Ebook Mathematics for physics A Guided Tour for Graduate Students. An engagingly written account of mathematical tools and ideas, this book provides a graduate-level introduction to the mathematics used in research in physics. The first half of the book focuses on the traditional mathematical methods of physics: differential and integral equations, Fourier series and the calculus of variations. The second half contains an introduction to more advanced subjects, including differential geometry, topology and complex variables.
This page intentionally left blank Mathematics for Physics A Guided Tour for Graduate Students An engagingly written account of mathematical tools and ideas, this book provides a graduate-level introduction to the mathematics used in research in physics The first half of the book focuses on the traditional mathematical methods of physics: differential and integral equations, Fourier series and the calculus of variations The second half contains an introduction to more advanced subjects, including differential geometry, topology and complex variables The authors’ exposition avoids excess rigour whilst explaining subtle but important points often glossed over in more elementary texts The topics are illustrated at every stage by carefully chosen examples, exercises and problems drawn from realistic physics settings These make it useful both as a textbook in advanced courses and for self-study Password-protected solutions to the exercises are available to instructors at www.cambridge.org/9780521854030 michael st o n e is a Professor in the Department of Physics at the University of Illinois at Urbana-Champaign He has worked on quantum field theory, superconductivity, the quantum Hall effect and quantum computing paul goldb a r t is a Professor in the Department of Physics at the University of Illinois at Urbana-Champaign, where he directs the Institute for Condensed Matter Theory His research ranges widely over the field of condensed matter physics, including soft matter, disordered systems, nanoscience and superconductivity MATHEMATICS FOR PHYSICS A Guided Tour for Graduate Students MICHAEL STONE University of Illinois at Urbana-Champaign and PAUL GOLDBART University of Illinois at Urbana-Champaign CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521854030 © M Stone and P Goldbart 2009 This publication is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 2009 ISBN-13 978-0-511-59516-5 eBook (EBL) ISBN-13 978-0-521-85403-0 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate To the memory of Mike’s mother, Aileen Stone: × = 81 To Paul’s mother and father, Carole and Colin Goldbart Contents Preface Acknowledgments page xi xiii Calculus of variations 1.1 What is it good for? 1.2 Functionals 1.3 Lagrangian mechanics 1.4 Variable endpoints 1.5 Lagrange multipliers 1.6 Maximum or minimum? 1.7 Further exercises and problems 1 10 27 32 36 38 Function spaces 2.1 Motivation 2.2 Norms and inner products 2.3 Linear operators and distributions 2.4 Further exercises and problems 50 50 51 66 76 Linear ordinary differential equations 3.1 Existence and uniqueness of solutions 3.2 Normal form 3.3 Inhomogeneous equations 3.4 Singular points 3.5 Further exercises and problems 86 86 93 94 97 98 Linear differential operators 4.1 Formal vs concrete operators 4.2 The adjoint operator 4.3 Completeness of eigenfunctions 4.4 Further exercises and problems 101 101 104 117 132 Green