Friday 4pm:
Friday 5pm:
Friday 6pm:
Friday 7pm:
Michaelmas term
Lent term
Easter term
Vector calculus:
Reminder of grad, div, curl, Laplacian, the divergence theorem, Stokes'
theorem, Green's theorem. Vector differential operators in orthogonal curvilinear
coordinates.
Integral transforms:
Fourier transforms; relation to Fourier series, simple properties and examples,
the delta function, convolution theorem and Parseval's theorem treated
heuristically, application to diffusion equation. Laplace transforms; simple
properties and examples, application to linear differential equations.
Matrices:
- N-dimensional vector space, matrices, scalar product, transformation
of basis vectors.
- Quadratic and Hermitian forms, quadric surfaces. Eigenvalues and eigenvectors
of a matrix; degenerate case, stationary property of eigenvalues. Orthogonal
and unitary transformations.
Ordinary differential equations:
- Power series of a complex variable; circle of convergence
- Homogeneous equations; solution by series (without full discussion of
logarithmic singularities), exemplified by Legendre's equation.
- Inhomogeneous equations; solution by variation of parameters, introduction
to Green's functions.
- Sturm-Liouville theory; self-adjoint operators, eigenfunctions and eigenvalues,
reality of eigenvalues and orthogonality of eigenfunctions. Eigenfunction
expansions and determination of coefficients. Legendre polynomials; orthogonality.
Calculus of Variations:
- Euler's equations and example, e.g. geodesics on a surface.
- Variational principles; Fermat's principle; Hamilton's principle and
deduction of Lagrange's equation, illustrated by a system with
.
Variational principle for the lowest eigenvalue and for higher eigenvalues (Rayleigh-Ritz).
Partial differential equations:
- Solution by separation of variables of Laplace's equation in two dimensions
in polar coordinates, and spherical polar coordinates; axisymmetric case
and Legendre polynomials again.
- Solution of Poisson's equation as an integral. Uniqueness theory for
Poisson's equation with Dirichlet boundary conditions. Green's identity.
Green's function for Laplace's equation with simple boundary conditions
using the method of images. Applications to electrostatic fields and steady
heat flow.
Cartesian Tensors:
Suffix notation. Transformation laws, addition, multiplication, contraction.
Isotropic tensors, symmetric and antisymmetric tensors. Vector products,
grad, div and curl in tensor form. Principal axes and diagonalization.
Tensor fields, e.g. conductivity, susceptibility.
Analytic functions and contour integration:
- Cauchy-Riemann equations; rational functions and exp(z).
Zeros, poles and essential singularities.
- Integration along a path; elementary properties. Cauchy's theorem; proof
by Cauchy-Riemann equations and divergence theorem in 2D. Integral of f'(z);
Cauchy's formula for f(z). Calculus of residues; examples
of contour integration; point at infinity, residue at infinity; multi-valued
functions, branch points, log(z). Inverse Laplace transforms
(Bromwich integral).
Numerical Methods (Lent term)
The aim of this course is to introduce numerical techniques that can be used on computers, rather than to provide a detailed treatment of accuracy or stability.
Algebraic equations:
- Rounding and truncation errors.
- Iterative methods for approximate solution of algebraic and transcendental
equations; first- and second-order convergence; solution by binary choppings
and linear interpolation.
- Solution of simultaneous linear equations by Gaussian elimination with
interchange . Tri-diagonal matrices.
Integration:
Quadrature by trapezoidal and Simpson's rule.
Ordinary differential equations:
- Initial value problems; predictor-corrector method and use of Runge-Kutta
method; elementary discussion of stability.
- Two-point boundary problems; reduction to simultaneous equations.
Partial differential equations:
- Representation of derivatives by finite differences. Diffusion equation;
solution by finite differences, stability condition for explicit scheme.
Laplace's equation: finite difference form; brief account of relaxation
and over-relaxation.
Mathematical Methods III (Easter term)
Small Oscillations:
Small oscillations and equilibrium; normal modes, normal coordinates, examples,
e.g. vibration of linear molecules such as CO2.
Group Theory:
- Idea of an algebra of symmetry operations; symmetry operations on a square.
Definition of a group; group table. Subgroups; homomorphic and isomorphic
groups.
- Representation of groups; reducible and irreducible representations;
basic theorems of representation theory. Classes, characters. Examples
of character tables of point groups. Applications in molecular physics.
Mathematical Methods for Physicists, G.B. Arfken and H.J. Weber (4th edn Academic Press 1995)
Mathematical Methods in the Physical Sciences, M.L. Boas (2nd edn Wiley 1983) - (Yuk! - M.)
Mathematical methods in Physics and Engineering, J.W. Dettman (Dover 1988)
Groups, Representations and Physics, H.F. Jones (2nd edition, IOP 1998)
Advanced Engineering Mathematics, E. Kreyszig (7th edition, Wiley 1993)
How To Use Groups, J.W. Leech (Chapman and Hall 1993 - out of print)
Mathematical Methods of Physics, J. Matthews and R.L. Walker (2nd edition, Benjamin/Cummings 1970)
Mathematical Methods for Physics and Engineering, K.F. Riley, M.P. Hobson and S.J. Bence (Cambridge University Press, 1998)
Michaelmas term: 11th and
25th November, W 2.15-4.15, Arts School Room A
Lent term: 10th March, M 2.15-4.15,
Arts School Room A
Easter term: 28th April and
13th May, W 2.15-4.15, Arts School Room A
These classes are designed to help you develop mathematical skills in solving problems, and so strengthen your knowledge of key material from IB coursework. This is valuable regardless of whether you intend to pursue theoretical topics in Part II (for which attendance at these courses is assumed). Although primarily intended for those taking Mathematics as well as Advanced Physics in IB, most of of the classes have also been found helpful by those taking the Mathematical Concepts in Physics course.
Vectors and Vector Fields: Simplification of problems using vectors. Vector area. Divergence theorem and Stokes' theorem. Derivation of physical law from postulates.
Fourier Series and Eigenfunction Expansions: Eigenfunction analysis: the stretched string of changing length. Systems with few degrees of freedom: the problem with beer. Fourier analysis: the diffusion equation.
Waves and Fourier Transforms: Fourier transforms: a problem of diffusion. Propagation of a wavepacket with dispersion.
Spherical Harmonics in Electromagnetism and Fluid Mechanics: Electrostatics; expansion in multipoles. Polarization of a sphere. Field inside a spherical capacitor. Fluid Mechanics; motion of a sphere under water.
Introduction to Tensors: Tensors in electrostatics. Suffix notation, summation convention. The outer product. Use of principal axes. The force on a dipole in a field.
The Structure of Electromagnetic Theory: Charges and currents in vacuo. Polarisation and Magnetisation. Fields and their energy.
Variational Calculus in Physics: Functionals. Euler's equation. Analytical dynamics. Method of undetermined multipliers. The mathematical shepherd.
Further Applications of Vector and Tensor Calculus: The antisymmetric tensor. Rigid body rotations, principal axes, precession. Stress and strain tensors. Transformation laws.
Mathematical Methods Applied to Quantum Mechanics. Methods of
earlier sheets illustrated in the context of Quantum Mechanics - Fourier
Series, quantum waves and their dispersion, variational basis of Schrödinger's
equation and the connection with the eigenvalue problem.
