Hassani S. Mathematical Methods: For Students of Physics and Related Fields
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xii Preface
History of Mathematics archive at www-groups.dcs.st-and.ac.uk:80.
Simmons, G. Calculus Gems, McGraw-Hill, New York, 1992.
Gamow, G. The Great Physicists: From Galileo to Einstein, Dover, New York,
1961.
Although extreme care was taken to correct all the misprints, it is very
unlikely that I have been able to catch all of them. I shall be most grateful to
those readers kind enough to bring to my attention any remaining mistakes,
typographical or otherwise. Please feel free to contact me.
Sadri Hassani
Department of Physics, Illinois State University, Normal, Illinois
Note to the Reader
“Why,” said the Dodo, “the best way to ex-
plainitistodoit.”
—Lewis Carroll
Probably the best advice I can give you is, if you want to learn mathematics
and physics, “Just do it!” As a first step, read the material in a chapter
carefully, tracing the logical steps leading to important results. As a (very
important) second step, make sure you can reproduce these logical steps, as
well as all the relevant examples in the chapter, with the book closed.No
amount of following other people’s logic—whether in a book or in a lecture—
can help you learn as much as a single logical step that you have taken yourself.
Finally, do as many problems at the end of each chapter as your devotion and
dedication to this subject allows!
Whether you are a physics or an engineering student, almost all the ma-
terial you learn in this book will become handy in the rest of your academic
training. Eventually, you are going to take courses in mechanics, electro-
magnetic theory, strength of materials, heat and thermodynamics, quantum
mechanics, etc. A solid background of the mathematical methods at the level
of presentation of this book will go a long way toward your deeper under-
standing of these subjects.
As you strive to grasp the (sometimes) difficult concepts, glance at the his-
torical notes to appreciate the efforts of the past mathematicians and physi-
cists as they struggled through a maze of uncharted territories in search of
the correct “path,” a path that demands courage, perseverance, self-sacrifice,
and devotion.
At the end of most chapters, you will find a short list of references that you
may want to consult for further reading. In addition to these specific refer-
ences, as a general companion, I frequently refer to my more advanced book,
Mathematical Physics: A Modern Introduction to Its Foundations, Springer-
Verlag, 1999, which is abbreviated as [Has 99]. There are many other excellent
books on the market; however, my own ignorance of their content and the par-
allelism in the pedagogy of my two books are the only reasons for singling out
[Has 99].
Contents
Preface to Second Edition vii
Preface ix
Note to the Reader xiii
I Coordinates and Calculus 1
1 Coordinate Systems and Vectors 3
1.1 VectorsinaPlaneandinSpace.................. 3
1.1.1 DotProduct ........................ 5
1.1.2 Vector or Cross Product . . . . . . . . . . . . . . . . . . 7
1.2 CoordinateSystems ........................ 11
1.3 Vectors in Different Coordinate Systems . . . . . . . . . . . . . 16
1.3.1 FieldsandPotentials.................... 21
1.3.2 CrossProduct ....................... 28
1.4 Relations Among Unit Vectors . . . . . . . . . . . . . . . . . . 31
1.5 Problems .............................. 37
2 Differentiation 43
2.1 TheDerivative ........................... 44
2.2 PartialDerivatives ......................... 47
2.2.1 Definition, Notation, and Basic Properties . . . . . . . . 47
2.2.2 Differentials......................... 53
2.2.3 ChainRule ......................... 55
2.2.4 HomogeneousFunctions.................. 57
2.3 ElementsofLength,Area,andVolume.............. 59
2.3.1 Elements in a Cartesian Coordinate System . . . . . . . 60
2.3.2 Elements in a Spherical Coordinate System . . . . . . . 62
2.3.3 Elements in a Cylindrical Coordinate System . . . . . . 65
2.4 Problems .............................. 68
xvi CONTENTS
3 Integration: Formalism 77
3.1 “
”Means“
um” ......................... 77
3.2 PropertiesofIntegral........................ 81
3.2.1 ChangeofDummyVariable................ 82
3.2.2 Linearity .......................... 82
3.2.3 Interchange of Limits . . . . . . . . . . . . . . . . . . . 82
