Mathematical tools for physics, james nearing

www.TheSolutionManual.com www.TheSolutionManual.com Mathematical Tools for Physics by James Nearing Physics Department University of Miami jnearing@miami.edu www.physics.miami.edu/nearing/mathmethods/ Copyright 2003, James Nearing Permission to copy for individual or classroom use is granted QA 37.2 Rev May, 2010 www.TheSolutionManual.com Contents iii Bibliography v Basic Stuff Fourier Series Examples Computing Fourier Series Choice of Basis Musical Notes Periodically Forced ODE’s Return to Parseval Gibbs Phenomenon Trigonometry Parametric Differentiation Gaussian Integrals erf and Gamma Differentiating Integrals Polar Coordinates Sketching Graphs Infinite Series Vector Spaces 24 Operators and Matrices 144 The Idea of an Operator Definition of an Operator Examples of Operators Matrix Multiplication Inverses Rotations, 3-d Areas, Volumes, Determinants Matrices as Operators Eigenvalues and Eigenvectors Change of Basis Summation Convention Can you Diagonalize a Matrix? Eigenvalues and Google Special Operators 53 Complex Numbers Some Functions Applications of Euler’s Formula Geometry Series of cosines Logarithms Mapping Differential Equations 124 The Underlying Idea Axioms Examples of Vector Spaces Linear Independence Norms Scalar Product Bases and Scalar Products Gram-Schmidt Orthogonalization Cauchy-Schwartz inequality Infinite Dimensions The Basics Deriving Taylor Series Convergence Series of Series Power series, two variables Stirling’s Approximation Useful Tricks Diffraction Checking Results Complex Algebra 101 68 Linear Constant-Coefficient Forced Oscillations Series Solutions Some General Methods Trigonometry via ODE’s Green’s Functions Separation of Variables Circuits Simultaneous Equations Simultaneous ODE’s Legendre’s Equation Asymptotic Behavior Multivariable Calculus 180 Partial Derivatives Chain Rule Differentials Geometric Interpretation Gradient Electrostatics Plane Polar Coordinates Cylindrical, Spherical Coordinates Vectors: Cylindrical, Spherical Bases i www.TheSolutionManual.com Introduction 13 Vector Calculus Gradient in other Coordinates Maxima, Minima, Saddles Lagrange Multipliers Solid Angle Rainbow 214 Fluid Flow Vector Derivatives Computing the divergence Integral Representation of Curl The Gradient Shorter Cut for div and curl Identities for Vector Operators Applications to Gravity Gravitational Potential Index Notation More Complicated Potentials 10 Partial Differential Equations 14 Complex Variables 15 Fourier Analysis 243 16 Calculus of Variations 384 Examples Functional Derivatives Brachistochrone Fermat’s Principle Electric Fields Discrete Version Classical Mechanics Endpoint Variation Kinks Second Order 268 Interpolation Solving equations Differentiation Integration Differential Equations Fitting of Data Euclidean Fit Differentiating noisy data Partial Differential Equations 12 Tensors 371 Fourier Transform Convolution Theorem Time-Series Analysis Derivatives Green’s Functions Sine and Cosine Transforms Wiener-Khinchine Theorem The Heat Equation Separation of Variables Oscillating Temperatures Spatial Temperature Distributions Specified Heat Flow Electrostatics Cylindrical Coordinates 11 Numerical Analysis 348 Differentiation Integration Power (Laurent) Series Core Properties Branch Points Cauchy’s Residue Theorem Branch Points Other Integrals Other Results 17 Densities and Distributions 295 410 Density Functionals Generalization Delta-function Notation Alternate Approach Differential Equations Using Fourier Transforms More Dimensions Examples Components Relations between Tensors Birefringence Non-Orthogonal Bases Manifolds and Fields Coordinate Bases Basis Change Index ii 430 www.TheSolutionManual.com Vector Calculus 326 Integrals Line Integrals Gauss’s Theorem Stokes’ Theorem Reynolds Transport Theorem Fields as Vector Spaces Introduction I wrote this text for a one semester course at the sophomore-junior level Our experience with students taking our junior physics courses is that even if they’ve had the mathematical prerequisites, they usually need more experience using the mathematics to handle it efficiently and to possess usable intuition about the processes involved If you’ve seen infinite series in a calculus course, you may have no idea that they’re good for anything If you’ve taken a differential equations course, which of the scores of techniques that you’ve seen are really used a lot? The world is (at least) three dimensional so you clearly need to understand multiple integrals, but will