Mathematical methods for students of physics and related fields~tqw~ darksiderg kho tài liệu bách khoa

Mathematical Methods Sadri Hassani Mathematical Methods For Students of Physics and Related Fields 123 Sadri Hassani IIlinois State University Normal, IL USA hassani@entropy.phy.ilstu.edu ISBN: 978-0-387-09503-5 e-ISBN: 978-0-387-09504-2 Library of Congress Control Number: 2008935523 c Springer Science+Business Media, LLC 2009 All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights Printed on acid-free paper springer.com To my wife, Sarah, and to my children, Dane Arash and Daisy Bita Preface to the Second Edition In this new edition, which is a substantially revised version of the old one, I have added five new chapters: Vectors in Relativity (Chapter 8), Tensor Analysis (Chapter 17), Integral Transforms (Chapter 29), Calculus of Variations (Chapter 30), and Probability Theory (Chapter 32) The discussion of vectors in Part II, especially the introduction of the inner product, offered the opportunity to present the special theory of relativity, which unfortunately, in most undergraduate physics curricula receives little attention While the main motivation for this chapter was vectors, I grabbed the opportunity to develop the Lorentz transformation and Minkowski distance, the bedrocks of the special theory of relativity, from first principles The short section, Vectors and Indices, at the end of Chapter of the first edition, was too short to demonstrate the importance of what the indices are really used for, tensors So, I expanded that short section into a somewhat comprehensive discussion of tensors Chapter 17, Tensor Analysis, takes a fresh look at vector transformations introduced in the earlier discussion of vectors, and shows the necessity of classifying them into the covariant and contravariant categories It then introduces tensors based on—and as a generalization of—the transformation properties of covariant and contravariant vectors In light of these transformation properties, the Kronecker delta, introduced earlier in the book, takes on a new look, and a natural and extremely useful generalization of it is introduced leading to the Levi-Civita symbol A discussion of connections and metrics motivates a four-dimensional treatment of Maxwell’s equations and a manifest unification of electric and magnetic fields The chapter ends with Riemann curvature tensor and its place in Einstein’s general relativity The Fourier series treatment alone does not justice to the many applications in which aperiodic functions are to be represented Fourier transform is a powerful tool to represent functions in such a way that the solution to many (partial) differential equations can be obtained elegantly and succinctly Chapter 29, Integral Transforms, shows the power of Fourier transform in many illustrations including the calculation of Green’s functions for Laplace, heat, and wave differential operators Laplace transforms, which are useful in solving initial-value problems, are also included viii Preface to Second Edition The Dirac delta function, about which there is a comprehensive discussion in the book, allows a very smooth transition from multivariable calculus to the Calculus of Variations, the subject of Chapter 30 This chapter takes an intuitive approach to the subject: replace the sum by an integral and the Kronecker delta by the Dirac delta function, and you get from multivariable calculus to the calculus of variations! Well, the transition may not be as simple as this, but the heart of the intuitive approach is Once the transition is made and the master Euler-Lagrange equation is derived, many examples, including some with constraint (which use the Lagrange multiplier technique), and some from electromagnetism and mechanics are presented Probability Theory is essential for quantum mechanics and thermodynamics This is the subject of Chapter 32 Starting with the basic notion of the probability space, whose prerequisite is an understanding of elementary set theory, which is also included, the notion of random variables and its connection to probability is introduced, average and variance are defined, and binomial, Poisson, and normal distributions are discussed in some detail Aside from the above major changes, I have also incorporated some other important changes including the rearrangement of some chapters, adding new sections and subsections to some existing chapters (for instance, the dynamics of fluids in Chapter 15), correcting all the mistakes, both typographic and conceptual, to which I have been directed by many readers of the first edition, and adding more problems at the end of each chapter Stylistically, I thought splitting the sometimes very long chapters into smaller ones and collecting the related chapters into Parts make the reading of the text smoother I hope I was not wrong! I would like to thank the many instructors, students, and general readers who communicated to me comments, suggestions, and errors they found in the book Among those, I especially thank Dan Holland for the many discussions we have had about the book, Rafael Benguria and Gebhard Gră ubl for pointing out some important historical and conceptual mistakes, and Ali Erdem and