functions 5.1 Inhomogeneous linear equations 5.2 Constructing Green functions 140 140 141 vii viii 5.3 5.4 5.5 5.6 5.7 Contents Applications of Lagrange’s identity Eigenfunction expansions Analytic properties of Green functions Locality and the Gelfand–Dikii equation Further exercises and problems 150 153 155 165 167 Partial differential equations 6.1 Classification of PDEs 6.2 Cauchy data 6.3 Wave equation 6.4 Heat equation 6.5 Potential theory 6.6 Further exercises and problems 174 174 176 181 196 201 224 The mathematics of real waves 7.1 Dispersive waves 7.2 Making waves 7.3 Nonlinear waves 7.4 Solitons 7.5 Further exercises and problems 231 231 242 246 255 260 Special functions 8.1 Curvilinear coordinates 8.2 Spherical harmonics 8.3 Bessel functions 8.4 Singular endpoints 8.5 Further exercises and problems 264 264 270 278 298 305 Integral equations 9.1 Illustrations 9.2 Classification of integral equations 9.3 Integral transforms 9.4 Separable kernels 9.5 Singular integral equations 9.6 Wiener–Hopf equations I 9.7 Some functional analysis 9.8 Series solutions 9.9 Further exercises and problems 311 311 312 313 321 323 327 332 338 342 10 Vectors and tensors 10.1 Covariant and contravariant vectors 10.2 Tensors 10.3 Cartesian tensors 10.4 Further exercises and problems 347 347 350 362 372 792 Appendix B Exercise B.6: By taking a suitable limit in Exercise B.3 show that, when acting on ∞ smooth functions f such that −∞ |f | dx is finite, we have H(Hf ) = −f , where ∞ P π (Hf )(x) = −∞ f (x ) dx x−x defines the Hilbert transform of a function on the real line (Because H gives zero when ∞ acting on a constant, some condition, such as −∞ |f | dx being finite, is necessary if H is to be invertible.) B.4 The Poisson summation formula Suppose that f (x) is a smooth function that tends rapidly to zero at infinity Then the series ∞ f (x + nL) F(x) = (B.56) n=−∞ converges to a smooth function of period L It therefore has a Fourier expansion ∞ am e−2πimx/L F(x) = (B.57) m=−∞ We can compute the Fourier coefficients am by integrating term-by-term am = = L L F(x) e2π imx/L dx ∞ L n=−∞ ∞ L f (x + nL) e2πimx/L dx = L = f (2πm/L) L −∞ f (x) e2πimx/L dx (B.58) Thus ∞ ∞ f (x + nL) = n=−∞ f (2π m/L)e−2πimx/L L m=−∞ (B.59) When we set x = 0, this last equation becomes ∞ ∞ f (2π m/L) f (nL) = L n=−∞ m=−∞ (B.60) Fourier series and integrals 793 The equality of this pair of doubly infinite sums is known as the Poisson summation formula Example: As the Fourier transform of a Gaussian is another Gaussian, the Poisson formula with L = applied to f (x) = exp(−κx2 ) gives ∞ e−κm = m=−∞ ∞ π κ e−m π /κ , (B.61) m=−∞ and (rather more usefully) applied to exp(− 12 tx2 + ixθ) gives ∞ +inθ e− tn = n=−∞ ∞ 2π t e− 2t (θ+2πn) (B.62) n=−∞ The last identity is known as Jacobi’s imaginary transformation It reflects the equivalence of the eigenmode expansion and the method-of-images solution of the diffusion equation ∂ 2ϕ ∂ϕ = ∂t ∂x (B.63) on the unit circle Notice that when t is small the sum on the right-hand side converges very slowly, whereas the sum on the left converges very rapidly The opposite is true for large t The conversion of a slowly converging series into a rapidly converging one is a standard application of the Poisson summation formula It is the prototype of many duality maps that exchange a physical model with a large coupling constant for one with weak coupling If we take the limit t → in (B.62), the