everything be rectangular? How you learn intuition? When you’ve finished a problem and your answer agrees with the back of the book or with your friends or even a teacher, you’re not done The way get an intuitive understanding of the mathematics and of the physics is to analyze your solution thoroughly Does it make sense? There are almost always several parameters that enter the problem, so what happens to your solution when you push these parameters to their limits? In a mechanics problem, what if one mass is much larger than another? Does your solution the right thing? In electromagnetism, if you make a couple of parameters equal to each other does it reduce everything to a simple, special case? When you’re doing a surface integral should the answer be positive or negative and does your answer agree? When you address these questions to every problem you ever solve, you several things First, you’ll find your own mistakes before someone else does Second, you acquire an intuition about how the equations ought to behave and how the world that they describe ought to behave Third, It makes all your later efforts easier because you will then have some clue about why the equations work the way they It reifies the algebra Does it take extra time? Of course It will however be some of the most valuable extra time you can spend Is it only the students in my classes, or is it a widespread phenomenon that no one is willing to sketch a graph? (“Pulling teeth” is the clich´e that comes to mind.) Maybe you’ve never been taught that there are a few basic methods that work, so look at section 1.8 And keep referring to it This is one of those basic tools that is far more important than you’ve ever been told It is astounding how many problems become simpler after you’ve sketched a graph Also, until you’ve sketched some graphs of functions you really don’t know how they behave When I taught this course I didn’t everything that I’m presenting here The two chapters, Numerical Analysis and Tensors, were not in my one semester course, and I didn’t cover all of the topics along the way Several more chapters were added after the class was over, so this is now far beyond a one semester text There is enough here to select from if this is a course text, but if you are reading it on your own then you can move through it as you please, though you will find that the first five chapters are used more in the later parts than are chapters six and seven Chapters 8, 9, and 13 form a sort of package I’ve tried to use examples that are not all repetitions of the ones in traditional physics texts but that provide practice in the same tools that you need in that context The pdf file that I’ve placed online is hyperlinked, so that you can click on an equation or section reference to go to that point in the text To return, there’s a Previous View button at the top or bottom of the reader or a keyboard shortcut to the same thing [Command← on Mac, Alt← on Windows, Control← on Linux-GNU] The index pages are hyperlinked, and the contents also appear in the bookmark window iii www.TheSolutionManual.com 2008 A change in notation in this edition: For polar and cylindrical coordinate systems it is common to use theta for the polar angle in one and phi for the polar angle in the other I had tried to make them the same (θ) to avoid confusion, but probably made it less rather than more helpful because it differed from the spherical azimuthal coordinate In this edition all three systems (plane polar, cylindrical, spherical) use phi as φ = tan−1 (y/x) In line integrals it is common to use ds for an element of length, and many authors will use dS for an element of area I have tried to avoid this confusion by sticking to d and dA respectively (with rare exceptions) In many of the chapters there are “exercises” that precede the “problems.” These are supposed to be simpler and mostly designed to establish some of the definitions that appeared in the text This text is now available in print from Dover Publishers They have agreed that the electronic version will remain available online iv www.TheSolutionManual.com I chose this font for the display versions of the text because it appears better on the screen than does the more common Times font The choice of available mathematics fonts is more limited I’d like to thank the students who found some, but probably not all, of the mistakes in the text Also Howard Gordon, who used it in his course and provided me with many suggestions for improvements Prof Joseph Tenn of Sonoma State University