Prerequisites
This course is offered to students taking either or both of Physics A and Physics B, but who are not taking 'Mathematics' in NST Part IB.
This course requires the material covered in the NST Part IA Mathematics and Physics for Natural Scientists courses, and uses examples showing how the mathematical methods introduced can be utilised in a range of physical problems. Fluency with integration of the tools from NST Part IA Mathematics with the topics taught in NST Part IA Physics is required.
Learning Outcomes and Assessment
In conjunction with the material from 'Mathematics' in NST Part IA, this course provides the mathematics required for Physics A and Physics B in NST Part IB, and the core and option lecture courses in Part II Physics. However, some of the topics required in Part II Physics TP1 and TP2 are not covered here.
Synopsis
Vector and Scalar fields in Cartesian coordinates: Basic definitions of scalar and vector fields. Line, surface, and volume integrals. Grad, Div and Curl. \( \nabla\) and Laplacian operators. Divergence Theorem, Stokes' Theorem, and Green's Theorem. Conservative fields. Maxwell's equations as example of vector differential operators.
Cylindrical, Spherical, and Curvilinear coordinate systems: Basic definitions of cylindrical and spherical coordinate systems. Application to scalar and vector fields. Curvilinear coordinate systems. Vector differential operators in cylindrical, spherical, and curvilinear coordinate systems.
Variational Principles: Lagrange multipliers. Euler–Lagrange equation.
Fourier Series: Fourier series of periodic functions using trigonometric functions. Discontinuities and Gibbs phenomenon. Even and odd functions. Fourier series in complex form. Solving one-dimensional differential equations using Fourier series. Notions of completeness and orthogonality.
Fourier Transforms: Definition. Symmetry considerations. Fourier transforms of differentials. The Dirac delta function. Convolution. Green's functions. Parseval's theorem.
Differential Equations: Laplace’s equation, Poisson's equation, the diffusion equation, the wave equation, Helmholtz equation, Schrödinger's equation. Separation of variables in Cartesian, cylindrical polar, and spherical polar coordinate systems. Summary of common differential equations and orthogonal functions. Examples, including Bessel, Legendre, Hermite functions etc. Analogy between function expansions and geometrical vector expansions: orthogonality and completeness. Convergence of power series. Power series expansions and solution of ordinary differential equations. Legendre polynomials, Bessel functions, Hermite polynomials and Spherical Harmonics illustrated by examples. Brief summary of Sturm–Liouville theory.
Matrices and Tensors: Basic matrix algebra. Determinants. Special matrix types, including Hermitian matrices. Eigenvalues, eigenvectors and diagonalization. Basic concept of a tensor. Summation convention: Kronecker delta and Levi–Civita symbol.
BOOKS
Mathematical Methods in the Physical Sciences, Boas M L (3rd edn, Wiley 2006)
Mathematical Methods for Physics and Engineering, Riley K F, Hobson M P and Bence S J (3rd edn, CUP 2006)