right-hand side approaches a sum of delta functions, and so gives us the useful identity 2π ∞ ∞ einx = n=−∞ δ(x + 2π n) (B.64) n=−∞ The right-hand side of (B.64) is sometimes called the “Dirac comb” Gauss sums The Poisson sum formula ∞ e−κm = m=−∞ π κ ∞ e−m π /κ (B.65) m=−∞ remains valid for complex κ, provided that Re κ > We can therefore consider the special case κ = iπ p + , q (B.66) 794 Appendix B where is a positive real number and p and q are positive integers whose product pq we assume to be even We investigate what happens to (B.65) as → The left-hand side of (B.65) can be decomposed into the double sum ∞ q−1 e−iπ(p/q)(r+mq) e− (r+mq)2 (B.67) m=−∞ r=0 Because pq is even, each term in e−iπ(p/q)(r+mq) is independent of m At the same time, the small limit of the infinite sum ∞ e− (r+mq)2 , (B.68) m=−∞ being a Riemann sum for the integral ∞ −∞ e− q2 m2 π q dm = , (B.69) becomes independent of r, and so a common factor of all terms in the finite sum over r If is small, we can make the replacement, κ −1 = − iπp/q − iπ p/q → 2 , 2 + π p /q π p /q (B.70) after which, the right-hand side contains the double sum ∞ p−1 eiπ(q/p)(r+mp) e− (q2 /p2 )(r+mp)2 (B.71) m=−∞ r=0 Again each term in eiπ(q/p)(r+mp) is independent of m, and ∞ e− (q2 /p2 )(r+mp)2 m=−∞ ∞ e− q2 m2 = e−iπ/4 q p → −∞ dm = q π (B.72) becomes independent of r Also lim →0 π κ (B.73) √ Thus, after cancelling the common factor of (1/q) π/ , we find that √ q q−1 e r=0 −iπ(p/q)r =e −iπ/4 √ p p−1 eiπ(q/p)r , r=0 pq even (B.74) Fourier series and integrals 795 This Poisson-summation-like equality of finite sums is known as the Landsberg–Schaar identity No purely algebraic proof is known Gauss considered the special case p = 2, in which case we get √ q q−1 e−2π ir /q r=0 = e−iπ/4 √ (1 + eiπq/2 ) (B.75) or, more explicitly q−1 e−2π ir /q r=0 ⎧ √ ⎪ (1 − i) q, ⎪ ⎪ ⎪ ⎨√q, = ⎪ 0, ⎪ ⎪ ⎪ ⎩ √ −i q, q = (mod 4), q = (mod 4), q = (mod 4), (B.76) q = (mod 4) The complex conjugate result is perhaps slightly prettier: q−1 e2π ir /q r=0 ⎧ √ ⎪ (1 + i) q, q = (mod 4), ⎪ ⎪ ⎪ √ ⎨ q, q = (mod 4), = ⎪ 0, q = (mod 4), ⎪ ⎪ ⎪ ⎩√ q = (mod 4) i q, (B.77) Gauss used these sums to prove the law of quadratic reciprocity Exercise B.7: By applying the Poisson summation formula to the Fourier transform pair f (x) = e− where |x| −ixθ e , and f (k) = 2 , + (k − θ)2 > 0, deduce that ∞ sinh = cosh − cos(θ − θ ) n=−∞ 2 + (θ − θ + 2π n)2 (B.78) Hence show that the Poisson kernel is equivalent to an infinite periodic sum of Lorentzians 2π − r2 − 2r cos(θ − θ ) + r =− π ∞ n=−∞ (ln r)2 ln r + (θ − θ + 2π n)2 References This is a list of books mentioned by name in the text They are not all in print, but should be found in any substantial physics library The publication data refer to the most recently available printings R Abrahams, J E Marsden, T Ratiu, Foundations of Mechanics (Benjamin Cummings, 2nd edition, 1978) V I Arnold, Mathematical Methods of Classical Mechanics (Springer Verlag, 2nd edition, 1997) J A de Azcárraga, J M Izquierdo, Lie groups, Lie Algebras, Cohomology and some Applications in Physics (Cambridge University Press, 1995) G Baym, Lectures on Quantum Mechanics (Addison Wesley, 1969) C M Bender, S A Orszag, Advanced Mathematical Methods for Scientists and Engineers (Springer Verlag, 1999) G Birkhoff, S MacLane, Survey of Modern Algebra (A K Peters, 1997) M Born, E Wolf, Principles of Optics (Cambridge University Press, 7th edition, 1999) A Erdélyi, W