has given me many very helpful ideas, correcting mistakes, improving notation, and suggesting ways to help the students Bibliography Mathematical Methods for Physics and Engineering by Riley, Hobson, and Bence Cambridge University Press For the quantity of well-written material here, it is surprisingly inexpensive in paperback Mathematical Methods in the Physical Sciences by Boas John Wiley Publ About the right level and with a very useful selection of topics If you know everything in here, you’ll find all your upper level courses much easier Mathematical Methods for Physicists by Arfken and Weber Academic Press level, but it is sufficiently thorough that will be a valuable reference work later At a more advanced Mathematical Methods by Hassani Springer At the same level as this text with many of the same topics, but said differently It is always useful to get a second viewpoint because it’s commonly the second one that makes sense — in whichever order you read them Schaum’s Outlines by various There are many good and inexpensive books in this series: for example, “Complex Variables,” “Advanced Calculus,” “German Grammar.” Amazon lists hundreds Visual Complex Analysis by Needham, Oxford University Press The title tells you the emphasis Here the geometry is paramount, but the traditional material is present too It’s actually fun to read (Well, I think so anyway.) The Schaum text provides a complementary image of the subject Complex Analysis for Mathematics and Engineering by Mathews and Howell Jones and Bartlett Press Another very good choice for a text on complex variables Despite the title, mathematicians should find nothing wanting here Applied Analysis by Lanczos Dover Publications This publisher has a large selection of moderately priced, high quality books More discursive than most books on numerical analysis, and shows great insight into the subject Linear Differential Operators by Lanczos Dover publications insights and unusual ways to look at the subject As always with this author, useful Numerical Methods that (usually) Work by Acton Mathematical Association of America tools with more than the usual discussion of what can (and will) go wrong Practical Numerical Recipes by Press et al Cambridge Press The standard current compendium surveying techniques and theory, with programs in one or another language A Brief on Tensor Analysis by James Simmonds Springer will recommend To anyone Under any circumstances Linear Algebra Done Right by Axler Springer Linear Algebra Done Wrong by Treil its own sake This is the only text on tensors that I Don’t let the title turn you away It’s pretty good (online at Brown University) Linear Algebra not just for Advanced mathematical methods for scientists and engineers by Bender and Orszag Springer Material you won’t find anywhere else, with clear examples “ a sleazy approximation that provides v www.TheSolutionManual.com Mathematical Methods in Physics by Mathews and Walker More sophisticated in its approach to the subject, but it has some beautiful insights It’s considered a standard, though now hard to obtain good physical insight into what’s going on in some system is far more useful than an unintelligible exact result.” Probability Theory: A Concise Course by Rozanov Dover way in 148 pages Clear and explicit and cheap Starts at the beginning and goes a long Calculus of Variations by MacCluer Pearson Both clear and rigorous, showing how many different types of problems come under this rubric, even “ operations research, a field begun by mathematicians, almost immediately abandoned to other disciplines once the field was determined to be useful and profitable.” Special Functions and Their Applications by Lebedev Dover functions developed in order to be useful, not just for sport The most important of the special www.TheSolutionManual.com The Penguin Dictionary of Curious and Interesting Geometry by Wells Penguin Just for fun If your heart beats faster at the sight of the Pythagorean Theorem, wait ’til you’ve seen Morley’s Theorem, or Napoleon’s, or when you first encounter an unduloid in its native habitat vi 17—Densities and Distributions 580 17.10 Interpret the functional derivative of the functional in Eq (18): δ δ[φ] δφ Despite appearances, this actually makes sense Ans: δ(x) 17.12 Verify the derivation of Eq (35) Also examine this solution for the cases that x0 is very large and that it is very small 17.13 Fill in the steps in section 17.7 leading to the Green’s function for g − k g = δ 17.14 Derive the analog of Eq (38) for the