To re-inforce students' prior knowledge of mathematical techniques To introduce new mathematical techniques for physics modules To enhance students' problem-solving abilities through structured application of these techniques in physics

(LO1) At the end of the module the student should be able to: Have knowledge of a range of mathematical techniques necessary for physics and astrophysics programmes Be able to apply these mathematical techniques in a range of physics and astrophysics programmes

(S1) Numeracy/computational skills - Reason with numbers/mathematical concepts

(S2) Numeracy/computational skills - Problem solving

Overview Integral and differential vector calculus: Scalar and vector fields Scalar and vector field functions Polar coordinate systems Derivation of the gradient, divergence and curl functions Examples of these operations including their physical significance Vector operations in polar coordinate systems Stoke’s theorem with examples Gauss’ theorem with examples Line, surface and volume elements in circular, spherical and cylindrical polar coordinates Line, surface and volume integrals in different coordinate systems - applications  Vectors and Matrices Real and complex vectors, linear independence, basis, scalar product, orthonormal basis. Revision of matrices. Sum, product, transposition. Symmetric and antisymmetric matrices.  Trace and determinant of square matrices. Laplace expansion theorem. Row echelon form of a matrix. Rank of a matrix. Application to vectors (coplanarity, collinearity). Systems of linear equations, Gaussian elimination. Inversion o f matrices using row operations. E igenvalues and eigenvectors of matrices. Complex and degenerate eigenvalues. Real symmetric matrices and diagonalisation. Orthogonal transformations and orthogonal matrices. Applications: rotational motion, inertia tensor. Applications Application: rotational motion, inertia tensor Hermitian scalar product of complex vectors. Hermitian matrices and diagonalization. Unitary transformations and unitary matrices. Application: quantum mechanics. Revision of Taylor's theorem, Taylor's theorem with remainder. Revision of infinite sums and series. Ratio test. Radius of convergences of power series. Revision of Taylor series. Generating Taylor series from known Taylor series by substitution and differentiation.

Teaching Method 1 – Lectures delivered online.
Description: Lecture to entire cohort on all course topics over VLE.
Attendance Recorded: Yes

Teaching Method 2 - Workshops delivered in person on campus.
Description: Weekly problem-solving classes to learn together with guidance from staff and receive feedback.
Attendance Recorded: Yes
Notes: = 12 x 1-hour workshops.