TECHNOLOGICAL AND CULTURAL TRANSFER OF AFRICAN IRONMAKING INTO THE ...
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1 TECHNOLOGICAL AND CULTURAL TRANSFER OF AFRICAN IRONMAKING INTO THE AMERICAS AND THE RELATIONSHIP TO SLAVE RESISTANCE Paper presented to Rediscovering America 1492 - 1992 Conference at Louisiana State University Department of Foreign Language and Literatures Baton Rouge, Louisiana February 28, 1992 Included in Rediscovering America: National, Cultural, and Disciplinary Boundaries Re-examined published in 1993 by Louisiana State University Dept. of Foreign Language and Literatures Prepared for Afrigeneas online papers on slavery, January 2000 Revised for Allies for Freedom website publication February 2007 by Jean Libby jlibby@alum.
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Mathematics for Physics I
Michael Stone
and
Paul Goldbart
PIMANDER-CASAUBON
Alexandria• Florence• Londonii
Copyrightc 2000-2008 M. Stone, P. Goldbart .
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 LaboratoryofPhysics, University ofIllinois,
1110 West Green Street, Urbana, IL 61801, USA.Preface
These notes were prepared for the first semester of a year-long mathematical
methods course for begining graduate students in physics. The emphasis is
on linear operators and stresses the analogy between such operators acting
on function spaces and matrices acting on finite dimensional spaces. The op-
erator language then provides a unified framework for investigating ordinary
and partial differential equations, and integral equations.
The mathematical prerequisites for the course are a sound grasp of un-
dergraduate calculus (including the vector calculus needed forelectricity and
magnetism courses), linear algebra (the more the better), and competence
at complex arithmetic. Fourier sums and integrals, as well as basic ordinary
differential equation theory receive a quick review, but it would help if the
reader had some prior experience to build on. Contour integration is not
required.
iiiiv PREFACEContents
Preface iii
1 Calculus of Variations 1
1.1 What is it good for? . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 The functional derivative . . . . . . . . . . . . . . . . . 2
1.2.2 The Euler-Lagrange equation . . . . . . . . . . . . . . 3
1.2.3 Some applications . . . . . . . . . . . . . . . . . . . . . 4
1.2.4 First integral . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 One degree of freedom . . . . . . . . . . . . . . . . . . 12
1.3.2 Noether’s theorem . . . . . . . . . . . . . . . . . . . . 15
1.3.3 Many degrees of freedom . . . . . . . . . . . . . . . . . 18
1.3.4 Continuous systems . . . . . . . . . . . . . . . . . . . . 19
1.4 Variable End Points. . . . . . . . . . . . . . . . . . . . . . . . 29
1.5 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . 36
1.6 Maximum or Minimum? . . . . . . . . . . . . . . . . . . . . . 40
1.7 Further Exercises and Problems . . . . . . . . . . . . . . . . . 42
2 Function Spaces 55
2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.1.1 Functions as vectors . . . . . . . . . . . . . . . . . . . 56
2.2 Norms and Inner Products . . . . . . . . . . . . . . . . . . . . 57
2.2.1 Norms and convergence . . . . . . . . . . . . . . . . . . 57
2.2.2 Norms from integrals . . . . . . . . . . . . . . . . . . . 59
2.2.3 Hilbert space . . . . . . . . . . . . . . . . . . . . . . . 61
2.2.4 Orthogonal polynomials . . . . . . . . . . . . . . . . . 69