case x < y 17.15 Calculate the contribution of the second exponential factor leading to Eq (40) 17.16 Starting with the formulation in Eq (23), what is the result of δ and of δ on a test function? Draw sketches of a typical δn , δn , and δn 17.17 If ρ(r ) = qa2 ∂ δ(r )/∂z , compute the potential and sketch the charge density You should express your answer in spherical coordinates as well as rectangular, perhaps commenting on the nature of the results and relating it to functions you have encountered before You can this calculation in either rectangular or spherical coordinates Ans: (2qa2 /4π )P2 (cos θ)/r3 17.18 What is a picture of the charge density ρ(r ) = qa2 ∂ δ(r )/∂x∂y? (Planar quadrupole) What is the potential for this case? 17.19 In Eq (45) I was not at all explicit about which variables are kept constant in each partial derivative Sort this out for both ∂/∂z and for ∂/∂uz www.TheSolutionManual.com 17.11 Repeat the derivation of Eq (33) but with less labor, selecting the form of the function g to simplify the work In the discussion following this equation, reread the comments on this subject 17—Densities and Distributions 581 17.21 Calculate the mean, variance, skewness, and the kurtosis excess for the density f (g) = A[δ(g) + δ(g − g0 ) + δ(g − xg0 )] See how these results vary with the parameter x Ans: skewness = 2−3/2 (1 + x)(x − 2)(2x − 1)/ − x + x2 kurt excess = −3 + 3/4 + x4 + (1 − x)4 / − x + x2 17.22 Calculate the potential of a linear quadrupole as in Eq (48) Also, what is the potential of the planar array mentioned there? You should be able to express the first of these in terms of familiar objects 17.23 (If this seems out of place, it’s used in the next problems.) The unit square, < x < and < y < 1, has area dx dy = over the limits of x and y Now change the variables to u = 12 (x + y) v =x−y and and evaluate the integral, du dv over the square, showing that you get the same answer You have only to work out all the limits Draw a picture This is a special example of how to change multiple variables of integration The single variable integral generalizes from f (x) dx = f (x) dx du du to where ∂(x, y) = det ∂(u, v) f (x, y) dx dy = ∂x ∂u ∂y ∂u f (x, y) ∂x ∂v ∂y ∂v For the given change from x, y to u, v show that this Jacobian determinant is one See for example maths.abdn.ac.uk/˜igc/tch/ma2001/notes/node77.html ∂(x, y) du dv ∂(u, v) www.TheSolutionManual.com 17.20 Use the results of the problem 16, showing graphs of δn and its derivatives Look again at the statements leading up to Eq (31), that g is continuous, and ask what would happen if it is not Think of the right hand side of Eq (30) as a δn too in this case, and draw a graph of the left side of the same equation if gn is assumed to change very fast, approaching a discontinuous function as n → ∞ Demonstrate by looking at the graphs of the left and right side of the equation that this can’t be a solution and so that g must be continuous as claimed 17—Densities and Distributions 582 x = (g1 + g2 ) and y = g1 − g2 then the fraction of these coordinates between x and x + dx and y and y + dy is f (g1 )f (g2 )dx dy = f (x + y/2)f (x − y/2)dx dy Note where the result of the preceding problem is used here For fixed x, integrate over all y and that gives you the fraction between x and x + dx That is the distribution function for (g1 + g2√)/2 Ans: Another Gaussian with the same mean and with rms deviation from the mean decreased by a factor 17.25 Same problem as the preceding one, but the initial function is f (g) = a/π + g2 (−∞ < g < ∞) a2 In this case however, you don’t have to evaluate the mean and the rms deviation Show why not Ans: The result reproduces the original f (g) exactly, with no change in the spread These two problems illustrate examples of “stable distributions,” for which the distribution of the average of two variables has the same form as the original distribution, changing at most the widths There are an infinite number of other stable distributions, but there are only three that have simple and explicit forms These examples show two of them 17.26 Same problem as the preceding two, but the initial function is f (g) = 1/gmax for < g < gmax 17.27 In the same way as defined in Eq (10), what is the functional derivative of Eq (5)? 