Mathematics for Physicists: (Manchester Physics Series)

Mathematics for Physicists is a relatively short volume covering all the essential mathematics needed for a typical first degree in physics, from a starting point that is compatible with modern school mathematics syllabuses. Early chapters deliberately overlap with senior school mathematics, to a degree that will depend on the background of the individual reader, who may quickly skip over those topics with which he or she is already familiar. The rest of the book covers the mathematics that is usually compulsory for all students in their first two years of a typical university physics degree, plus a little more. There are worked examples throughout the text, and chapter-end problem sets. Mathematics for Physicists features: Interfaces with modern school mathematics syllabuses All topics usually taught in the first two years of a physics degree Worked examples throughout Problems in every chapter, with answers to selected questions at the end of the book and full solutions on a website This text will be an excellent resource for undergraduate students in physics and a quick reference guide for more advanced students, as well as being appropriate for students in other physical sciences, such as astronomy, chemistry and earth sciences.

Table of Contents:
Editors’ preface to the Manchester Physics Series xi Authors’ preface xiii Notes and website information xv 1 Real numbers, variables and functions 1 1.1 Real numbers 1 1.1.1 Rules of arithmetic: rational and irrational numbers 1 1.1.2 Factors, powers and rationalisation 4 1.1.3 Number systems 6 1.2 Real variables 9 1.2.1 Rules of elementary algebra 9 1.2.2 Proof of the irrationality of 2 11 1.2.3 Formulas, identities and equations 11 1.2.4 The binomial theorem 13 1.2.5 Absolute values and inequalities 17 1.3 Functions, graphs and co-ordinates 20 1.3.1 Functions 20 1.3.2 Cartesian co-ordinates 23 Problems 1 28 2 Some basic functions and equations 31 2.1 Algebraic functions 31 2.1.1 Polynomials 31 2.1.2 Rational functions and partial fractions 37 2.1.3 Algebraic and transcendental functions 41 2.2 Trigonometric functions 41 2.2.1 Angles and polar co-ordinates 41 2.2.2 Sine and cosine 44 2.2.3 More trigonometric functions 46 2.2.4 Trigonometric identities and equations 48 2.2.5 Sine and cosine rules 51 2.3 Logarithms and exponentials 53 2.3.1 The laws of logarithms 54 2.3.2 Exponential function 56 2.3.3 Hyperbolic functions 60 2.4 Conic sections 63 Problems 2 68 3 Differential calculus 71 3.1 Limits and continuity 71 3.1.1 Limits 71 3.1.2 Continuity 75 3.2 Differentiation 77 3.2.1 Differentiability 78 3.2.2 Some standard derivatives 80 3.3 General methods 82 3.3.1 Product rule 83 3.3.2 Quotient rule 83 3.3.3 Reciprocal relation 84 3.3.4 Chain rule 86 3.3.5 More standard derivatives 87 3.3.6 Implicit functions 89 3.4 Higher derivatives and stationary points 90 3.4.1 Stationary points 92 3.5 Curve sketching 95 Problems 3 98 4 Integral calculus 101 4.1 Indefinite integrals 101 4.2 Definite integrals 104 4.2.1 Integrals and areas 105 4.2.2 Riemann integration 108 4.3 Change of variables and substitutions 111 4.3.1 Change of variables 111 4.3.2 Products of sines and cosines 113 4.3.3 Logarithmic integration 115 4.3.4 Partial fractions 116 4.3.5 More standard integrals 117 4.3.6 Tangent substitutions 118 4.3.7 Symmetric and antisymmetric integrals 119 4.4 Integration by parts 120 4.5 Numerical integration 123 4.6 Improper integrals 126 4.6.1 Infinite integrals 126 4.6.2 Singular integrals 129 4.7 Applications of integration 132 4.7.1 Work done by a varying force 132 4.7.2 The length of a curve 133 4.7.3 Surfaces and volumes of revolution 134 4.7.4 Moments of inertia 136 Problems 4 137 5 Series and expansions 143 5.1 Series 143 5.2 Convergence of infinite series 146 5.3 Taylor’s theorem and its applications 149 5.3.1 Taylor’s theorem 149 5.3.2 Small changes and l’Hˆopital’s rule 150 5.3.3 Newton’s method 152 5.3.4 Approximation errors: Euler’s number 153 5.4 Series expansions 153 5.4.1 Taylor and Maclaurin series 154 5.4.2 Operations with series 157 5.5 Proof of d’Alembert’s ratio test 161 5.5.1 Positive series 161 5.5.2 General series 162 5.6 Alternating and other series 163 Problems 5 165 6 Complex numbers and variables 169 6.1 Complex numbers 169 6.2 Complex plane: Argand diagrams 172 6.3 Complex variables and series 176 6.3.1 Proof of the ratio test for complex series 179 6.4 Euler’s formula 180 6.4.1 Powers and roots 182 6.4.2 Exponentials and logarithms 184 6.4.3 De Moivre’s theorem 185 6.4.4 Summation of series and evaluation of integrals 187 Problems 6 189 7 Partial differentiation 191 7.1 Partial derivatives 191 7.2 Differentials 193 7.2.1 Two standard results 195 7.2.2 Exact differentials 197 7.2.3 The chain rule 198 7.2.4 Homogeneous functions and Euler’s theorem 199 7.3 Change of variables 200 7.4 Taylor series 203 7.5 Stationary points 206 *7.6 Lagrange multipliers 209 7.7 Differentiation