2.3 Linear Operators and Distributions . . . . . . . . . . . . . . . 74
vvi CONTENTS
2.3.1 Linear operators . . . . . . . . . . . . . . . . . . . . . 74
2.3.2 Distributions and test-functions . . . . . . . . . . . . . 77
2.4 Further Exercises and Problems . . . . . . . . . . . . . . . . . 85
3 Linear Ordinary Differential Equations 95
3.1 Existence and Uniqueness of Solutions . . . . . . . . . . . . . 95
3.1.1 Flows for first-order equations . . . . . . . . . . . . . . 95
3.1.2 Linear independence . . . . . . . . . . . . . . . . . . . 97
3.1.3 The Wronskian . . . . . . . . . . . . . . . . . . . . . . 98
3.2 Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.3 Inhomogeneous Equations . . . . . . . . . . . . . . . . . . . . 103
3.3.1 Particular integral and complementary function . . . . 103
3.3.2 Variation of parameters . . . . . . . . . . . . . . . . . 104
3.4 Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.4.1 Regular singular points . . . . . . . . . . . . . . . . . . 106
3.5 Further Exercises and Problems . . . . . . . . . . . . . . . . . 108
4 Linear Differential Operators 111
4.1 Formal vs. Concrete Operators . . . . . . . . . . . . . . . . . 111
4.1.1 The algebra of formal operators . . . . . . . . . . . . . 111
4.1.2 Concrete operators . . . . . . . . . . . . . . . . . . . . 113
4.2 The Adjoint Operator . . . . . . . . . . . . . . . . . . . . . . 114
4.2.1 The formal adjoint . . . . . . . . . . . . . . . . . . . . 114
4.2.2 A simple eigenvalue problem . . . . . . . . . . . . . . . 119
4.2.3 Adjoint boundary conditions . . . . . . . . . . . . . . . 121
4.2.4 Self-adjoint boundary conditions . . . . . . . . . . . . . 122
4.3 Completeness of Eigenfunctions . . . . . . . . . . . . . . . . . 128
4.3.1 Discrete spectrum . . . . . . . . . . . . . . . . . . . . . 129
4.3.2 Continuous spectrum . . . . . . . . . . . . . . . . . . . 135
4.4 Further Exercises and Problems . . . . . . . . . . . . . . . . . 145
5 Green Functions 155
5.1 Inhomogeneous Linear equations . . . . . . . . . . . . . . . . . 155
5.1.1 Fredholm alternative . . . . . . . . . . . . . . . . . . . 155
5.2 Constructing Green Functions . . . . . . . . . . . . . . . . . . 156
5.2.1 Sturm-Liouville equation . . . . . . . . . . . . . . . . . 157
5.2.2 Initial-value problems . . . . . . . . . . . . . . . . . . . 160
5.2.3 Modified Green function . . . . . . . . . . . . . . . . . 165CONTENTS vii
5.3 Applications of Lagrange’s Identity . . . . . . . . . . . . . . . 167
5.3.1 Hermiticity of Green function . . . . . . . . . . . . . . 167
5.3.2 Inhomogeneous boundary conditions . . . . . . . . . . 168
5.4 Eigenfunction Expansions . . . . . . . . . . . . . . . . . . . . 170
5.5 Analytic Properties of Green Functions . . . . . . . . . . . . . 171
5.5.1 Causality implies analyticity . . . . . . . . . . . . . . . 172
5.5.2 Plemelj formulæ . . . . . . . . . . . . . . . . . . . . . . 176
5.5.3 Resolvent operator . . . . . . . . . . . . . . . . . . . . 179
5.6 Locality and the Gelfand-Dikii equation . . . . . . . . . . . . 184
5.7 Further Exercises and problems . . . . . . . . . . . . . . . . . 185
6 Partial Differential Equations 193
6.1 Classification of PDE’s . . . . . . . . . . . . . . . . . . . . . . 193
6.2 Cauchy Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
6.2.1 Characteristics and first-order equations . . . . . . . . 197
6.2.2 Second-order hyperbolic equations . . . . . . . . . . . . 198
6.3 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 200
