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Mathematics for Physics II Mathematics for Physics II A set of lecture notes by Michael Stone PIMANDER-CASAUBON Alexandria Florence London • • ii Copyright c 2001,2002,2003 M. Stone. All rights reserved. No part of this material can be reproduced, stored or transmitted without the written permission of the author. For information contact: Michael Stone, Loomis Laboratory of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801, USA. Preface These notes cover the material from the second half of a two-semester se- quence of mathematical methods courses given to first year physics graduate students at the University of Illinois. They consist of three loosely connected parts: i) an introduction to modern “calculus on manifolds”, the exterior differential calculus, and algebraic topology; ii) an introduction to group rep- resentation theory and its physical applications; iii) a fairly standard course on complex variables. iii iv PREFACE Contents Preface iii 1 Tensors in Euclidean Space 1 1.1 CovariantandContravariantVectors . 1 1.2 Tensors .............................. 4 1.3 CartesianTensors. .. .. 18 1.4 Further Exercises and Problems . 29 2 Differential Calculus on Manifolds 33 2.1 VectorandCovectorFields. 33 2.2 DifferentiatingTensors . 39 2.3 ExteriorCalculus . .. .. 48 2.4 PhysicalApplications. 54 2.5 CovariantDerivatives. 63 2.6 Further Exercises and Problems . 70 3 Integration on Manifolds 75 3.1 BasicNotions ........................... 75 3.2 Integrating p-Forms........................ 79 3.3 Stokes’Theorem ......................... 84 3.4 Applications............................ 87 3.5 ExercisesandProblems. .105 4 An Introduction to Topology 115 4.1 Homeomorphism and Diffeomorphism . 116 4.2 Cohomology............................117 4.3 Homology .............................122 4.4 DeRham’sTheorem .. .. .138 v vi CONTENTS 4.5 Poincar´eDuality . .142 4.6 CharacteristicClasses. .147 4.7 HodgeTheoryandtheMorseIndex . 154 5 Groups and Group Representations 171 5.1 BasicIdeas ............................171 5.2 Representations . .179 5.3 PhysicsApplications . .192 5.4 Further Exercises and Problems . 201 6 Lie Groups 207 6.1 MatrixGroups ..........................207 6.2 GeometryofSU(2) . .. .. .213 6.3 LieAlgebras............................234 6.4 Further Exercises and Problems . 253 7 The Geometry of Fibre Bundles 257 7.1 FibreBundles ..........................257 7.2 PhysicsExamples . .259 7.3 WorkingintheTotalSpace . .274 8 Complex Analysis I 291 8.1 Cauchy-Riemannequations. 291 8.2 Complex Integration: Cauchy and Stokes . 303 8.3 Applications............................312 8.4 Applications of Cauchy’s Theorem . 318 8.5 Meromorphic functions and the Winding-Number . 334 8.6 AnalyticFunctionsandTopology . 337 8.7 Further Exercises and Problems . 353 9 Complex Analysis II 359 9.1 ContourIntegrationTechnology . 359 9.2 The Schwarz Reflection Principle . 370 9.3 Partial-Fraction and Product Expansions . 381 9.4 Wiener-HopfEquationsII . .387 9.5 Further Exercises and Problems . 396 CONTENTS vii 10 Special Functions II 401 10.1 TheGammaFunction . .401 10.2 LinearDifferentialEquations. 406 10.3 Solving ODE’s via Contour integrals . 414 10.4 AsymptoticExpansions. .421 10.5 EllipticFunctions . .432 10.6 Further Exercises and Problems . 439 viii CONTENTS Chapter 1 Tensors in Euclidean Space In this chapter we explain how a vector space V gives rise to a family of associated tensor spaces, and how mathematical objects such as linear maps or quadratic forms should be understood as being elements of these spaces. We then apply these ideas to physics. We make extensive use of notions and notations from the appendix on linear algebra, so it may help to review that material before we begin. 1.1 Covariant and Contravariant Vectors When we have a vector space V over R, and e , e ,..., e and e0 , e0 ,..., e0 { 1 2 n} { 1 2 n} are both bases for V , then we may expand each of the basis vectors eµ in terms of the eµ0 as µ eν = aν eµ0 . (1.1) We are here, as usual, using the Einstein summation convention that repeated indices are to be summed over. Written out in full for a three-dimensional space, the expansion would be 1 2 3 e1 = a1e10 + a1e20 + a1e30 , 1 2 3 e2 = a2e10 + a2e20 + a2e30 , 1 2 3 e3 = a3e10 + a3e20 + a3e30 . We could also have expanded the eµ0 in terms of the eµ as 1 µ eν0 =(a− )ν eµ0 . (1.2) 1 2 CHAPTER 1. TENSORS IN EUCLIDEAN SPACE µ 1 µ As the notation implies, the matrices of coefficients aν and (a− )ν