix diagonalisation, matrix inversion and Gaussian elimination. 
• Explain the concepts and properties of scalar and vector fields and the properties of div, grad and curl and be able to calculate the divergence and curl of vector fields in various coordinate systems.
• Calculate line, surface/flux and volume integrals in various coordinate systems and relate them to the divergence and curl through the Divergence Theorem and Stokes’ Theorem.

Linear Algebra: 9 lectures

Basic vector spaces, basis vectors. Orthogonality. Products of vectors. Extending vectors beyond 3D. Matrix algebra and applications of linear algebra. Matrix types and properties. The determinant and its uses with matrices. Gauss elimination and solving linear equations. Matrix inversion and diagonalisation. Determination of eigenvalues and eigenvectors of a matrix and their applications to solving coupled ordinary differential equations.

Vector operators: div, grad and curl: 6 lectures

Scalar and vector fields. Definition and uses of the gradient operator. The method of Lagrange multipliers. Definitions of divergence and curl. Combinations of div, grad and curl; theorems. The Laplacian. Vector operators in cylindrical and spherical polar coordinates.

The Divergence Theorem, Stokes Theorem, conservative forces : 7 lectures

Line integrals of scalar and vector fields. Surface integrals and flux of vector fields. Integral expression for divergence. Divergence theorem and its uses. Conservation laws; continuity equation. Integral expression for curl. Stokes' theorem and its uses. Definition of conservative field. Relation to potentials.

* Other

10% Weekly online 

10% Tutorial Work/attendance 

Feedback will be offered by tutors on students’ written solutions to weekly examples sheets, and model answers will be issued. Interactive feedback will be offered during the Workshop sessions.

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

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

Mathematical methods for physics and engineering    Riley, K. F.; Hobson, M. P.; Bence, S. J, Cambridge University Press, 2006, ISBN: 0521861535

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…

Div, grad, curl, and all that: an informal text on vector calculus, Schey, H. M., W. W. Norton, 2005, ISBN: 0393925161

Calculus: early transcendentals, Stewart, James, Brooks/Cole, 2011, ISBN: 0538498870

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