6.3.1 d’Alembert’s Solution. . . . . . . . . . . . . . . . . . . 200
6.3.2 Fourier’s Solution . . . . . . . . . . . . . . . . . . . . . 205
6.3.3 Causal Green Function . . . . . . . . . . . . . . . . . . 206
6.3.4 Odd vs. Even Dimensions . . . . . . . . . . . . . . . . 212
6.4 Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
6.4.1 Heat Kernel . . . . . . . . . . . . . . . . . . . . . . . . 217
6.4.2 Causal Green Function . . . . . . . . . . . . . . . . . . 219
6.4.3 Duhamel’s Principle . . . . . . . . . . . . . . . . . . . 221
6.5 Potential Theory . . . . . . . . . . . . . . . . . . . . . . . . . 223
6.5.1 Uniqueness and existence of solutions . . . . . . . . . . 224
6.5.2 Separation of Variables . . . . . . . . . . . . . . . . . . 228
6.5.3 Eigenfunction Expansions . . . . . . . . . . . . . . . . 237
6.5.4 Green Functions . . . . . . . . . . . . . . . . . . . . . 239
6.5.5 Boundary-value problems . . . . . . . . . . . . . . . . 241
6.5.6 Kirchhoff vs. Huygens . . . . . . . . . . . . . . . . . . 245
6.6 Further Exercises and problems . . . . . . . . . . . . . . . . . 249
7 The Mathematics of Real Waves 257
7.1 Dispersive waves . . . . . . . . . . . . . . . . . . . . . . . . . 257
7.1.1 Ocean Waves . . . . . . . . . . . . . . . . . . . . . . . 257
7.1.2 Group Velocity . . . . . . . . . . . . . . . . . . . . . . 261viii CONTENTS
7.1.3 Wakes . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
7.1.4 Hamilton’s Theory of Rays . . . . . . . . . . . . . . . . 266
7.2 Making Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
7.2.1 Rayleigh’s Equation . . . . . . . . . . . . . . . . . . . 269
7.3 Non-linear Waves . . . . . . . . . . . . . . . . . . . . . . . . . 274
7.3.1 Sound in Air . . . . . . . . . . . . . . . . . . . . . . . 274
7.3.2 Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
7.3.3 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . 282
7.4 Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
7.5 Further Exercises and Problems . . . . . . . . . . . . . . . . . 289
8 Special Functions 295
8.1 Curvilinear Co-ordinates . . . . . . . . . . . . . . . . . . . . . 295
8.1.1 Div, Grad and Curl in Curvilinear Co-ordinates . . . . 298
8.1.2 The Laplacian in Curvilinear Co-ordinates . . . . . . . 301
8.2 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . 302
8.2.1 Legendre Polynomials . . . . . . . . . . . . . . . . . . 302
8.2.2 Axisymmetric potential problems . . . . . . . . . . . . 305
8.2.3 General spherical harmonics . . . . . . . . . . . . . . . 308
8.3 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . 311
8.3.1 Cylindrical Bessel Functions . . . . . . . . . . . . . . . 312
8.3.2 Orthogonality and Completeness . . . . . . . . . . . . 320
8.3.3 Modified Bessel Functions . . . . . . . . . . . . . . . . 325
8.3.4 Spherical Bessel Functions . . . . . . . . . . . . . . . . 328
8.4 Singular Endpoints . . . . . . . . . . . . . . . . . . . . . . . . 332
8.4.1 Weyl’s Theorem . . . . . . . . . . . . . . . . . . . . . . 332
8.5 Further Exercises and Problems . . . . . . . . . . . . . . . . . 340
9 Integral Equations 347
9.1 Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
9.2 Classification of Integral Equations . . . . . . . . . . . . . . . 348
9.3 Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . 350
9.3.1 Fourier Methods . . . . . . . . . . . . . . . . . . . . . 350
9.3.2 Laplace Transform Methods . . . . . . . . . . . . . . . 352
9.4 Separable Kernels . . . . . . . . . . . . . . . . . . . . . . . . . 358