Magnus, F Oberhettinger, F G Tricomi, Higher Transcendental Functions (in vols), and Tables of Integral Transforms (2 vols), known collectively as “The Bateman Manuscript Project” (McGraw-Hill, 1953) F G Friedlander, M Joshi, Introduction to the Theory of Distributions (Cambridge University Press, 2nd edition, 1999) P R Halmos, Finite Dimensional Vector Spaces (Springer Verlag, 1993) J L Lagrange, Analytic Mechanics (English translation, Springer Verlag, 2001) L D Landau, E M Lifshitz, Quantum Mechanics (Non-relativistic Theory) (Pergamon Press, 1981) M J Lighthill, Generalized Functions (Cambridge University Press, 1960) M J Lighthill, Waves in Fluids (Cambridge University Press, 2002) J C Maxwell, A Treatise on Electricity and Magnetism (Dover reprint edition, 1954) C W Misner, K S Thorne, J A Wheeler, Gravitation (W H Freeman, 1973) P M Morse, H Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953) N I Muskhelishvili, Singular Integral Equations (Dover, reprint of 2nd edition, 1992) R G Newton, Scattering Theory of Waves and Particles (Dover, reprint of 2nd edition, 2002) A Perelomov, Generalized Coherent States and their Applications (Springer-Verlag, 1986) M Reed, B Simon, Methods of Modern Mathematical Physics (4 vols.) (Academic Press, 1978–80) L I Schiff, Quantum Mechanics (McGraw-Hill, 3rd edition, 1968) M Spivak, A Comprehensive Introduction to Differential Geometry (5 vols.) (Publish or Perish, 2nd edition, 1979) 797 798 References I Stackgold, Boundary Value Problems of Mathematical Physics, Vols I and II (SIAM, 2000) G N Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, 2nd edition, 1955) E T Whittaker, G N Watson, A Course of Modern Analysis (Cambridge University Press, 4th edition, 1996) H Weyl, The Classical Groups (Princeton University Press, reprint edition, 1997) Index p-chain, 457, 618 p-cycle, 618 p-form, 389 Abel’s equation, 315 Abel’s theorem, 633 action principle, 10, 25 addition theorem for elliptic functions, 735 for spherical harmonics, 278 adjoint boundary conditions, 110 dependence on inner product, 754 formal, 104 of linear map, 753 adjugate matrix, 763, 764 Airy’s equation, 729 algebraic geometry, 357 analytic function, 51 analytic signal, 684 angular momentum, 12, 14, 15 anti-derivation, 390 apodization, 790 approximation paraxial, 293 WKB, 285 associated Legendre functions, 276 atlas, 377 Atwood machine, 11 axial gauge, 22 axisymmetric potential, 273 Banach space, 55 Bargmann, Valentine, 622 Bargmann-Fock space, 107, 622 basis dual, 748 of vector space, 745 Bergman space, 622 Bergman, Stefan, 622 Bernoulli numbers, 689 Bernoulli’s equation, 25, 31, 232, 252 Bernoulli, Daniel, Bernoulli, Johan, Berry’s phase, 581 Bessel function cylindrical, 279 modified, 291 spherical, 294 Beta function, 708 Betti number, 449, 460, 654 Bianchi identity, 409 block diagonal, 775 Bochner Laplacian, 496 Bogomolnyi equation, 445 Borcherds, Richard, 741 Borel–Weil–Bott theorem, 583 boundary conditions adjoint, 110 Dirichlet, Neumann and Cauchy, 175, 612 free fluid surface, 31 homogeneous, 103 inhomogeneous, 103, 151 mixed, 290 natural, 28 self-adjoint, 111 singular, 298 brachistochrone, branch cut, 159, 650 branch point, 650 branching rules, 523, 570 Brouwer degree, 426, 489 bulk modulus, 365 bundle cotangent, 399 tangent, 376 trivial, 576 vector, 376 Caldeira–Leggett model, 146, 156, 189, 219 Calugareanu relation, 438 Cartan algebra, 565 Cartan, Élie, 379, 545 Casimir