are inverses of each other: µ 1 ν 1 µ ν µ aν (a− )σ =(a− )ν aσ = δσ . (1.3) µ If we know the components x of a vector x in the eµ basis then the compo- µ nents x0 of x in the eµ0 basis are obtained from µ ν ν µ x = x0 eµ0 = x eν =(x aν ) eµ0 (1.4) µ µ ν by comparing the coefficients of eµ0 . We find that x0 = aν x . Observe how µ µ the eµ and the x transform in “opposite” directions. The components x are therefore said to transform contravariantly. Associated with the vector space V is its dual space V ∗, whose elements µ are covectors, i.e. linear maps f : V R. If f V ∗ and x = x e , we use → ∈ µ the linearity property to evaluate f(x) as µ µ µ f(x)= f(x eµ)= x f(eµ)= x fµ. (1.5) Here, the set of numbers fµ = f(eµ) are the components of the covector f. If µ we change basis so that eν = aν eµ0 then µ µ µ fν = f(eν )= f(aν eµ0 )= aν f(eµ0 )= aν fµ0 . (1.6) µ We conclude that fν = aν fµ0 . The fµ components transform in the same man- ner as the basis. They are therefore said to transform covariantly. In physics it is traditional to call the the set of numbers xµ with upstairs indices (the components of) a contravariant vector. Similarly, the set of numbers fµ with downstairs indices is called (the components of) a covariant vector. Thus, contravariant vectors are elements of V and covariant vectors are elements of V ∗. The relationship between V and V ∗ is one of mutual duality, and to mathematicians it is only a matter of convenience which space is V and which space is V ∗. The evaluation of f V ∗ on x V is therefore often ∈ ∈ written as a “pairing” (f, x), which gives equal status to the objects being put togther to get a number. A physics example of such a mutually dual pair is provided by the space of displacements x and the space of wave-numbers 1 k. The units of x and k are different (meters versus meters− ). There is therefore no meaning to “x + k,” and x and k are not elements of the same vector space. The “dot” in expressions such as ik x ψ(x)= e · (1.7) 1.1. COVARIANT AND CONTRAVARIANT VECTORS 3 cannot be a true inner product (which requires the objects it links to be in the same vector space) but is instead a pairing (k, x) k(x)= k xµ. (1.8) ≡ µ In describing the physical world we usually give priority to the space in which we live, breathe and move, and so treat it as being “V ”. The displacement vector x then becomes the contravariant vector, and the Fourier-space wave- number k, being the more abstract quantity, becomes the covariant covector. Our vector space may come equipped with a metric that is derived from a non-degenerate inner product. We regard the inner product as being a bilinear form g : V V R, so the length x of a vector x is g(x, x). The set of numbers × → k k p gµν = g(eµ, eν) (1.9) comprises the (components of) the metric tensor. In terms of them, the inner of product x, y of pair of vectors x = xµe and y = yµe becomes h i µ µ x, y g(x, y)= g xµyν. (1.10) h i ≡ µν Real-valued inner products are always symmetric, so g(x, y) = g(y, x) and gµν = gνµ. As the product is non-degenerate, the matrix gµν has an inverse, which is traditionally written as gµν. Thus νλ λν λ gµν g = g gνµ = δµ. (1.11) The additional structure provided by the metric permits us to identify V with V ∗. The identification is possible, because, given any f V ∗, we can ∈ find a vector f V such that ∈ f(x)= f, x . (1.12) e h i We obtain f by solving the equation e ν e fµ = gµνf (1.13) ν νµ to get f = g fµ. We may now drop thee tilde and identify f with f, and hence V with V ∗. When we do this, we say that the covariant components µ fµ are relatede to the contravariant components f by raising e µ µν f = g fν, (1.14) 4 CHAPTER 1. TENSORS IN EUCLIDEAN SPACE or lowering ν fµ = gµνf , (1.15) the index µ using the metric tensor. Bear in mind that this V ∼= V ∗ identi- fication depends crucially on the metric. A different metric will, in general, identify an f V ∗ with a completely different f V . ∈ ∈ We may play this game in the Euclidean space En with its “dot” inner product. Given a vector x and a basis e fore which g = e e , we can µ µν µ · ν define two sets of components for the same vector. Firstly the coefficients xµ appearing in the basis expansion µ x = x eµ, (1.16) and secondly the “components” x = e x = g(e , x)= g(e , xνe )= g(e , e )xν = g xν (1.17) µ µ · µ µ ν µ ν µν of x along the basis vectors. These two set of numbers are then respectively called the contravariant and covariant components of the vector x.
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