9.4.1 Eigenvalue problem . . . . . . . . . . . . . . . . . . . . 358
9.4.2 Inhomogeneous problem . . . . . . . . . . . . . . . . . 359
9.5 Singular Integral Equations . . . . . . . . . . . . . . . . . . . 361CONTENTS ix
9.5.1 Solution via Tchebychef Polynomials . . . . . . . . . . 361
9.6 Wiener-Hopf equations I . . . . . . . . . . . . . . . . . . . . . 365
9.7 Some Functional Analysis . . . . . . . . . . . . . . . . . . . . 370
9.7.1 Bounded and Compact Operators . . . . . . . . . . . . 371
9.7.2 Closed Operators . . . . . . . . . . . . . . . . . . . . . 374
9.8 Series Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 378
9.8.1 Liouville-Neumann-Born Series . . . . . . . . . . . . . 378
9.8.2 Fredholm Series . . . . . . . . . . . . . . . . . . . . . . 378
9.9 Further Exercises and Problems . . . . . . . . . . . . . . . . . 382
A Linear Algebra Review 387
A.1 Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
A.1.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . 387
A.1.2 Bases and components . . . . . . . . . . . . . . . . . . 388
A.2 Linear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
A.2.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 390
A.2.2 Range-nullspace theorem . . . . . . . . . . . . . . . . . 391
A.2.3 The dual space . . . . . . . . . . . . . . . . . . . . . . 392
A.3 Inner-Product Spaces . . . . . . . . . . . . . . . . . . . . . . . 393
A.3.1 Inner products . . . . . . . . . . . . . . . . . . . . . . 393
A.3.2 Euclidean vectors . . . . . . . . . . . . . . . . . . . . . 395
A.3.3 Bra and ket vectors . . . . . . . . . . . . . . . . . . . . 395
A.3.4 Adjoint operator . . . . . . . . . . . . . . . . . . . . . 397
A.4 Sums and Differences of Vector Spaces . . . . . . . . . . . . . 398
A.4.1 Direct sums . . . . . . . . . . . . . . . . . . . . . . . . 398
A.4.2 Quotient spaces . . . . . . . . . . . . . . . . . . . . . . 399
A.4.3 Projection-operator decompositions . . . . . . . . . . . 400
A.5 Inhomogeneous Linear Equations . . . . . . . . . . . . . . . . 400
A.5.1 Rank and index . . . . . . . . . . . . . . . . . . . . . . 401
A.5.2 Fredholm alternative . . . . . . . . . . . . . . . . . . . 403
A.6 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
A.6.1 Skew-symmetric n-linear Forms . . . . . . . . . . . . . 403
A.6.2 The adjugate matrix . . . . . . . . . . . . . . . . . . . 406
A.6.3 Differentiating determinants . . . . . . . . . . . . . . . 408
A.7 Diagonalization and Canonical Forms . . . . . . . . . . . . . . 409
A.7.1 Diagonalizing linear maps . . . . . . . . . . . . . . . . 409
A.7.2 Diagonalizing quadratic forms . . . . . . . . . . . . . . 415
A.7.3 Block-diagonalizing symplectic forms . . . . . . . . . . 418x CONTENTS
B Fourier Series and Integrals. 423
B.1 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
B.1.1 Finite Fourier series . . . . . . . . . . . . . . . . . . . . 423
B.1.2 Continuum limit . . . . . . . . . . . . . . . . . . . . . 425
B.2 Fourier Integral Transforms . . . . . . . . . . . . . . . . . . . 428
B.2.1 Inversion formula . . . . . . . . . . . . . . . . . . . . . 428
B.2.2 The Riemann-Lebesgue lemma . . . . . . . . . . . . . 430
B.3 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
B.3.1 The convolution theorem . . . . . . . . . . . . . . . . . 432
B.3.2 Apodization and Gibbs’ phenomenon . . . . . . . . . . 432
B.4 The Poisson Summation Formula . . . . . . . . . . . . . . . . 438
C Bibliography 441

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