operator, 560 799 800 CAT scan, 318 catenary, 6, 35 Cauchy data, 176 sequence, 54 Cayley’s identity, 100 law of inertia, 774 theorem, 764 theorem for groups, 502 chain complex, 459 characteristic, 247 chart, 377 Christoffel symbols, 404 Cicero, Marcus Tullius, 423 closed form, 392, 399 set, 336 co-kernel, 755 coordinates cartesian, 362 Conformal, see coordinates, isothermal cylindrical, 266 generalized, 10 isothermal, 657 light-cone, 183 orthogonal curvilinear, 264 plane polar, 12, 265 spherical polar, 265 co-root vector, 567 cohomology, 454 commutator, 382 complementary function, 94, 152 space, 755 complete normed vector space, 54 orthonormal set, 60, 117, 120 set of functions, 56 set of states, 747 completeness condition, 78, 117, 123, 124, 130, 131 completion process, 58 complex algebraic curve, 654 complex differentiable, 607 complex projective space, 357, 428 components covariant and contravariant, 750 of matrix, 746 physical versus coordinate, 267 conjugate hermitian, see adjoint operator, 753 connection rule, 285 Index conservation law and symmetry, 14 global, 21 local, 21 constraint, 22, 33 holonomic versus anholonomic, 385 contour, 616 convergence in Lp , 54 in the mean, 52 of functions, 52 pointwise, 52 uniform, 52 convolution, 786 Cornu spiral, 223, 675 coset, 755 covector, 348, 748 critical mass, 296 cup product, 473 curl as a differential form, 391 in curvilinear coordinates, 268 cycloid, d’Alembert, Jean le Ronde, 10, 182 d’Angelo, John, 610 D-bar problem, 620 Darboux coordinates, 400, 402, 584 theorem, 400 de Rham’s theorem, 470 de Rham, Georges, 454 deficiency indices, 113 degree-genus relation, 655 dense set, 58, 60, 103 density of states, 124 derivation, 386, 394 derivative complex, 608 convective, 447 covariant, 403 exterior, 390 functional, half, 194 Lie, 386 of delta function, 68 of determinant, 765 weak, 74 descent equations, 604 determinant derivative of, 765 elementary definition, 759 powerful definition, 761 diagonalizing matrix, 766 Index quadratic form, 772 symplectic form, 775 Dido, queen of Carthage, diffeomorphism, 450 diffraction, 223 dilogarithm, 642 dimensional regularization, 216, 642 Dirac comb, 793 Dirac gamma matrices, 549 Dirac notation, 747, 751 bra and ket vectors, 751 dagger (†) map, 751 direct sum, 754 Dirichlet principle, 203 problem, 202 dispersion, 231 equation, 233 relation, 677 distribution involutive, 384 of tangent fields, 383 distributions delta function, 70 elements of L2 as, 73 Heaviside, 84, 201 principal part, 75, 673 theory of, 71 divergence theorem, 18 divergence, in curvilinear coordinates, 267 domain, 608 of dependence, 187 of differential operator, 103 of function, 52 dual space, 71, 748 eigenvalue, 51, 766 as Lagrange multiplier, 34 eigenvector, 766 Eilenberger equation, 173 elliptic function, 653, 735 elliptic modular function, 739 embedding, 655 endpoint fixed, singular, 298 variable, 27 energy as first integral, 15 density, 21 internal, 25 field, 24 enthalpy, 25 entire function, 638, 644 801 entropy, 34 specific, 25 equivalence class, 54, 755 equivalence relation, 501 equivalent sequence, 58 essential singularity, 638, 644 Euler angles, 385, 409, 542 character, 463, 486, 654 class, 482 Euler’s equation, 26 Euler–Lagrange equation, Euler–Maclaurin sum formula, 690 Euler–Mascheroni constant, 280, 710 exact form, 392 exact sequence, 463 long, 467 short, 464, 467 exponential map, 538 Faraday, Michael, 22 Fermat’s little