Thomas Ferguson for reading multiple chapters of the book, catching many mistakes, and suggesting ways to improve the presentation of the material Jerome Brozek meticulously and diligently read most of the book and found numerous errors Although a lawyer by profession, Mr Brozek, as a hobby, has a keen interest in mathematical physics I thank him for this interest and for putting it to use on my book Last but not least, I want to thank my family, especially my wife Sarah for her unwavering support S.H Normal, IL January, 2008 Preface Innocent light-minded men, who think that astronomy can be learnt by looking at the stars without knowledge of mathematics will, in the next life, be birds —Plato, Timaeos This book is intended to help bridge the wide gap separating the level of mathematical sophistication expected of students of introductory physics from that expected of students of advanced courses of undergraduate physics and engineering While nothing beyond simple calculus is required for introductory physics courses taken by physics, engineering, and chemistry majors, the next level of courses—both in physics and engineering—already demands a readiness for such intricate and sophisticated concepts as divergence, curl, and Stokes’ theorem It is the aim of this book to make the transition between these two levels of exposure as smooth as possible Level and Pedagogy I believe that the best pedagogy to teach mathematics to beginning students of physics and engineering (even mathematics, although some of my mathematical colleagues may disagree with me) is to introduce and use the concepts in a multitude of applied settings This method is not unlike teaching a language to a child: it is by repeated usage—by the parents or the teacher—of the same word in different circumstances that a child learns the meaning of the word, and by repeated active (and sometimes wrong) usage of words that the child learns to use them in a sentence And what better place to use the language of mathematics than in Nature itself in the context of physics I start with the familiar notion of, say, a derivative or an integral, but interpret it entirely in terms of physical ideas Thus, a derivative is a means by which one obtains velocity from position vectors or acceleration from velocity vectors, and integral is a means by which one obtains the gravitational or electric field of a large number of charged or massive particles If concepts (e.g., infinite series) not succumb easily to physical interpretation, then I immediately subjugate the physical x Preface situation to the mathematical concepts (e.g., multipole expansion of electric potential) Because of my belief in this pedagogy, I have kept formalism to a bare minimum After all, a child needs no knowledge of the formalism of his or her language (i.e., grammar) to be able to read and write Similarly, a novice in physics or engineering needs to see a lot of examples in which mathematics is used to be able to “speak the language.” And I have spared no effort to provide these examples throughout the book Of course, formalism, at some stage, becomes important Just as grammar is taught at a higher stage of a child’s education (say, in high school), mathematical formalism is to be taught at a higher stage of education of physics and engineering students (possibly in advanced undergraduate or graduate classes) Features The unique features of this book, which set it apart from the existing textbooks, are • the inseparable treatments of physical and mathematical concepts, • the large number of original illustrative examples, • the accessibility of the book to sophomores and juniors in physics and engineering programs, and • the large number of historical notes on people and ideas All mathematical concepts in the book are either introduced as a natural tool for expressing some physical concept or, upon their introduction, immediately used in a physical setting Thus, for example, differential equations are not treated as some mathematical equalities seeking solutions, but rather as a statement about the laws of Nature (e.g., the second law of motion) whose solutions describe the behavior of a physical system Almost all examples and problems in this book come directly from physical situations in mechanics, electromagnetism, and, to a lesser extent, quantum mechanics and thermodynamics Although the examples are drawn from physics, they are conceptually at such an introductory level that students of engineering and chemistry will have no difficulty benefiting from the mathematical discussion involved in them Most mathematical-methods books are written for readers with a higher level of sophistication than a sophomore or junior physics or engineering student This book is directly and precisely targeted at sophomores and juniors, and seven years of teaching it to such an audience have proved both the need for such a book and the adequacy of its level My experience with sophomores and juniors has shown that peppering the mathematical topics with a bit of history makes the subject more enticing It also gives a little boost to the motivation of many students, which at times can Preface run very low The history of ideas removes the myth that all mathematical concepts