17.28 Rederive Eq (27) by choosing an explicit delta sequence, δn (x) www.TheSolutionManual.com 17.24 In problem you found the mean and variance for a Gaussian, f (g) = Ae−B(g−g0 ) Interpreting this as a distribution of grades in a class, what is the resulting distribution of the average of any two students? That is, given this function for all students, what is the resulting distribution of (g1 + g2 )/2? What is the mean of this and what is the root-mean-square deviation from the mean? How these compare to the original distribution? To this, note that f (g)dg is the fraction of students in the interval g to g + dg, so f (g1 )f (g2 )dg1 dg2 is the fraction for both Now make the change of variables Abramowitz and Stegun, 9, 65, 461 absolute convergence, 37 acceleration, 82 Adams methods, 375 stable 378 alternating symbol, 228, 307, 308 alternating tensor, 206, 408, 435, 436 Λ, 205 analytic, 476 core properties 477 function 471 angular momentum, 187–190, 194–196, 212, 396, 540, 543 angular velocity, 188, 212 annulus, 475 anti-commutator, 231 Antihermitian, 223 antisymmetric, 206, 207, 223, 407–409, 436 area element, 248 area mass density, 246 area vector, 282, 289 arithmetic mean, 155 as the crow flies, 172 asphalt, 531 associative law, 161, 163 asteroid, 279 asymptotic, 42 atlas, 488 autocorrelation function, 516 backwards iteration, 378 barn, 264 basis, 129, 135, 138, 143, 155, 166–176, 191–219, 337, 343, 402–433 change 216, 429 cylindrical 252 Hermite 229 reciprocal 416 spherical 252 Bernoulli, 527 Bessel, 96, 98, 121, 246, 273, 470, 499 β = 1/kT , 262, 276 bilinear concomitant, 136, 138 billiard ball, 276 binomial: coefficient, 40, 43 series 31, 40, 49, 59, 83, 474 birefringence, 409 blizzard, 245 Bolivia, 488 boundary condition, 101, 136–142, 303, 324, 326, 329, 330, 334, 343, 347, 353, 525 boundary layer, 126 boundary value, 495 boundary value problem, 329 brachistochrone, 441, 527 branch point, 477, 486–488, 492, 493 Brazil, 488 bulk modulus, 148 • CaCO3 , 409 • 583 www.TheSolutionManual.com Index calcite, 409 Calculus of Variations, 521 capacitance, 534 capacitor, 106 Cauchy, 478, 495 Cauchy sequence, 177, 184 Cauchy-Schwartz inequality, 170, 174–176, 178 Cayley-Hamilton Theorem, 230 center of mass, 559 central limit theorem, 44 chain rule, 27, 238, 239, 271 characteristic equation, 213, 220, 221 characteristic polynomial, 230 charge density, 574 charged rings, 52 Chebyshev polynomial, 184 checkerboard, 344 chemistry, 185 circuit, 106, 157 circulation density, 448 clarinet, 141 Coast and Geodetic Survey, 244 coaxial cable, 534 coefficient of expansion, 521 combinatorial factor, 40, 59 commutative law, 161 commutator, 231 comparison test, 34–36 complete, 177 complete the square, 38, 505 complex variable, 469–495, 504 584 complex: conjugate, 73, 136, 170, 172, 470 exponential 69, 71, 79, 81, 83, 90, 135, 145 integral 472 logarithm 76 mapping 77, 84 number 67, 486 polar form 71, 73 square root 68, 75 trigonometry 73 component, 129, 148, 166–167, 178, 186, 195, 402–434 not 425 operator 191–222 composition, 200, 210, 435 conductivity, 321, 322, 323, 350, 401 conservative field, 450 constraint, 258, 260, 262 continuity condition, 303, 316 contour integral, 472, 477, 478, 504, 511, 573, 575 examples 479–486, 492 contour map, 244 contravariant, 417–428, 431 convergence, 176 convergence test, 35 convergence: absolute, 37 non 361 speed 152, 326, 351, 362 convolution, 506, 517 coordinate, 296, 422–434 cylindrical 345 change 239 www.TheSolutionManual.com Index cylindrical 247, 252, 293, 298 non-orthogonal 425, 428 polar 246 rectangular 246, 290 spherical 247, 252, 293, 298 correlation matrix, 383 cosh, 2, 72, 121, 137, 324, 526, 532 cosine transform, 513 Court of Appeals, 172 covariant, 417–431 critical damping, 120 cross product, 307 cross section, 264–269 absorption 264 scattering 264 crystal, 190, 398, 416 curl, 287, 294, 308, 452, 456, 457 components 295, 297 current density, 305 cyclic permutation, 307 cycloid, 529, 553 • d3 r, 248 data compression, 381 daughter, 122 ∇, 246, 254, 292, 296, 526 components 297 identities 298 ∇2 , 302, 345 ∇i , 308 δ(x), 565 δij , 173, 201, 228, 317 delta function, 562, 567 delta sequence, 566 585 delta-functional, 565 density, 556 derivative, 11, 191, 453, 469 derivative, numerical, 363–365 determinant, 109–115, 203–215, 225, 411 of composition 210 diagonal, 211 diagonalize, 258 diamonds, 277 dielectric tensor, 408 difference equation, 377 differential, 235–243, 254, 526 differential equation, 33, 70, 562 Greens’ function 571 inhomogeneous 569 constant coefficient 87, 377 eigenvector 214 Fourier 135, 146 Green’s function 102 indicial equation 98 inhomogeneous 91 linear, homogeneous 87 matrix 220 numerical solution 372–378 ordinary 87–115 partial 321–345, 386–389 separation of variables 104 series solution 95 