theorem, 503 Feynman path integral, 435 fibre, 576 fibre bundle, 381 field covector, 380 tangent vector, 377 field theory, 17 first integral, flow barotropic, 25 incompressible, 31, 609 irrotational, 25, 609 Lagrangian versus Eulerian description, 25 of tangent vector field, 382 foliation, 383 form n-linear, 760 closed, 399 quadratic, 772 symplectic, 774 Fourier series, 57, 781 Fourier, Joseph, 185 Fredholm determinant, 341 equation, 312 operator, 333, 485 series, 341 Fredholm alternative for differential operators, 140 for integral equations, 323 for systems of linear equations, 759 Fresnel integrals, 674 Friedel sum rule, 139 802 Frobenius’ integrability theorem, 384 reciprocity theorem, 529 Frobenius–Schur indicator, 527 Fréchet derivative, see functional derivative function space, 51 normed, 51 functional definition, derivative, local, Gauss quadrature, 77 linking number, 436 sum, 795 Gauss–Bruhat decomposition, 697 Gauss–Bonnet theorem, 483, 600 Gaussian elimination, 329 Gelfand–Dikii equation, 166, 173 Gell-Mann “λ” matrices, 563 generalized functions, see distributions generating function for Legendre polynomials, 274 for Bessel functions, 281 for Chern character, 480 genus, 654 geometric phase, see Berry’s phase geometric quantization, 583 Gibbs’ phenomenon, 788 gradient as a covector, 380 in curvilinear coordinates, 267 Gram-Schmidt procedure, 62, 271 Grassmann, Herman, 359 Green function analyticity of causal, 155 causal, 146, 186, 197 construction of, 141 modified, 149, 154 symmetry of, 150 Green, George, 368 group velocity, 234 Haar measure, 552 half-range Fourier series, 782 hanging chain, see catenary Hankel function, 280 spherical, 295 harmonic conjugate, 608 harmonic oscillator, 120 Haydock recursion, 80 Heaviside function, 84 Index Helmholtz decomposition, 205 equation, 221, 280 Helmholtz–Hodge decomposition, 226 hermitian differential operator, see formally self-adjoint operator matrix, 104, 767, 768 hermitian conjugate matrix, 754 operator, see adjoint heterojunction, 114 Hilbert space, 55 rigged, 73 Hilbert transform, 683, 791, 792 Hodge “ ” map, 396, 658 decomposition, 228, 485 theory, 483 Hodge, William, 483 homeomorphism, 450 homology group, 460 homotopy, 432, 548 class, 432 Hopf bundle, see monopole bundle index, 434, 544 map, 430, 542, 544 horocycles, 662 hydraulic jump, 251 ideal, 556 identity delta function as, 67 matrix, 747 image space, 748 images, method of, 219 immersion, 655 index of operator, 759 index theorem, 486, 695, 698 indicial equation, 98 induced metric, 421 induced representation, 528 inequality Cauchy–Schwarz Bunyakovsky, 55 triangle, 53, 54, 56 inertial wave, 262 infimum, 51 infinitesimal homotopy relation, 394 integral equation Fredholm, 313 Volterra, 313 integral kernel, 67 Index interior multiplication, 393 intersection form, 474 Jacobi identity, 401, 555 Jordan form, 711, 770 jump condition, 143 Kelvin wedge, 238 kernel, 747 Killing field, 388 form, 557 Killing, William, 388 Kirchhoff approximation, 222 Korteweg de Vries (KdV) equation, 102 Kramer’s degeneracy, 533 Kramers–Kronig relation, 161 Lagrange interpolation formula, 77 Lagrange multiplier, 33 as eigenvalue, 34 Lagrange’s identity, 104 Lagrange’s theorem, 500 Lagrange, Joseph-Louis, 10 Lagrangian, 10 density, 18 Lamé constants, 365 Lanczos algorithm, 80 Landsberg–Schaar identity, 795 Laplace development, 763 Laplace–Beltrami operator, 485 Laplacian acting on vector field, 226, 