are clear cut, and come into being as a finished and polished product It reveals to the students that ideas, just like artistic masterpieces, are molded into perfection in the hands of many generations of mathematicians and physicists Use of Computer Algebra As soon as one applies the mathematical concepts to real-world situations, one encounters the impossibility of finding a solution in “closed form.” One is thus forced to use approximations and numerical methods of calculation Computer algebra is especially suited for many of the examples and problems in this book Because of the variety of the computer algebra softwares available on the market, and the diversity in the preference of one software over another among instructors, I have left any discussion of computers out of this book Instead, all computer and numerical chapters, examples, and problems are collected in Mathematical Methods Using Mathematica R , a relatively self-contained companion volume that uses Mathematica R By separating the computer-intensive topics from the text, I have made it possible for the instructor to use his or her judgment in deciding how much and in what format the use of computers should enter his or her pedagogy The usage of Mathematica R in the accompanying companion volume is only a reflection of my limited familiarity with the broader field of symbolic manipulations on the computers Instructors using other symbolic algebra programs such as Maple R and Macsyma R may generate their own examples or translate the Mathematica R commands of the companion volume into their favorite language Acknowledgments I would like to thank all my PHY 217 students at Illinois State University who gave me a considerable amount of feedback I am grateful to Thomas von Foerster, Executive Editor of Mathematics, Physics and Engineering at Springer-Verlag New York, Inc., for being very patient and supportive of the project as soon as he took over its editorship Finally, I thank my wife, Sarah, my son, Dane, and my daughter, Daisy, for their understanding and support Unless otherwise indicated, all biographical sketches have been taken from the following sources: Kline, M Mathematical Thought: From Ancient to Modern Times, Vols 1–3, Oxford University Press, New York, 1972 xi Index Abel, 331, 503, 675 Abel, Niels Henrik biography, 326 absolute differential, 463 Acceleration, 44 acceleration components spherical coordinates, 353 active transformation, 178 addition of velocities relativistic law of, 246 adjoint DO, 571–573 advanced Green’s function, 712 affine connection, 462–464, 470 algebra fundamental theorem, 478, 511 amplitude complex, 486 angle as integral, 345 solid, 344–350 total, 349 total, 346 angular momentum, 28 conservation, 351 central force, 354 angular momentum operator, 412 spherical coordinates, 435 antisymmetric tensor, 452 Arago, 321 Archimedes, 47 Archimedes, of Syracuse biography, 81 area element primary, 60 astragalus, 781 astrophysics, 415 Poisson equation, 415 attractor, 766 strange, 778 autonomous, 767 average, 790, 801 azimuth, 12 azimuthal angle, 12 symmetry, 607 azimuthal equation, 549 Barrow, 87, 96, 97, 481 Barrow, Isaac biography, 47 basin of attraction, 766 basis, 16 for plane, 175 orthonormal, 186 standard, 216 Bayes’ theorem, 788 Beltrami identity, 731 Bernoulli, 272, 294, 320, 321, 326, 642 Bernoulli’s FODE, 560 Bessel, 322 Bessel differential equation, 548, 641 recursion relation, 644 Bessel equation Liouville substitution, 589 Bessel function, 333–335, 644 addition theorem, 651 confluent hypergeometric, 652 expansion in, 653–654 physical examples, 654–656 generating function, 651 integral representation, 652 Laplace’s equation, 642–654 order negative integer, 646 orthogonality relation, 649 properties, 646–652 recurrence relation, 646 second kind, 645 Bessel’s integral, 652 818 INDEX Bessel, Friedrich Wilhelm biography, 641 beta function, 320 Bezout, 210 Bianchi identity, 470 bifurcation, 757 Hopf, 770 period-doubling, 757 binding energy, 582 binomial probability distribution, 792–797 binomial theorem, 265 biography Abel, Niels Henrik, 326 Archimedes, of Syracuse, 81 Barrow, Isaac, 47 Bessel, Friedrich Wilhelm, 641 Biot, Jean-Baptiste, 115 Cauchy, Augustin-Louis, 503 Cavalieri, Bonaventura, 90 Cavendish, Henry, 23 Cayley, Arthur, 192 Coulomb, Charles, 23 d’Alembert, Jean Le Rond, 548 Descartes, Rene, 15 Dirac, Paul Adrien Maurice, 151 Euler, Leonhard, 321 Fermat, Pierre de , 15 Fourier, Joseph, 304 Gauss, Johann Carl Friedrich, 330 Gibbs, Josiah Willard, 381 Hamilton, William R., 10 Heaviside, Oliver, 382 Hermite, Charles, 674 Jacobi, Carl Gustav Jacob, 326 Kepler, Johannes, 579 Laplace, Pierre Simon de, 593 Legendre, Adrien-Marie, 617 Leibniz, Gottfried Wilhelm, 103 Maxwell, James Clerk, 419 Newton, Isaac, 96 Savart, Felix, 115 Stokes, George Gabriel, 398 Sylvester, James Joseph, 210 Taylor, Brook, 294 Wallis, John, 90 Biot, Jean-Baptiste biography, 115 Biot–Savart law, 30 circuit, 111 general, 110 bipolar coordinates, 73 three-dimensional, 74, 438 Boltzmann, 419 Bose-Einstein statistics, 792 boundary condition, 542 Dirichlet, 593 Neumann, 593 boundary conditions periodic, 574 separated, 574 brachistochrone, 731 Bromwich contour, 722 butterfly effect, 771 calculus