Physical Mathematics 2010 Dr Peter Boyle School of Physics and Astronomy (JCMB 4407) [email protected] This version: September 23, 2010 Abstract These are the lecture notes to accompany the Physical Mathematics lecture course. They are a modified version of the previous years notes which written by Dr Alisdair Hart. Contents 1 Organisation 3 1.1 Books . 3 1.2 On the web . 3 1.3 Workshops . 3 2 The wave equation 4 2.1 Separation of variables . 4 2.2 Solving the ODE’s . 5 2.3 Boundary and initial conditions . 5 2.3.1 BCs & orthogonality of normal modes . 6 2.3.2 Completeness of normal modes . 6 2.3.3 Boundary conditions (in x)............................ 7 2.3.4 Normal mode decomposition . 8 2.3.5 Initial conditions (in t).............................. 8 2.3.6 A quick example . 9 3 Fourier Series 10 3.1 Overview . 10 3.2 The Fourier expansion . 10 3.3 Orthogonality . 11 3.4 Calculating the Fourier coefficients . 11 3.5 Even and odd expansions . 11 3.5.1 Expanding an even function . 12 3.5.2 Expanding an odd function . 12 3.6 Periodic extension, or what happens outside the range? . 12 3.7 Complex Fourier Series . 14 3.7.1 Example . 15 3.8 Comparing real and complex Fourier expansions . 16 3.9 Differentiation and integration . 16 3.10 Gibbs phenomenon . 17 1 4 Functions as vectors 18 4.1 Linear spaces . 18 4.1.1 Gramm-Schmidt orthogonalisation . 18 4.1.2 Matrix mechanics . 18 4.2 Scalar product of functions . 18 4.2.1 Scalar product → integral . 19 4.3 Inner products, orthogonality, orthonormality and Fourier series . 19 4.3.1 Example: complex Fourier series . 20 4.3.2 Normalised basis functions . 21 5 Fourier transforms 22 5.0.3 Relation to Fourier Series . 22 5.0.4 FT conventions . 22 5.1 Some simple examples of FTs . 22 5.1.1 The top-hat . 22 5.2 The Gaussian . 24 5.2.1 Gaussian integral . 24 5.2.2 Fourier transform of Gaussian . 25 5.3 Reciprocal relations between a function and its F.T . 26 5.4 FTs as generalised Fourier series . 27 5.5 Differentiating Fourier transforms . 28 5.6 The Dirac delta function . 28 5.6.1 Definition . 28 5.6.2 Sifting property . 29 5.6.3 The delta function as a limiting case . 29 5.6.4 Proving the sifting property . 29 5.6.5 Some other properties of the Dirac function . 30 5.6.6 FT and integral representation of δ(x) ..................... 30 5.7 Convolution . 31 5.7.1 Examples of convolution . 32 5.8 The convolution theorem . 32 5.8.1 Examples of the convolution theorem . 33 5.9 Physical applications of Fourier transforms . 33 5.9.1 Far-field optical diffraction patterns . 33 6 Vectors (review) 35 6.1 Scalars and Vectors . 35 6.2 Why vector notation . 35 6.3 Scalar product . 35 6.4 Cartesian coordinates . 35 6.4.1 Example: parallel and perpendicular pieces . 36 6.4.2 Example: uniform gravitational field . 36 6.4.3 Example: Equation of a plane . 36 6.4.4 Example: Plane waves . 37 6.5 Cross product . 38 2 7 Vector calculus 39 7.1 Volume, Surface, and Line integrals . 39 7.1.1 Volume integral . 39 7.2 Line integrals in physics . 40 7.2.1 Example: work done . 41 7.2.2 Strategy for evaluating line integrals . 41 7.2.3 Example: Chief Brody’s shark cage . 42 7.2.4 Example: Line integral . 43 7.3 Surface integrals . 44 7.3.1 Strategy for evaluating surface integrals . 44 7.3.2 Example: surface integral over cube . 45 7.4 Partial derivatives . 45 7.5 Gradient operator . 46 7.5.1 Example: Electrostatic field . 46 7.5.2 Example: Gravitational field . 47 7.6 Divergence of a vector function . 47 7.6.1 Example: Oil from a well . 47 7.6.2 Vector identities . 48 7.7 Integral theorems . 48 7.7.1 Divergence theorem . 48 7.8 Curl of a vector function . 49 7.8.1 Stoke’s theorem . 51 7.9 Conservatism, irrotationalism and the scalar potential . 52 7.10 Laplacian . 52 7.11 Physical applications examples . 53 7.11.1 Action on a plane wave . 53 7.11.2 Gaussian pill boxes . 53 7.11.3 Point charges & δ-functions . 54 8 PDE’s 57 8.1 Fourier Transforms in 2D and 3D ............................ 58 8.1.1 Derivatives of Fourier Transforms . 58 8.2 Solving PDE’s by Fourier transform . 59 8.3 Relation to Green function solution . 60 9 Curvilinear coordinate systems 61 9.1 Circular (or plane) polar coordinates . 61 9.1.1 Area integrals . 62 9.2 Cylindrical polar coordinates . 63 9.3 Spherical polar coordinates . 63 9.4 Differential operators . 65 9.4.1 Gradient . ..