simultaneous 111, 214 singular point 96 trigonometry 101, 121 vector space 168 differential operator, 292 differentiation, 472, 509, 524 diffraction, 46, 62, 81 www.TheSolutionManual.com Index dimension, 166, 191, 200 dimensional analysis, 50, 484 dipole moment density, 410 Dirac, 564 direct basis, 416, 426, 436 disaster, 375 dispersion, 269, 277, 387, 393 dissipation, 387, 388, 393 distribution, 564 divergence, 287–308, 323, 437 divergence theorem, 445 divergence: components, 289 cylindrical coordinates 293, 297 integral form 289 dog-catcher, 64 domain, 21, 190, 191, 398 Doppler effect, 61 drumhead, 163, 172, 246, 272 dry friction, 88, 123 dumbbell, 195, 226 dx, 236 • eigenvalue, 137, 212, 224, 384 eigenvector, 211, 221, 222, 224, 229, 384, 413 electric circuit, 105 electric dipole, 60, 274, 318 electric field, 53, 55, 190, 245, 311, 313, 338, 346, 352, 433, 533 electron mass, 313 electrostatic potential, 338 electrostatics, 52, 251, 345, 534 ellipse, 461 ellipsoid, 288 586 EMF, 456 endpoint variation, 541 energy density, 313, 315, 408 entire function, 495 ijk , 228 equal tempered, 140 equipotential, 244, 258 erf, 8, 24, 28, 390 error function, 8, 24, 25, 59 essential singularity, 474, 496 Euclidean fit, 381, 394 Euler, 524, 536 constant 25 formula 70, 72, 79, 485 method 372, 373, 374, 387, 393 Euler-Lagrange, 526, 544 extrapolation, 390 • Faraday, 455 Fermat, 530, 547 filter circuit, 157 Flat Earth Society, 306 Florida, 244, 326 fluid: equilibrium, 315 expansion 287, 289 flow 286 flow rate 281, 282, 305 rotating 287 flute, 141 focal length, 551 focus, 522, 550 focus of lens, 63 focus of mirror, 63 www.TheSolutionManual.com Index 587 • Gauss, 379, 493 Gauss elimination, 208, 232 Gauss reduction, 204 Gauss’s Theorem, 298, 444, 533 Gaussian, 41, 44, 505, 508, 559, 579 Gaussian integration, 390 Gaussian: integrals, 6–8, 24, 42 integration 370–372, 392 quadrature 371 generalized function, 562, 568 generating function, 125 geodesic, 522, 527 geometric optics, 266, 269 geometric series, 30, 38, 45 Gibbs phenomenon, 132, 149 glycerine, 545 Google, 222 grade density, 559 gradient, 243–246, 253, 296, 315, 426, 428, 444, 449, 457, 526 covariant components 426 cylindrical coordinates 253 spherical coordinates 254 Gram-Schmidt, 174, 178, 184 graphs, 21–23 grating, 81 gravitational energy, 315 gravitational: energy, 315, 316 field 253, 273, 299, 312 potential 244, 274, 301, 338 Green’s function, 102–104, 121, 124, 510, 562, 574 Gamma function, 8–9, 25, 41 • Fourier series, 33, 128, 131–149, 166, 173, 178, 224, 324–345, 502 bases 135 basis 174 best fit 256 does it work? 143 fundamental theorem 136 sound of 141 square wave 132 Fourier sine, cosine, 513 Fourier transform, 502–503, 513, 572, 574 frames, 166 frequency, 507 Frobenius series, 33, 96–99, 121 function, 229, 421, 423, 435, 477 df 236, 237 addition 161 composition 435 definition of 164, 397 elementary 8, 9, 30 linear 190 operator 186 functional, 399–418, 523, 558–565 functional derivative, 526, 536, 540 functional: derivative, 560 integral 560 3-linear 408 bilinear 400, 402, 405, 407 multilinear 400 n-linear 401 representation 399, 405, 417 fundamental theorem of calculus, 14 www.TheSolutionManual.com Index half-life, 122 Hamiltonian, 530, 543 harmonic, 140 harmonic oscillator, 70, 88–89, 95, 326, 346 Green’s function 562 critical 120, 220 damped 88, 123 energy 91 forced 91, 92, 144 Green’s function 102, 122 Greens’ function 510 resonance 145 harmonic resonance, 147 Hawaii, 521 heads, 10 heat flow, 321–338 heat flow vector, 322 heat wave, 327 heat, minimum, 271, 275 heated disk, 242 Helmholtz Decomposition, 457 Helmholtz-Hodge, 460 hemisemidemiquaver, 507 Hermite, 179, 228, 391 Hermitian, 223, 456 Hero, 26 Hessian, 256, 548 Hilbert Space, 177 history, blame, 565 horoscope, 379 Hulk, 64 Huygens, 46 hyperbolic functions, 2–5, 73, 121, 324, 331, 505 inverse • 588 Iceland spar, 409 ideal gas, 559 Idempotent, 223 impact parameter, 265 impedance, 107, 127 impulse, 102, 104 independence, 166, 380 index notation, 317, 318, 404, 417 index of refraction, 62, 268, 277, 442 indicial equation, 98 inductor, 106 inertia tensor, 223 inertia: moment of, 16, 195, 228, 250 tensor 190, 194–198, 396, 401, 408, 435 infallible, 50 infinite series, 79, 128 infinite-dimensional, 176 infinitesimal, 235, 237, 422 inflation, 57 instability, 361, 377, 387 integral, 12–19 contour 472 fractional 519 numerical 366–372 principal value 390 Riemann 12 Stieljes 16 surface 283, 289, 294 test 35, 334 intensity, 46–48, 50, 148, 269 interest rate, 57 interpolation, 357–392 