fundamental theorem, 87 calculus of residues, 525–536 canonical equations, 749 Cantor set, 776 Cardan, 481 Cartesian vector, 216 component, 216 Cauchy, 279, 326, 331, 594 Cauchy criterion, 261 Cauchy integral formula, 508–509 Cauchy, Augustin-Louis biography, 503 Cauchy–Goursat theorem, 505 Cauchy–Riemann conditions, 500 Cauchy–Riemann theorem, 501 Cavalieri, Bonaventura biography, 90 Cavendish, Henry biography, 23 Cayley, 192, 211 Cayley, Arthur biography, 192 center of mass, 21 central force, 354, 579–583 eccentricity, 581 central limit theorem, 809 centrifugal potential, 581 chain rule, 55–57 Champollion, 304 chaos, 753 theory systems obeying DE, 770–773 INDEX systems obeying iterated map, 763 universality, 773–778 coefficient, 173 cofactor, 205 collision relativistic, 250–253 column vector, 177 combination, 791 complement, 783 complex conjugate, 479 function analytic, 499 continuous, 499 regular point, 499 singular point, 499 singularity, 499 integral positive sense, 507 number, 478 absolute value, 479 argument, 483 imaginary part, 478 real part, 478 plane, 478 complex amplitude, 486 complex function, 497–511 derivative, 499–503 derivative as integral, 509–511 integration, 503–508 complex number Cartesian form, 478 Fourier series, 489 polar form, 482 roots, 486 complex numbers, 477–488 Cartesian form, 477–481 Fourier series, 488–491 polar form, 482–488 complex power series, 516 analyticity, 517 convergence circle, 517 differentiation, 517 integration, 517 uniform convergence, 517 complex series absolute convergence, 516 component, 176 Compton wavelength, 253 conditional probability, 786–789 819 conducting cylindrical can, 655 conductor electrical, 594 heat, 598 confluent hypergeometric function, 332– 333 connection affine, 462–464, 470 metric, 465–468 constraints, 360, 738 continuity equation, 378–381 differential form, 380 integral form, 380 contour, 505 Bromwich, 722 simple closed, 505 contractible to zero, 400 contraction, 451 contravariant vector, 445–447 convergence test, 267–272 convolution, 716 coordinate generalized, 741 coordinate system, 11–15 bipolar, 73 three-dimensional, 74, 438 Cartesian, 11, 12 cylindrical, 12 elliptic, 73, 213 elliptic cylindrical, 73, 213, 436 parabolic, 73 paraboloidal, 74, 437 polar, 11 prolate spheroidal, 74, 213, 437 spherical, 12 toroidal, 74, 213, 437 unit vector, 31–36 vector, 16–31 coordinate time, 239–240 Copernicus, 97, 417, 580 correlation probability, 803 cosine transform, 697 Coulomb, 744 Coulomb’s law, 22, 24 Coulomb, Charles biography, 23 covariance in probability, 803 820 INDEX covariant derivative, 464–465 covariant differential, 462–464 covariant vector, 445–447 Cramer, 210 Crelle, 326 cross product, 7, 28–31 as a tensor, 447 Levi-Civita symbols, 458 parallelepiped volume, 10 parallelogram area, curl curvilinear coordinates, 431–435 vector field, 391–398 current density, 379 and flux, 379 curvature, 468–471 scalar, 470 curve parametric equation, 61 primary, 59 curvilinear vector analysis, 423–435 curvilinear coordinates curl, 431–435 divergence, 427–431 gradient, 425–427 Laplacian, 435 cycloid, 732 d’Alembert, 273, 303 d’Alembert, Jean Le Rond biography, 548 d´Alembert, 743 damping factor, 311 DE first-order, 551–561 integrating factor, 553–555 linear, 556–561 second-order, 563–570 de Broglie, 666 de Moivre theorem, 485 del operator, 359 delta Kronecker, 442 delta function and Laplacian, 412 cylindrical, 160 derivative, 147, 159 Legendre expansion, 630 limit of sequence, 492 one-variable, 139–151 point sources, 144 polar, 156 representation, 491–492 spherical, 160 three-variable, 159–165 two-dimensional, 155 two-variable, 154–159 density, 45 current, 379 flux, 371–381 of states, 677 probability, 801 density function surface, 154 derivative, 44–46 covariant, 464–465 functional, 730 mixed, 52 normal, 593 partial, 47–59 time vector, 350–355 total, 86 Descartes, 46, 97, 103, 215, 417, 481, 482 Descartes, Rene biography, 15 determinant, 202–207, 222–227 parallelepiped volume, 10 differential, 53–54 absolute, 463 covariant, 462–464 exact, 553 differential equation Bessel, 548, 641 recursion relation, 644 second solution, 645–646 solutions, 642–645 confluent hypergeometric, 332 Hermite recursion relation, 668 hypergeometric, 328 Legendre, 608 second solution, 617–619 order of, 556 ordinary, 542 partial, 542 second-order linear adjoint, 572 integrating factor, 571 INDEX differential operator, 217, 576 diffusion equation, 661 time-dependent, 663 dimension, 176 fractal, 775–778 dipole approximation, 298 magnetic, 410 dipole moment, 298 dipole potential, 299 Dirac, 26 biography, 151 Dirac delta function in variational problems, 730 step function, 153 Dirac, Paul Adrien Maurice biography, 151 disjoint sets, 783 distance spacetime, 240–242 distribution, 146 normal, 806–809 sum of two, 807 divergence, 371–381 curvilinear coordinates, 427–431 spherical coordinates, 430 theorem, 374–378 vector field, 374 Doppler shift relativistic, 255 dot product, 5, 21 double del operation, 407–412 double factorial, 319 dummy index, 262 dynamical system autonomous, 767 nonautonomous, 767 eccentricity, 581 eigenvalue, 224 eigenvalue equation, 224 eigenvector, 224 Einstein, 215, 666 summation convention, 441 Einstein curvature tensor, 471 