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Outline:

This module aims to provide students with the mathematical foundations and understanding required for their studies in physics and astrophysics related degree programmes, and to give students experience and skills in mathematical manipulation and problem-solving. Topics include: differential equations, multiple integrals, matrices and linear transformations, vector operations.

Aims:

This module aims to:

• provide, together with PHAS0002, the mathematical foundations required for all the first year and some of the second year courses in the Physics and Astronomy programmes;
• prepare students for the second year Mathematics course PHAS0025 and MATH0043;
• give students further practice in mathematical manipulation and problem solving.

Intended Learning Outcomes:

By the end of the module the student should be confident in applying the mathematical methods covered to analytical treatments of problems in the physical sciences. They will also have acquired the necessary background to study advanced techniques in algebra and analysis.

Teaching and Learning Methodology:

This module is delivered via weekly lectures supplemented by a series of problem solving tutorials and additional discussion.

In addition to timetabled lecture and PST hours, it is expected that students engage in self-study in order to master the material. This can take the form, for example, of practicing example questions and further reading in textbooks and online.

Indicative Topics:

• Differential Equations: Ordinary first-order; separable; integrating factor method; perfect differential; Ordinary second order, homogeneous and non-homogeneous, including equal roots;
• Vector Operators: Gradient, divergence, curl and Laplacian operators; derivation of vector operators in cylindrical and spherical coordinates; triple vector products including differential operators;
• Multi-dimensional Integrals: Area and volume integrals; change of coordinates; area and volume elements in polar, cylindrical and spherical coordinates; parametrisation of curved surfaces; vector normal to a surface; integration of scalar functions over a curved surface; flux of a vector field through a curved surface; divergence and curl of vector fields; Gauss and Stokes integration theorems;
• Matrices and Linear Transformations: Matrix multiplication and addition; Finding the determinant, trace, transpose and inverse of a matrix; Properties of the matrices; Multiple matrices transformation; Definition of 2, 3 and higher order determinants in terms of row evaluation; Manipulation of determinants; Cramer’s rule for the solution of linear simultaneous equations; Eigenvalues and eigenvectors; Eigenvalues of unitary and Hermitian matrices; Real quadratic forms; Normal modes of oscillation. Revision of real 3-dimensional vectors; Complex linear vector spaces.