www.TheSolutionManual.com Index inverse transform, 504 iterative method, 100 • Jacobi, 231, 430 Jacobian, 581 • kinetic energy, 58, 272, 560 density 246 kinks, 544 Klein bottle, 448 kludge, 563 Kramer’s rule, 204, 232 Kronecker delta, 173, 201, 306 kurtosis, 559, 579 • L-R circuit, 106 Lagrange, 524, 536 Lagrange multiplier, 258–262, 275, 276, 383, 386 Lagrange multipliers, 540 Lagrangian, 539 Laguerre, 96, 391 Lanczos, 372 Laplace, 162, 337, 339, 457, 534, 553 Laplacian, 302, 345, 355 Laurent series, 473, 478, 496, 497 Lax-Friedrichs, 389, 393 Lax-Wendroff, 389, 393 least square, 255, 379, 381, 391 least upper bound, 170 Lebesgue, 15 Legendre, 60, 96, 115–118, 128, 179, 226, 274, 313, 355, 371, 393 Leibnitz, 238 589 length of curve, 439, 523 lens, 551 limit cycle, 361 line integral, 443, 451, 473 linear charge density, 55, 346 linear programming, 277 linear: difference equation, 377, 393 equations 26, 109 function 397 functional 399, 400 independence 166 transformation 190 linearity, 187, 189, 435 Lobatto integration, 393 logarithm, 3, 31, 76, 106, 490 longitude, latitude, 248 Lorentz force, 456 • m(V ), 557 magnetic field, 253, 272, 314, 420, 433 tensor 229, 408 magnetic flux, 455 magnetic monopole, 464 manifold, 421, 423, 431 mass density, 250, 299, 312 matrix, 172, 192–222, 380, 404 correlation 383 positive definite 258 as operator 209 column 169, 173, 193, 209, 227, 406 diagonal 211 diagonalize? 219 identity 201 multiplication 193, 200 scalar product 172 www.TheSolutionManual.com Index Maxwell, 409 Maxwell’s equations, 163, 453, 456, 465 Maxwell-Boltzmann, 559 mechanics, 539 messy and complicated, 93 metric tensor, 402, 418, 428, 431 microwave, 338 midpoint integration, 14, 368 mirage, 532 Morse Theory, 550 musical instrument, 140 Măobius strip, 448 • nth mean, 65 natural boundary, 501 New York, 172 Newton, 527, 539 gravity 298–304, 313, 315 method 359 Nilpotent, 223, 230 Noether’s theorem, 543 noise, 385, 392 norm, 169–176, 180, 256 normal modes, 113, 170 • Oops, 563 operator, 186, 190–222 components 192, 217 differential 137, 199 exponential 61, 274 inverse 202 rotation 186, 193, 202 translation 229 vector 298 590 optical path, 442, 531 optics, 530 order, 474 orientation, 204 Orthogonal, 223, 457 orthogonal coordinate, 248 orthogonal coordinates, 248, 293, 423 orthogonality, 130, 137, 256, 326, 348 orthogonalization, 174 orthonormal, 172, 173 oscillation, 89, 113, 115, 342 coupled 112, 124, 168, 214 damped 90 forced 95 temperature 326, 351 • panjandrum, 556 paraboloid, 258 parallel axis theorem, 197 parallelepiped, 225, 282, 436 parallelogram, 202, 205, 225 parallelogram identity, 180 parameters, 51 parametric differentiation, 5, 7, 25 Parseval’s identity, 142, 147, 156, 507 ∂i , 308, 317 partial integration, 19, 29, 136, 458, 525, 541, 560 ∂S, ∂V , 445 partition function, 262 Pascal’s triangle, 61 Pauli, 227 PDE, numerical, 386 Peano, 160 www.TheSolutionManual.com Index Index principal components, 384, 394 principal value, 390 probability, 10 product formula, 471 Pythagorean norm, 172 • quadratic equation, 63, 82 quadrupole, 60, 253, 273, 275, 318 quasi-equilibrium, 95 • radian, radioactivity, 122 rainbow, 266–269, 276 random variable, 385 range, 190, 398 ratio test, 34 rational number, 176 reality, 334, 357 reciprocal basis, 416, 422, 426, 436 reciprocal vector, 430 rectifier, 154 regular point, 95 regular singular point, 96 relation, 397 relativity, 58, 431 residue, 478, 512 residue theorem, 478, 479 resistor, 106, 271, 275 resonance, 145 Reynolds’ transport theorem, 455, 465 Riemann Integral, 12, 472 Riemann Surface, 486, 488, 489, 490, 498 Riemann-Stieljes integral, 16, 228 rigid body, 187, 194, 212, 396 www.TheSolutionManual.com pendulum, 168 periodic, 71, 339 periodic boundary condition, 343, 502, 513 perpendicular axis theorem, 227 Perron-Frobenius, 223 pinking shears, 489 pitfall, 361 Plateau, 545 Poisson, 301, 574 polar coordinates, 20, 82, 240, 273, 473, 540 polarizability, 190 polarizability tensor, 398, 410 polarization, 190, 269, 410 pole, 474, 480, 494 order 474, 475 polynomial, 160, 493 characteristic 230 Chebyshev 184 Hermite 179, 228, 391 Laguerre 391 Legendre 60, 117, 125, 179, 226, 274, 316, 318, 355, 371, 581 population density, 556 Postal Service, 171, 277 potential, 60, 301, 304, 316, 338–345, 533, 576 potential energy density, 272 power, 321, 334 power mean, 65 power series, 471, 524 power spectrum, 149 Poynting vector, 413 pre-Snell law, 276 pressure tensor, 401 prestissimo, 507 591 roots of unity, 75 rotation, 186, 190, 193, 202, 287, 294 