Einstein equation, 471 electric field, 104 point charge, 25 electrical conductor, 594 electrodynamics 821 Lagrangian density, 745 tensors, 459–461 element area, 59–68 Cartesian, 60–62 cylindrical, 65–68 length, 59–68 spherical, 62–64 volume, 59–68 elliptic coordinates, 73, 213 elliptic cylindrical coordinates, 73, 213, 436 elliptic functions, 322–326 elliptic integral complete, 324 first kind, 323 second kind, 323 empty set, 782 energy relativistic, 249 zero mass particle, 250 energy momentum tensor, 471 equation canonical, 749 Klein-Gordon, 747 error function, 322, 806 Euclid, 47, 80, 90 Euler, 272, 303, 326, 330, 503, 642, 743 Euler angles, 201 Euler equation, 483 Euler’s equation, 414 Euler, Leonhard biography, 321 Euler-Lagrange equation, 729–731, 734–736, 738, 739 event, 784 compound, 784 elementary, 784 random, 781 exact differential, 553 expectation value, 790 extremum problem, 727 gradient, 359–361 factorial double, 319 factorial function, 99, 318 Faraday, 26 Feigenbaum alpha, 774 Feigenbaum delta, 773 822 INDEX Feigenbaum numbers, 773–775 Fermat, Pierre de biography, 15 Fermi energy, 677 Fermi-Dirac statistics, 791 Feynman, 26 field, 21–28, 343 electric, 104 scalar, 343 spinor, 343 tensor, 343 vector, 343 field point, 25, 78 fine structure constant, 679 finite constraint problem, 739 fixed point iterated map, 755 stable, 756 flat space, 470 Florence Nightingale, 210 fluid dynamics, 413–415 flux, 365–369 density, 371–381 vector field, 365–369 FODE, 551–561 Bernoulli, 560 homogeneous, 560 integrating factor, 553–555 Lagrange, 561 linear, 556–561 normal, 551 integral of, 552 FOLDE, 556–561 explicit solution, 557 force central, 354 force density, 414 form factor, 702 four-acceleration, 248 four-momentum, 247–250 four-vector, 243 four-velocity, 247–250 Fourier, 115, 279, 322 Fourier series, 299–303 complex numbers, 488–491 to Fourier transform, 693–696 Fourier transform, 693–712 and derivatives, 702–703 and quark model, 702 application to DEs, 702–704 convolution theorem, 724 Coulomb potential charge distribution, 701 point charge, 700 definition, 695 examples, 698–702 Gaussian, 699 Green’s functions, 705–712 heat equation one-dimensional, 704 higher dimensions, 696 inverse, 695 of delta function, 698 properties, 696 Fourier, Joseph biography, 304 Fourier-Bessel series, 655 fractal, 777 fractal dimension, 775–778 free index, 440 frequency natural, 586 Frobenius method, 608–610, 693 function analytic isolated singularity, 525 principal part, 528 antiderivative, 87 as integral, 317–326 as power series, 327–335 Bessel, 333–335, 644 Laplace’s equation, 642–654 beta, 320 complex, 497–511 derivative, 499–503 residue, 526 complex hyperbolic, 502 complex trigonometric, 502 confluent hypergeometric, 332 delta point sources, 144 elliptic, 322–326 error, 322 even, 84 factorial, 318 gain, 586 gamma, 318–319 Stirling approximation, 319 harmonic, 501 homogeneous, 57–59 INDEX hypergeometric, 328–330 integral representation, 329 iterated map, 755 linear density, 143 logistic map, 755 odd, 84 periodic, 299 piecewise continuous, 82 primitive, 87 rational integral, 529–531 sequence, 274–279 series, 274–279 special, 550 transfer, 586 functional, 728 functional derivative, 730 fundamental theorem of algebra, 478 fundamental theorem of calculus, 87 G-orthogonal, 187, 219 matrix, 191, 222 space, 200 vector in space, 199 gain function, 586 Galileo, 26, 90, 97, 325 gamma function, 318–319 Stirling approximation, 319 gauge transformation, 418 Gauss, 279, 321, 326, 503, 617, 641 Gauss elimination, 231 Gauss’s law, 369 differential form, 378 integral form, 377 Gauss, Johann Carl Friedrich biography, 330 Gaussian Fourier transform of, 699 Gay–Lussac, 594 generalized coordinates, 741 generalized momentum, 748 generating function Hermite polynomials, 673 geodesic, 465 relativity, 466 sphere, 466 geometric series, 271 geometry and metric tensor, 456 823 distance formula, 241 Gibb’s phenomenon, 302 Gibbs, 370 Gibbs, Josiah Willard biography, 381 Goldbach, 320 gradient, 355–361, 445 components, 440 curvilinear coordinates, 425–427 normal to surface, 358 three dimensions, 357 two dimensions, 357 Gram–Schmidt process, 221 for space, 199 Grassmann, 382 Green, 210 Green’s function advanced, 712 differebtial eq for, 707 heat equation, 709–710 Laplacian, 708–709 Poisson equation, 709 retarded, 712 wave equation, 711–712 Green’s Functions, 705–712 Gregory, 272, 294 guided wave, 682–686 TE, 684 TEM, 685 TM, 684 Halley, 641 Hamilton, 369, 382 Hamilton, William R biography, 10 Hamiltonian, 747–749 harmonic oscillator quantum, 667 Hermite DE, 668 heat-conducting plate circular, 664–665 rectangular, 663–664 heat-conducting rod, 662–663 heat conductor, 598 heat equation, 543, 661–665 Green’s function, 709–710 one-dimensional, 704 heat transfer time-dependent, 663 824 INDEX heat-conducting rod, 662 Heaviside, 370 Heaviside, Oliver biography, 382 Heisenberg, 26, 151, 675 Heisenberg uncertainty relation, 699 Helmholtz Coil, 291–293 Helmholtz free energy, 54 Hermite DE recursion relation, 668 Hermite polynomial, 670 orthogonality, 672 Hermite polynomials, 229, 575 generating function, 673 Hermite, Charles biography, 674 HNOLDE, 575 characteristic polynomial, 576 homogeneous function, 57 homogeneous function, 