Mathematics for Physicists

Brian Martin was a full-time member of staff of the Department of Physics & Astronomy at UCL from 1968 to 1995, including a decade from 1994 to 2004 as Head of the Department. I retired in 2005 and now hold the title of Emeritus Professor of Physics. I have extensive experience of teaching undergraduate mathematics classes at all levels and experience of other universities via external examining for first degrees at Imperial College and Royal Holloway College London. I was also the external member of the General Board of the Department of Physics at Cambridge University that reviewed the whole academic programme of that department, including teaching.Graham Shaw is a full-time member of staff of the School of Physics & Astronomy at Manchester University and will retire in September 2009. I have extensive experience of teaching undergraduate physics and the associated mathematics, and have been a member of the department’s Teaching Committee and the Course Director of the Honours School of Mathematics and Physics for many years.

• Editors’ preface to the Manchester Physics Series xi Authors’ preface xiiiNotes and website information xv1 Real numbers, variables and functions 11.1 Real numbers 11.1.1 Rules of arithmetic: rational and irrational numbers 11.1.2 Factors, powers and rationalisation 41.1.3 Number systems 61.2 Real variables 91.2.1 Rules of elementary algebra 91.2.2 Proof of the irrationality of 2 111.2.3 Formulas, identities and equations 111.2.4 The binomial theorem 131.2.5 Absolute values and inequalities 171.3 Functions, graphs and co-ordinates 201.3.1 Functions 201.3.2 Cartesian co-ordinates 23Problems 1 282 Some basic functions and equations 312.1 Algebraic functions 312.1.1 Polynomials 312.1.2 Rational functions and partial fractions 372.1.3 Algebraic and transcendental functions 412.2 Trigonometric functions 412.2.1 Angles and polar co-ordinates 412.2.2 Sine and cosine 442.2.3 More trigonometric functions 462.2.4 Trigonometric identities and equations 482.2.5 Sine and cosine rules 512.3 Logarithms and exponentials 532.3.1 The laws of logarithms 542.3.2 Exponential function 562.3.3 Hyperbolic functions 602.4 Conic sections 63Problems 2 683 Differential calculus 713.1 Limits and continuity 713.1.1 Limits 713.1.2 Continuity 753.2 Differentiation 773.2.1 Differentiability 783.2.2 Some standard derivatives 803.3 General methods 823.3.1 Product rule 833.3.2 Quotient rule 833.3.3 Reciprocal relation 843.3.4 Chain rule 863.3.5 More standard derivatives 873.3.6 Implicit functions 893.4 Higher derivatives and stationary points 903.4.1 Stationary points 923.5 Curve sketching 95Problems 3 984 Integral calculus 1014.1 Indefinite integrals 1014.2 Definite integrals 1044.2.1 Integrals and areas 1054.2.2 Riemann integration 1084.3 Change of variables and substitutions 1114.3.1 Change of variables 1114.3.2 Products of sines and cosines 1134.3.3 Logarithmic integration 1154.3.4 Partial fractions 1164.3.5 More standard integrals 1174.3.6 Tangent substitutions 1184.3.7 Symmetric and antisymmetric integrals 1194.4 Integration by parts 1204.5 Numerical integration 1234.6 Improper integrals 1264.6.1 Infinite integrals 1264.6.2 Singular integrals 1294.7 Applications of integration 1324.7.1 Work done by a varying force 1324.7.2 The length of a curve 1334.7.3 Surfaces and volumes of revolution 1344.7.4 Moments of inertia 136Problems 4 1375 Series and expansions 1435.1 Series 1435.2 Convergence of infinite series 1465.3 Taylor’s theorem and its applications 1495.3.1 Taylor’s theorem 1495.3.2 Small changes and l’Hˆopital’s rule 1505.3.3 Newton’s method 1525.3.4 Approximation errors: Euler’s number 1535.4 Series expansions 1535.4.1 Taylor and Maclaurin series 1545.4.2 Operations with series 1575.5 Proof of d’Alembert’s ratio test 1615.5.1 Positive series 1615.5.2 General series 1625.6 Alternating and other series 163Problems 5 1656 Complex numbers and variables 1696.1 Complex numbers 1696.2 Complex plane: Argand diagrams 1726.3 Complex variables and series 1766.3.1 Proof of the ratio test for complex series 1796.4 Euler’s formula 1806.4.1 Powers and roots 1826.4.2 Exponentials and logarithms 1846.4.3 De Moivre’s theorem 1856.4.4 Summation of series and evaluation of integrals 187Problems 6 1897 Partial differentiation 1917.1 Partial derivatives 1917.2 Differentials 1937.2.1 Two standard results 1957.2.2 Exact differentials 1977.2.3 The chain rule 1987.2.4 Homogeneous functions and Euler’s theorem 1997.3 Change of variables 2007.4 Taylor series 2037.5 Stationary points 206*7.6 Lagrange multipliers 2097.7 Differentiation of integrals 211Problems 7 2148 Vectors 2198.1 Scalars and vectors 2198.1.1 