components 194 composition 201 roundoff error, 378, 391 rpncalculator, ruler, 521 Runge-Kutta, 372, 394 Runge-Kutta-Fehlberg, 374 • saddle, 258 saddle point, 254, 256, 275 scalar product, 134, 146, 169–176, 179, 180, 224, 307, 330, 340, 399, 415, 426 scattering, 264–269 scattering angle, 265, 276 Schwartz, 566 Schwarzenegger, 245 secant method, 362 Self-adjoint, 223 semiperimeter, 26 separated solution, 330 separation of variables, 104, 323–345 series, 30–46 of series 37 absolute convergence 37 common 31 comparison test 34, 36 convergence 34, 57 differential equation 95, 124 double 344, 373 examples 30 exponential 228 faster convergence 60 Frobenius 96–99, 473 592 geometric 228 hyperbolic sine 331 integral test 35 Laurent 473 power 31, 290, 357 ratio test 34 rearrange 37 secant 38, 57 telescoping 60, 154 two variables 38, 59 sheet, 488, 490 dσ/dΩ, 265–269, 276 similarity transformation, 217 simple closed curve, 478 simply-connected, 451, 452 Simpson, 359 Simpson’s rule, 368, 554 simultaneous equations, 109, 112 sine integral, 151, 157 sine transform, 513 singular perturbations, 126 singular point, 95 singularity, 474, 478, 492 sinh, 2, 72, 121, 137, 324, 526 sketching, 21 skewness, 559, 579 Snell, 530 snowplow, 122 soap bubbles, 545 Sobolev, 183, 566 solenoid, 252, 319 solid angle, 263, 265 space, 421 specific heat, 234, 321 www.TheSolutionManual.com Index spectral density, 515 speed of light, 523 Spiderman, 64 √ 2, 176 square-integrable, 162, 165 stable distribution, 582 stainless steel, 350 Stallone, 245 steady-state, 109, 127 step function, 562, 565 sterradian, 263 Stirling’s formula, 40, 59, 261 stock market, 128 Stokes’ Theorem, 298, 449, 453 straight line, 527 strain, 288 stress, 401 stress-strain, 398 string, 181 Sturm-Liouville, 136, 353 subspace, 457 sum of cosines, 83 Sumerians, summation by parts, 19 summation convention, 219, 228, 306, 417 sun, 315 superposition, 336 surface integral, 282, 285, 449, 453, 455 closed 289 examples 283 Symmetric, 223, 407, 456 • tails, 10 593 tanh, Taylor series, 31, 474, 476, 496 telescoping series, 60, 154 temperature, 262 temperature gradient, 531 temperature: distribution, 328, 332 expansion 521 of slab 324 oscillation 326 profile 326 tensor, 186, 288, 397 component 403 contravariant components 417 field 422 inertia 189, 194–198, 212, 226, 396, 401, 408, 435 metric 402, 428, 431 rank 400, 401, 402, 408, 409, 417 stress 401, 422 totally antisymmetric 408 transpose 406 test function, 564, 568 Texas A&M, 161 thermal expansion, 521 θ(x), 562 thin lens, 550 time average, 148 time of travel, 441 torque, 188, 212 tough integral, 54 trace, 172, 208, 232 www.TheSolutionManual.com Index transformation, 186 area 203 basis 429 composition 210 determinant 203, 436 electromagnetic field 433, 434, 437 linear 190 Lorentz 431–432, 434 matrix 430 similarity 186, 217, 218, 230 transient, 109 transport theorem, 455 transpose, 406, 407 trapezoidal integration, 368, 536 triangle area, 20, 26 triangle inequality, 169, 175, 176 trigonometric identities, 72, 75 trigonometry, triple scalar product, 318 • uncountable sum, 181 Unitary, 223 • variance, 380, 385, 386, 392, 559, 579 variational approximation, 534 vector space, 160–175, 176, 256, 421, 456, 560 axioms 161 basis 166 dimension 166 examples 161, 170, 179 not 164, 168 scalar product 169 subspace 165 theorems 182 594 vector: calculus, 420, 439 derivative 286 eigenvector 384 field 353, 420, 421 gradient 243, 245, 259 heat flow 333 identities 317, 533 operators 298 potential 452 unit 252, 273 visualization, 270 volume element, 248 volume of a sphere, 250, 273 • wave, 411 wave equation, 274, 321, 354, 387 Weierstrass, 469 Weierstrass-Erdmann, 545 weird behavior, 547 wet friction, 88 wheel-balancing, 212 Wiener-Khinchine theorem, 515 wild oscillation, 377 winding number, 487, 490 wine cellar, 328 work, 442 work-energy theorem, 443 • ζ(2), 36, 153 zeta function, 9, 33, 57 www.TheSolutionManual.com Index ...www.TheSolutionManual.com Mathematical Tools for Physics by James Nearing Physics Department University of Miami jnearing@miami.edu www.physics.miami.edu /nearing/ mathmethods/ Copyright 2003, James Nearing Permission... the numerator for small x is approximately 1, you immediately have that Γ(x) ∼ 1/x for small x (1.15) The integral definition, Eq (1.12), for the Gamma function is defined only for the case that... toss a coin It’s straight-forward to derive this from Stirling’s formula In fact it is just as easy to a version of it for which the coin is biased, or more generally, for any case that one of