57–59 homogeneous SOLDE exact, 571 Hooke, 97 Hopf bifurcation, 770 HSOLDE, 564 second solution, 568 Huygens, 97, 103 hydrogen atom, 677–680, 802 hyperbolic cosine, 290 hyperbolic sine, 290 hypergeometric function, 328–330 confluent, 332–333 integral representation, 329 identity matrix, 180 indeterminate form, 294–297 index free, 440 indicial equation, 609 induction mathematical, 265–266 inductive definition, 266 infinite series, 266–274 inner product, 218–222 positive definite, 187 Riemannian, 187 inner product matrix, 185 integral, 79 as function, 317–326 Bessel’s, 652 derivative of, 85–86 function of trigonometric, 534–536 indefinite, 87 line, 387–391 Mellin inversion, 722 rational function, 529–531 rational trigonometric, 532–534 integral transform, 693 kernel, 693 integrand, 80 integrating factor, 553–555 integration, 77–80 application Cartesian coordinates, 104–107, 112, 115–117 cylindrical coordinates, 107–109, 112–115, 118–119 double integrals, 115–122 electricity, 104–109 general, 91–96 gravity, 104–109 magnetostatics, 109–115 mechanics, 101–103 single integral, 101–115 spherical coordinates, 120–122 triple integrals, 122–128 Cauchy integral formula, 508–509 change of dummy variable, 82 complex function, 503–508 interchange of limits, 82 linearity, 82 parameter, 80 partition of range, 82 point, 79 properties, 81–89 region of, 79 small region, 83 symmetric range, 84 transformation of variable, 83 variable, 80 intersection, 783 inverse matrix, 203, 207 of a matrix, 180 inverse Fourier transform, 695 ionic crystal one-dimensional, 145 potential energy, 164 two-dimensional, 157 INDEX ISOLDE, 569 isoperimetric problem, 738 iterated map, 754–763 fixed point, 755 orbit, 755 Jacobi, 211, 331 Jacobi, Carl Gustav Jacob biography, 326 Jacobian, 207–210 in probability, 804 Jacobian matrix, 208 Kaluza, 215 Kepler, 89, 97 Kepler’s first law, 582 Kepler’s second law, 582 Kepler’s third law, 583 Kepler, Johannes biography, 579 kernel integral transform, 693 Klein-Gordon equation, 747 Koch snowflake, 777 Kronecker delta, 222, 442, 449, 489 Euclidean metric, 466 generalized, 452 Lagrange, 294, 304, 326, 330, 594, 617, 642 biography, 742 Lagrange identity, 572 Lagrange multiplier, 360, 738 Lagrangian, 740–745 interacting particles, 741 Klein-Gordon, 747 particle in EM field, 746 single particle, 741 Lagrangian density, 744–745 electrodynamics, 745 Laguerre polynomials, 230, 679 Laplace, 115, 304, 322, 326, 744 Laplace transform, 712–723 and differential equations, 718–721 Bromwich contour, 722 convolution, 716 cosine, 713 derivative, 717–718 first shift, 714 gamma function, 713 825 imaginary exponential, 713 integral, 717–718 inverse, 721–723 linearity, 714 Mellin inversion integral, 722 periodic functions, 716 properties, 713–717 second shift, 714 sine, 713 step function, 713 unit function, 713 Laplace’s equation, 411, 542, 546 Bessel functions, 642–654 Cartesian coordinates, 594–603 cylindrical coordinates, 639–656 Legendre polynomials, 610–617 radial equation, 619–622 solution uniqueness, 592 spherical coordinates, 607–634 uniqueness of solution, 592–593 Laplace, Pierre Simon de biography, 593 Laplacian, 411 and Dirac delta function, 412 curvilinear coordinates, 435 Green’s function, 708–709 Laurent series complex, 518–522 Lavoisier, 744 law of addition of velocities, 237 law of large numbers, 809 law of motion relativistic, 253–254 Legendre, 304, 326 Legendre equation, 575 recursion relation, 611 Legendre functions second kind, 618 Legendre polynomial, 228, 614, 616 expansion in, 628–630 physical examples, 631–634 generating function, 621 Laplace’s equation, 610–617 multipole expansion, 621 orthogonality, 625 parity, 622 properties, 622–628 recurrence relation, 623 Rodrigues formula, 626 826 INDEX Legendre polynomials, 229, 575 Legendre transformation, 54, 748 Legendre, Adrien-Marie biography, 617 Leibniz, 46, 87, 90, 97, 210, 272, 482 Leibniz, Gottfried Wilhelm biography, 103 length element primary, 59 Levi-Civita symbol, 453 Levi-Civita symbols cross product, 458 l’Hˆ opital’s rule, 294–297 limit cycle, 770 line integral, 387–391 linear combination, 173 linear dependence, 174 linear equation, 230–234 compatible, 231 echelon form, 232 homogeneous, 234 incompatible, 231 linear independence, 174 linear operator, 216 linear transformation, 216–218 Liouville substitution, 588 logistic map, 755 second iterate, 757 Lorentz gauge, 419 Lorentz transformation, 243–247 general, 244 in dimensions, 245 lowering indices, 457–459 Lyapunov exponent, 763 Maclaurin, 210, 272 Maclaurin series, 287 Madelung constant, 165 magnetic charge, 409 magnetic dipole moment, 410 magnetic field moving charge, 30 magnetic force current loop, 420 moving charge, 30 magnetic monopole, 409 manifold, 456, 469 map iterated, 754–763 marginal probability, 786–789 mathematical induction, 265–266 matrix, 177 G-orthogonal, 191, 222 space, 200 identity, 180 inner product, 185 inverse, 180, 203, 207 Jacobian, 208 metric, 185 multiplication rule, 442 orthogonal, 190 symmetric, 182 transformation in space, 195 transpose, 181 unit, 180 zero, 180 Maxwell, 26, 369, 382 Maxwell’s equations, 415–419 derivation of wave equation, 417 relation to relativity, 237 Maxwell, James Clerk biography, 419 Maxwell-Boltzmann statistics, 