of integrals 211 Problems 7 214 8 Vectors 219 8.1 Scalars and vectors 219 8.1.1 Vector algebra 220 8.1.2 Components of vectors: Cartesian co-ordinates 221 8.2 Products of vectors 225 8.2.1 Scalar product 225 8.2.2 Vector product 228 8.2.3 Triple products 231 8.2.4 Reciprocal vectors 236 8.3 Applications to geometry 238 8.3.1 Straight lines 238 8.3.2 Planes 241 8.4 Differentiation and integration 243 Problems 8 246 9 Determinants, Vectors and Matrices 249 9.1 Determinants 249 9.1.1 General properties of determinants 253 9.1.2 Homogeneous linear equations 257 9.2 Vectors in n Dimensions 260 9.2.1 Basis vectors 261 9.2.2 Scalar products 263 9.3 Matrices and linear transformations 265 9.3.1 Matrices 265 9.3.2 Linear transformations 270 9.3.3 Transpose, complex, and Hermitian conjugates 273 9.4 Square Matrices 274 9.4.1 Some special square matrices 274 9.4.2 The determinant of a matrix 276 9.4.3 Matrix inversion 278 9.4.4 Inhomogeneous simultaneous linear equations 282 Problems 9 284 10 Eigenvalues and eigenvectors 291 10.1 The eigenvalue equation 291 10.1.1 Properties of eigenvalues 293 10.1.2 Properties of eigenvectors 296 10.1.3 Hermitian matrices 299 10.2 Diagonalisation of matrices 302 10.2.1 Normal modes of oscillation 305 10.2.2 Quadratic forms 308 Problems 10 312 11 Line and multiple integrals 315 11.1 Line integrals 315 11.1.1 Line integrals in a plane 315 11.1.2 Integrals around closed contours and along arcs 319 11.1.3 Line integrals in three dimensions 321 11.2 Double integrals 323 11.2.1 Green’s theorem in the plane and perfect differentials 326 11.2.2 Other co-ordinate systems and change of variables 330 11.3 Curvilinear co-ordinates in three dimensions 333 11.3.1 Cylindrical and spherical polar co-ordinates 334 11.4 Triple or volume integrals 337 11.4.1 Change of variables 338 Problems 11 340 12 Vector calculus 345 12.1 Scalar and vector fields 345 12.1.1 Gradient of a scalar field 346 12.1.2 Div, grad and curl 349 12.1.3 Orthogonal curvilinear co-ordinates 352 12.2 Line, surface, and volume integrals 355 12.2.1 Line integrals 355 12.2.2 Conservative fields and potentials 359 12.2.3 Surface integrals 362 12.2.4 Volume integrals: moments of inertia 367 12.3 The divergence theorem 368 12.3.1 Proof of the divergence theorem and Green’s identities 369 12.3.2 Divergence in orthogonal curvilinear co-ordinates 372 12.3.3 Poisson’s equation and Gauss’ theorem 373 12.3.4 The continuity equation 376 12.4 Stokes’ theorem 377 12.4.1 Proof of Stokes’ theorem 378 12.4.2 Curl in curvilinear co-ordinates 380 12.4.3 Applications to electromagnetic fields 381 Problems 12 384 13 Fourier analysis 389 13.1 Fourier series 389 13.1.1 Fourier coefficients 390 13.1.2 Convergence 394 13.1.3 Change of period 398 13.1.4 Non-periodic functions 399 13.1.5 Integration and differentiation of Fourier series 401 13.1.6 Mean values and Parseval’s theorem 405 13.2 Complex Fourier series 407 13.2.1 Fourier expansions and vector spaces 409 13.3 Fourier transforms 410 13.3.1 Properties of Fourier transforms 414 13.3.2 The Dirac delta function 419 13.3.3 The convolution theorem 423 Problems 13 426 14 Ordinary differential equations 431 14.1 First-order equations 433 14.1.1 Direct integration 433 14.1.2 Separation of variables 434 14.1.3 Homogeneous equations 435 14.1.4 Exact equations 438 14.1.5 First-order linear equations 440 14.2 Linear ODEs with constant coefficients 441 14.2.1 Complementary functions 442 14.2.2 Particular integrals: method of undetermined coefficients 446 14.2.3 Particular integrals: the D-operator method 448 14.2.4 Laplace transforms 453 14.3 Euler’s equation 459 Problems 14 461 15 Series solutions of ordinary differential equations 465 15.1 Series solutions 465 15.1.1 Series solutions about a regular point 467 15.1.2 Series solutions about a regular singularity: Frobenius method 469 15.1.3 Polynomial solutions 475 15.2 Eigenvalue equations 478 15.3 Legendre’s equation 481 15.3.1 Legendre functions and Legendre polynomials 482 15.3.2 The generating function 487 15.3.3 Associated Legendre equation 490 15.3.4 Rodrigues’ formula 492 15.4 Bessel’s equation 494 15.4.1 Bessel functions 495 15.4.2 Properties of non-singular Bessel functions Jν (x) 499 Problems 15 502 16 Partial differential equations 507 16.1 Some important PDEs in physics 510 16.2 Separation of variables: Cartesian co-ordinates 511 16.2.1 The wave equation in one spatial dimension 512 16.2.2 The wave equation in three spatial dimensions 515 16.2.3 The diffusion equation in one spatial dimension 518 16.3 Separation of variables: polar co-ordinates 520 16.3.1 Plane-polar co-ordinates 520 16.3.2 Spherical polar co-ordinates 524 16.3.3 Cylindrical polar co-ordinates 529 16.4 The wave equation: d’Alembert’s solution 532 16.5 Euler equations 535 16.6 Boundary conditions and uniqueness 538 16.6.1 Laplace transforms 540 Problems 16 544 Answers to selected problems 549 Index 559

About the Author :
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.