269, 483 in curvilinear coordinates, 269 Lax pair, 102, 259 least upper bound, see supremum Legendre function, 679 Legendre function Qn (x), 719 Levi-Civita symbol, 361 Levinson’s theorem, 139 Lie algebra, 530 bracket, 382, 555 derivative, 386 Lie, Sophus, 530 limit-circle case, 298 line bundle, 576 linear dependence, 745 linear map, 746 linear operator bounded, 333 closable, 336 closed, 336 803 compact, 333 Fredholm, 333 Hilbert–Schmidt, 333 Liouville measure, 34 Liouville’s theorem, 91 Liouville–Neumann series, 339 Lipshitz’ formula, 689 Lobachevski geometry, 39, 42, 445, 662 LU decomposition, see Gaussian elimination manifold, 377 orientable, 417 parallelizable, 572 Riemann, 406 symplectic, 399 map anticonformal, 612 isogonal, 611 Möbius, 649 Maupertuis, Pierre de, 10 Maxwell’s equations, 22 Maxwell, James Clerk, 22 measure, Liouville, 34 Mehler’s formula, 65 metric tensor, 750 minimal polynomial equation, 767 modular group, 739 monodromy, 98, 711 monopole bundle, 583, 595 moonshine, monstrous, 500, 741 Morse function, 488 index, 488 Morse function, 487 multilinear form, 356 multipliers, undetermined, see Lagrange multipliers Möbius map, 649, 736 strip, 576 Neumann function, 279 Neumann’s formula, 680 Noether’s theorem, 17 Noether, Emmy, 14 non-degenerate, 750 norm Lp , 53 definition of, 53 sup, 53 null-space, see kernel Nyquist criterion, 155, 681 804 observables, 112 optical fibre, 292 orbit, of group action, 504 order of group, 499 orientable manifold, 417 orthogonal complement, 755 orthonormal set, 56 Pöschel–Teller equation, 122, 130, 135, 308, 717 pairing, 72, 348, 469, 749 Parseval’s theorem, 61, 785 particular integral, 94, 152 Pauli σ matrices, 430, 534 Peierls, Rudolph, 125, 296 period and de Rham’s theorem, 471 of elliptic function, 653 Peter–Weyl theorem, 552 Pfaffian system, 384 phase shift, 126, 301 phase space, 34 phase velocity, 234 Plücker relations, 360, 374 Plücker, Julius, 360 Plateau’s problem, 1, Plemelj formulæ, 678 Plemelj, Josip, 159 Poincaré disc, 42, 446, 662 duality, 487 lemma, 391, 450 Poincaré–Hopf theorem, 488 Poincaré–Bertrand theorem, 686 point ordinary, 97 regular singular, 97 singular, 97 singular endpoint, 298 Poisson bracket, 400 kernel, 211, 795 summation, 792 Poisson’s ratio, 367 pole, 621 polynomials Hermite, 64, 122 Legendre, 63, 271, 335 orthogonal, 62 Tchebychef, 65, 324 Pontryagin class, 482 Index pressure, 26 principal bundle, 576 principal-part integral, 75, 159, 323, 670 principle of least action, see action principle product cup, 473 direct, 506 group axioms, 498 inner, 749 of matrices, 747 tensor, 355 wedge, 358, 390 projective plane, 461 pseudo-momentum, 22 pseudo-potential, 304 quadratic form, 772 diagonalizing, 772 signature of, 774 quaternions, 534 quotient of vector spaces, 755 group, 500 space, 505 range, see image space range–null-space theorem, 748 rank column, 748 of Lie algebra, 566 of matrix, 748 of tensor, 350 Rayleigh–Ritz method, 119 realm, sublunary, 125 recurrence relation for Bessel functions, 282, 298 for orthogonal polynomials, 62 residue, 621 resolution of the identity, 515, 756, 769, 772 resolvent, 161 resonance, 128, 304 retraction, 451 Riemann P symbol, 714 sum, 617 surface, 651 Riemann–Lebesgue lemma, 785 Riemann–Hilbert problem, 329 Riesz–Fréchet theorem, 72, 76 Rodriguez’ formula, 63, 272, 719 rolling conditions, 385, 442 root vector, 564 Routhian, 14 Russian formula, 603 Index scalar product, see