791 mean, 790 Mellin inversion integral, 722 membrane, 686–687 metric connection, 465–468 relativity, 466 metric matrix, 185 metric tensor, 454–461 definition, 456 relativity, 458 minimal coupling, 749 Minkowski, 215 mode of oscillation, 682 Mă obius band, 366 moment quadrupole, 449 moment generating function, 790 binomial distribution, 793 Poisson distribution, 799 moment of inertia, 122 momentum generalized, 748 relativistic, 249 zero mass particle, 250 Monge, 115, 304 INDEX motion constant of, 552 multipole expansion, 297–299 Napoleon, 304, 593 natural frequency, 586 Neumann function, 645 Newton, 16, 26, 43, 46, 78, 87, 90, 103, 122, 272, 294, 317, 322, 326, 330, 481, 482, 548, 593 Newton, Isaac biography, 96 NOLDE, 575 nonautonomous, 767 normal distribution, 806–809 sum of two, 807 nth iterate, 760 ODE, 542 ODE and PDEs, 542–550 Olbers, 641 operator angular momentum, 412 spherical coordinates, 435 del, 359 differential, 576 linear, 217 linear, 216 orientable surface, 366 orthogonal matrix, 190 orthogonal polynomial standardization, 227 orthogonal polynomials, 227–230 orthonormal basis, 186 parabolic coordinates, 73 paraboloidal coordinates, 74, 437 parallel translation, 465 Parseval relation, 654 Parseval’s relation, 724 partial derivative, 47–59 particle in a box, 675 Pascal, 15, 103, 481 passive transformation, 178 PDE, 542 separation Cartesian coordinates, 544–546 cylindrical coordinates, 547–548 827 spherical coordinates, 548–550 PDE and ODE, 542–550 period-doubling, 757 periodic BC, 574 permutation, 791 phase space, 764–766 diagram, 764 trajectory, 764 Planck, 666 plane basis, 175 Poincar´e, 674 Poisson, 594 Poisson distribution, 797–800 Poisson equation, 411, 542 astrophysics, 415 Green’s function, 709 polar coordinates, 16 polar equation, 549 pole of order m, 528 simple, 528 polynomial Hermite, 229, 670 Laguerre, 230, 679 Legendre, 228, 229, 614, 616 Laplace’s equation, 610–617 orthogonal, 227–230 standardization, 227 position vector, 19 potential, 21–28, 399 centrifugal, 581 difference, 399 of a dipole, 299 potential energy, 553 power series, 283–299 continuity, 285 differential equations, 307 differentiation, 285 integration, 285 operations, 520 radius of convergence, 283 zero, 285 pressure, 46 primary curve, 59 primary surface, 60 probability average, 790 basic concepts, 781–792 binomial distribution, 792–797 828 INDEX conditional , 786–789 correlation, 803 covariance, 803 density, 801 expectation value, 790 independent random variable, 802 marginal , 786–789 mean, 790 moment generating function, 790 Poisson distribution, 797–800 sample space, 784–786 set theory, 782–784 standard deviation, 790 variance, 790 probability space, 784 prolate spheroidal coordinates, 74, 213, 437 proper time, 239–240 quadrupole moment, 449 quantization hydrogen atom, 679 quantum harmonic oscillator, 667–674 quantum mechanics angular momentum operator, 412 spherical coordinates, 435 quantum particle in a box, 675–677 quantum tunneling, 676 quaternions, 11 radial, 19 radial equation, 549 raising indices, 457–459 random event, 781 random variable continuous, 801–809 independent, 802 transformation, 804–806 rate of change, 44 ratio test Waring, 273 recursion relation, 308, 610 relativistic collision, 250–253 relativistic energy, 249 relativistic law of motion, 253–254 relativistic momentum, 249 relativity geodesic, 466 metric connection, 466 metric tensor, 458 principle, 238 special, 237 residue, 526 calculus, 525–536 residue theorem, 527 definite integral rational function, 529 rational trigonometric, 532 trigonometric function, 534 retarded Green’s function, 712 Ricci tensor, 470 Riemann, 321 Riemann curvature tensor, 468–471 Riemann zeta function, 269 Riemannian manifold, 456 right-hand rule, 392 rigid transformation, 190 Rodrigues formula, 626 Rosetta stone, 304 row vector, 181 sample space, 784–786 Savart, Felix biography, 115 scalar curvature, 470 scalar function, 445 Schră odinger, 675 biography, 666 Schră odinger equation, 543, 546, 666–680 time-independent, 666 Schwarz inequality, 185, 220 Schwinger, 26 second iterate, 757 second variation, 735–738 self-similarity, 775 separated boundary conditions, 574 separation of time, 543 separatrix, 766 sequence, 259–262 bounded, 261 convergence, 260 Cauchy criterion, 261 divergence, 260 functions, 274–279 limit, 260 monotone decreasing, 261 monotone increasing, 261 partial sum, 259, 267 series, 266–274 INDEX alternating test, 270 application to DE, 307–311 complex, 518 Laurent, 518–522 Taylor, 518–522 convergence absolute, 268 comparison test, 268 conditional, 272 generalized ratio test, 270 integral test, 268 n-th term test, 267 ratio test, 269 convergent grouping, 273 rearranging, 273 familiar functions, 287–291 Fourier, 299–303 complex numbers, 489 Fourier–Bessel, 655 functions, 274–279 uniform convergence, 276 geometric, 271 harmonic order p, 269 Laurent complex, 518 Maclaurin, 287 binomial function, 288 complex, 518 exponential function, 287 hyperbolic function, 289 logarithmic function, 291 trigonometric function, 287 operations on, 273–274 power, 283–299 differential equations, 307 Taylor, 286–287 complex, 518 multivariable, 305–307 uniform convergence differentiation, 278 integration, 278 uniformly convergent, 277–279 