Vector algebra 2208.1.2 Components of vectors: Cartesian co-ordinates 2218.2 Products of vectors 2258.2.1 Scalar product 2258.2.2 Vector product 2288.2.3 Triple products 2318.2.4 Reciprocal vectors 2368.3 Applications to geometry 2388.3.1 Straight lines 2388.3.2 Planes 2418.4 Differentiation and integration 243Problems 8 2469 Determinants, Vectors and Matrices 2499.1 Determinants 2499.1.1 General properties of determinants 2539.1.2 Homogeneous linear equations 2579.2 Vectors in n Dimensions 2609.2.1 Basis vectors 2619.2.2 Scalar products 2639.3 Matrices and linear transformations 2659.3.1 Matrices 2659.3.2 Linear transformations 2709.3.3 Transpose, complex, and Hermitian conjugates 2739.4 Square Matrices 2749.4.1 Some special square matrices 2749.4.2 The determinant of a matrix 2769.4.3 Matrix inversion 2789.4.4 Inhomogeneous simultaneous linear equations 282Problems 9 28410 Eigenvalues and eigenvectors 29110.1 The eigenvalue equation 29110.1.1 Properties of eigenvalues 29310.1.2 Properties of eigenvectors 29610.1.3 Hermitian matrices 29910.2 Diagonalisation of matrices 30210.2.1 Normal modes of oscillation 30510.2.2 Quadratic forms 308Problems 10 31211 Line and multiple integrals 31511.1 Line integrals 31511.1.1 Line integrals in a plane 31511.1.2 Integrals around closed contours and along arcs 31911.1.3 Line integrals in three dimensions 32111.2 Double integrals 32311.2.1 Green’s theorem in the plane and perfect differentials 32611.2.2 Other co-ordinate systems and change of variables 33011.3 Curvilinear co-ordinates in three dimensions 33311.3.1 Cylindrical and spherical polar co-ordinates 33411.4 Triple or volume integrals 33711.4.1 Change of variables 338Problems 11 34012 Vector calculus 34512.1 Scalar and vector fields 34512.1.1 Gradient of a scalar field 34612.1.2 Div, grad and curl 34912.1.3 Orthogonal curvilinear co-ordinates 35212.2 Line, surface, and volume integrals 35512.2.1 Line integrals 35512.2.2 Conservative fields and potentials 35912.2.3 Surface integrals 36212.2.4 Volume integrals: moments of inertia 36712.3 The divergence theorem 36812.3.1 Proof of the divergence theorem and Green’s identities 36912.3.2 Divergence in orthogonal curvilinear co-ordinates 37212.3.3 Poisson’s equation and Gauss’ theorem 37312.3.4 The continuity equation 37612.4 Stokes’ theorem 37712.4.1 Proof of Stokes’ theorem 37812.4.2 Curl in curvilinear co-ordinates 38012.4.3 Applications to electromagnetic fields 381Problems 12 38413 Fourier analysis 38913.1 Fourier series 38913.1.1 Fourier coefficients 39013.1.2 Convergence 39413.1.3 Change of period 39813.1.4 Non-periodic functions 39913.1.5 Integration and differentiation of Fourier series 40113.1.6 Mean values and Parseval’s theorem 40513.2 Complex Fourier series 40713.2.1 Fourier expansions and vector spaces 40913.3 Fourier transforms 41013.3.1 Properties of Fourier transforms 41413.3.2 The Dirac delta function 41913.3.3 The convolution theorem 423Problems 13 42614 Ordinary differential equations 43114.1 First-order equations 43314.1.1 Direct integration 43314.1.2 Separation of variables 43414.1.3 Homogeneous equations 43514.1.4 Exact equations 43814.1.5 First-order linear equations 44014.2 Linear ODEs with constant coefficients 44114.2.1 Complementary functions 44214.2.2 Particular integrals: method of undetermined coefficients 44614.2.3 Particular integrals: the D-operator method 44814.2.4 Laplace transforms 45314.3 Euler’s equation 459Problems 14 46115 Series solutions of ordinary differential equations 46515.1 Series solutions 46515.1.1 Series solutions about a regular point 46715.1.2 Series solutions about a regular singularity: Frobenius method 46915.1.3 Polynomial solutions 47515.2 Eigenvalue equations 47815.3 Legendre’s equation 48115.3.1 Legendre functions and Legendre polynomials 48215.3.2 The generating function 48715.3.3 Associated Legendre equation 49015.3.4 Rodrigues’ formula 49215.4 Bessel’s equation 49415.4.1 Bessel functions 49515.4.2 Properties of non-singular Bessel functions Jν (x) 499Problems 15 50216 Partial differential equations 50716.1 Some important PDEs in physics 51016.2 Separation of variables: Cartesian co-ordinates 51116.2.1 The wave equation in one spatial dimension 51216.2.2 The wave equation in three spatial dimensions 51516.2.3 The diffusion equation in one spatial dimension 51816.3 Separation of variables: polar co-ordinates 52016.3.1 Plane-polar co-ordinates 52016.3.2 Spherical polar co-ordinates 52416.3.3 Cylindrical polar co-ordinates 52916.4 The wave equation: d’Alembert’s solution 53216.5 Euler equations 53516.6 Boundary conditions and uniqueness 53816.6.1 Laplace transforms 540Problems 16 544Answers to selected problems 549Index 559