product, inner scattering length, 301 Schwartz space, 71 Schwartz, Laurent, 71 Schwarzian derivative, 100 Scott Russell, John, 257 section, 577 of bundle, 381 Seeley coefficients, 167 self-adjoint extension, 113 self-adjoint matrix, 767 self-adjoint operator formally, 105 truly, 111 seminorm, 71 Serret–Frenet relations, 442 sesquilinear, 749 sextant, 546 shear modulus, 366 sheet, 651 simplex, 455 simplicial complex, 456 singular endpoint, 298 singular integral equations, 323 skew-symmetric form, see symplectic form skyrmion, 428 soap film, soliton, 102, 255 space Lp , 53 Banach, 55 Hilbert, 55 homogeneous, 505 of C n functions, 51 of test functions, 71 retractable, 451 spanning set, 745 spectrum continuous, 117, 299 discrete, 117 point, see discrete spectrum spherical harmonics, 276 spinor, 430, 545 stereographic map, 428, 648 Stokes’ line, 734 phenomenon, 727 theorem, 422 strain tensor, 389 stream-function, 609 streamline, 609 string sliding, 28 vibrating, 18 805 structure constants, 536 Sturm–Liouville operator, 36, 37, 68, 106 supremum, 51 symmetric differential operator, see formally self-adjoint operator symplectic form, 399, 774 tangent bundle, 376 space, 376 tantrix, 438 Taylor column, 262 tensor cartesian, 362 curvature, 406 elastic constants, 41 energy–momentum, 21 isotropic, 363 metric, 750 momentum flux, 26 strain, 40, 364, 389 stress, 41, 364 torsion, 406 test function, 71 theorem Abel, 633 addition for spherical harmonics, 278 Blasius, 628 Cayley’s, 764 Darboux, 400 de Rham, 470 Frobenius integrability, 384 Frobenius’ reciprocity, 529 Gauss–Bonnet, 483, 600 Green’s, 218 Lagrange, 500 Morse index, 488 Peter–Weyl, 552 Picard, 644 Poincaré–Hopf, 488 Poincaré–Bertrand, 686 range-null-space, 748 residue, 620 Riemann mapping, 613 Riesz–Fréchet, 72, 76 Stokes, 422 Weierstrass approximation, 63 Weyl’s, 298 Theta function, 647 tidal bore, see hydraulic jump topological current, 433 806 torsion in homology, 462 of curve, 442 tensor, 406 transfom Hilbert, 683 transform Fourier, 313, 784 Fourier–Bessel, see Hankel Hankel, 288 Hilbert, 791, 792 Laplace, 314, 315 Legendre, 14, 26 Mellin, 314 Mellin sine, 212 Radon, 318 variational principle, 119 variety, 357 Segre, 357 vector bundle, 403 Laplacian, 226, 269, 483 vector space definition, 744 velocity potential, 25, 609 as Lagrange multiplier, 36 vielbein, 404 orthonormal, 409, 477 volume form, 422 vorticity, 26, 45 Index wake, 237 ship, 238 wave drag, 237 equation, 181 momentum, see pseudo-momentum non-linear, 246 shock, 249 surface, 30, 231 transverse, 19 Weber’s disc, 289 Weierstrass ℘ function, 736 approximation theorem, 63 weight, 564 Weitzenböck formula, 496 Weyl’s identity, 533, 778 theorem, 298 Weyl, Hermann, 125, 298 Wiener–Hopf integral equations, 327 sum equations, 692 winding number, 426 Wronskian, 89 and linear dependence, 91 in Green function, 143 Wulff construction, 46 Young’s modulus, 366 ... blank Mathematics for Physics A Guided Tour for Graduate Students An engagingly written account of mathematical tools and ideas, this book provides a graduate-level introduction to the mathematics. .. including soft matter, disordered systems, nanoscience and superconductivity MATHEMATICS FOR PHYSICS A Guided Tour for Graduate Students MICHAEL STONE University of Illinois at Urbana-Champaign... in print format 2009 ISBN-13 978-0-511-59516-5 eBook (EBL) ISBN-13 978-0-521-85403-0 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external