set theory, 782–784 complement, 783 difference, 783 disjoint sets, 783 intersection, 783 829 union, 782 Venn diagrams, 783 sine transform, 697 soap film problem, 733 SOLDE, 563–570 basis of solutions, 565 central force, 579 constant coefficient, 575–587 homogeneous, 576–583 inhomogeneous, 583–587 homogeneous, 564 second solution, 567–569 inhomogeneous general solution, 569–570 Kepler problem, 580 linearity, 564–565 normal form, 563 singular point, 563 superposition, 564–565 superposition principle, 564 uniqueness of solution, 564–565 uniqueness theorem, 565 variation of constants, 569 Wronskian, 566–567 solid angle, 344–350 total, 349 source point, 25, 79 space dimension, 11 flat, 470 point, 11 probability, 784 spacetime distance, 240–242 being zero, 242 span, 175 special functions, 550 standard basis, 216 standard deviation, 790 statistical independence, 788 statistics Bose-Einstein, 792 Fermi-Dirac, 791 Maxwell-Boltzmann, 791 stellar equilibrium, 415 step function, 152–153 Dirac delta function, 153 Laplace transform, 713 Stifel, 481 Stirling, 320 Stirling approximation, 319, 792, 808 830 INDEX Stokes’ theorem, 391–398 Stokes, George Gabriel biography, 398 strange attractor, 778 Sturm–Liouville system, 574 Sturm-Liouville equation, 574 subset, 782 success excess, 793 summation, 262–266 superposition principle, 25, 564 surface primary, 60 Sylvester, and Cayley, 192 Sylvester, James Joseph biography, 210 symmetric matrix, 182 symmetric tensor, 452 Taylor series, 286–287 complex, 518–522 multivariable, 305–307 Taylor, Brook biography, 294 tensor, 447–454 addition, 450 algebraic properties, 450–452 contraction, 451 differentiation, 462–468 Einstein curvature, 471 electrodynamics, 459–461 energy momentum, 471 Levi-Civita symbols, 453 metric, 454–461 definition, 456 relativity, 458 multiplication, 451 numerical, 452–454 rank of, 448 Ricci, 470 Riemann curvature, 468–471 symmetrization, 452 torsion, 463 terminal velocity, 559 theorem central limit, 809 time coordinate, 239–240 proper, 239–240 time constant, 586 toroidal coordinates, 74, 213, 437 torque, 28 torsion tensor, 463 transfer function, 586 transform cosine, 697 Fourier, 693–712 and quark model, 702 application to DEs, 702–704 convolution theorem, 724 examples, 698–702 Gaussian, 699 Green’s functions, 705–712 heat equation in 1D, 704 inverse, 695 of delta function, 698 properties, 696 integral, 693 Laplace, 712–723 and differential equations, 718– 721 Bromwich contour, 722 convolution, 716 cosine, 713 derivative, 717–718 first shift, 714 gamma function, 713 imaginary exponential, 713 integral, 717–718 inverse, 721–723 linearity, 714 Mellin inversion integral, 722 periodic functions, 716 properties, 713–717 second shift, 714 sine, 713 step function, 713 unit function, 713 sine, 697 transformation active, 178 coordinate, 13 differentiation, 197 gauge, 418 Legendre, 54, 748 linear, 216 Lorentz, 243–247 matrix in space, 195 orthogonal, 442 INDEX passive, 178 rigid, 190 transient term, 586 transpose of a matrix, 181 transposition, 181 properties, 182 triangle inequality, 480 tunneling, 676 uncertainty relation, 699 uniform convergence Weierstrass M-test, 276 uniformly convergent series, 277–279 union, 782 unit matrix, 180 unit vectors, universal set, 782 partition, 785 Van de Graff, 117 Vandermonde, 210 variable random continuous, 801–809 transformation, 804–806 variance, 790, 801 variational problem, 728–740 constraints, 738–740 several dependent variables, 734 several independent variables, 734 soap film, 733 vector Cartesian component, 216 n-dimensional, 216 column, 177 component, 176 contravariant, 445–447 coordinate system, 16–31 covariant, 445–447 cross product, 7–10 field conservative, 398–404 curl, 391–398 flux, 365–369 G-orthogonal, 219 in space, 199 indices, 439–471 831 inner product, 182–191, 198–202 plane, 3–10, 174–191 position, 19 row, 181 space, 3–10, 192–207 time derivative, 350–355 transformation, 194–198 transformation of components, 176– 182 transformation properties, 441–445 unit, vector analysis curvilinear, 423–435 vector field conservative curl, 400 curl of, 394 divergence, 374 vector potential, 408 vector space, 173, 215–227 velocity, 44 terminal, 559 Venn diagrams, 783 vibrating membrane, 686–687 Vieta, 481 Wallis, 97, 293, 321, 326 Wallis, John biography, 90 wave equation, 543, 680–687 advanced Green’s function, 712 from Maxwell’s equations, 417 Green’s function, 711–712 retarded Green’s function, 712 wave guide, 682–686 cylindrical, 686 longitudinal part, 682 rectangular, 685 transverse part, 682 weight function, 227 Wheatstone, 382 Wronskian, 566–567 Yukawa potential, 700 zero zero zero zeta mass, 250 matrix, 180 spacetime distance, 242 function, 269 ... separating the level of mathematical sophistication expected of students of introductory physics from that expected of students of advanced courses of undergraduate physics and engineering While.. .Mathematical Methods Sadri Hassani Mathematical Methods For Students of Physics and Related Fields 123 Sadri Hassani IIlinois State University... required for introductory physics courses taken by physics, engineering, and chemistry majors, the next level of courses—both in physics and engineering—already demands a readiness for such intricate