3.2.4 Partition of Range of Integration . . . . . . . . . . . . . 82
3.2.5 Transformation of Integration Variable . . . . . . . . . . 83
3.2.6 Small Region of Integration . . . . . . . . . . . . . . . . 83
3.2.7 IntegralandAbsoluteValue................ 84
3.2.8 SymmetricRangeofIntegration ............. 84
3.2.9 DifferentiatinganIntegral................. 85
3.2.10 Fundamental Theorem of Calculus . . . . . . . . . . . . 87
3.3 Guidelines for Calculating Integrals . . . . . . . . . . . . . . . . 91
3.3.1 ReductiontoSingleIntegrals ............... 92
3.3.2 Components of Integrals of Vector Functions . . . . . . 95
3.4 Problems .............................. 98
4 Integration: Applications 101
4.1 SingleIntegrals...........................101
4.1.1 An Example from Mechanics . . . . . . . . . . . . . . . 101
4.1.2 Examples from Electrostatics and Gravity . . . . . . . . 104
4.1.3 Examples from Magnetostatics . . . . . . . . . . . . . . 109
4.2 Applications:DoubleIntegrals ..................115
4.2.1 Cartesian Coordinates . . . . . . . . . . . . . . . . . . . 115
4.2.2 CylindricalCoordinates ..................118
4.2.3 Spherical Coordinates . . . . . . . . . . . . . . . . . . . 120
4.3 Applications:TripleIntegrals ...................122
4.4 Problems ..............................128
5 Dirac Delta Function 139
5.1 One-VariableCase .........................139
5.1.1 LinearDensitiesofPoints .................143
5.1.2 Properties of the Delta Function . . . . . . . . . . . . . 145
5.1.3 TheStepFunction .....................152
5.2 Two-VariableCase .........................154
5.3 Three-VariableCase ........................159
5.4 Problems ..............................166
II Algebra of Vectors 171
6 Planar and Spatial Vectors 173
6.1 VectorsinaPlaneRevisited....................174
6.1.1 Transformation of Components . . . . . . . . . . . . . . 176
6.1.2 InnerProduct........................182
CONTENTS xvii
6.1.3 Orthogonal Transformation . . . . . . . . . . . . . . . . 190
6.2 VectorsinSpace ..........................192
6.2.1 Transformation of Vectors . . . . . . . . . . . . . . . . . 194
6.2.2 InnerProduct........................198
6.3 Determinant.............................202
6.4 TheJacobian ............................207
6.5 Problems ..............................211
7 Finite-Dimensional Vector Spaces 215
7.1 LinearTransformations ......................216
7.2 InnerProduct............................218
7.3 TheDeterminant..........................222
7.4 EigenvectorsandEigenvalues ...................224
7.5 Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . 227
7.6 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . 230
7.7 Problems ..............................234
8 Vectors in Relativity 237
8.1 ProperandCoordinateTime ...................239
8.2 SpacetimeDistance.........................240
8.3 Lorentz Transformation . . . . . . . . . . . . . . . . . . . . . . 243
8.4 Four-Velocity and Four-Momentum . . . . . . . . . . . . . . . . 247
8.4.1 Relativistic Collisions . . . . . . . . . . . . . . . . . . . 250
8.4.2 SecondLawofMotion...................253
8.5 Problems ..............................254
III Infinite Series 257
9 Infinite Series 259
9.1 Infinite Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 259
9.2 Summations.............................262
9.2.1 Mathematical Induction . . . . . . . . . . . . . . . . . . 265
9.3 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
9.3.1 TestsforConvergence ...................267
9.3.2 OperationsonSeries....................273
9.4 Sequences and Series of Functions . . . . . . . . . . . . . . . . 274
9.4.1 Properties of Uniformly Convergent Series . . . . . . . . 277
9.5 Problems ..............................279
10 Application of Common Series 283
10.1PowerSeries.............................283
10.1.1 TaylorSeries ........................286
10.2 Series for Some Familiar Functions . . . . . . . . . . . . . . . . 287
10.3HelmholtzCoil ...........................291
10.4 Indeterminate Forms and L’Hˆopital’s Rule . . . . . . . . . . . . 294
xviii CONTENTS
10.5MultipoleExpansion........................297
10.6FourierSeries............................299
10.7 Multivariable Taylor Series . . . . . . . . . . . . . . . . . . . . 305
10.8 Application to Differential Equations . . . . . . . . . . . . . . . 307
10.9Problems ..............................311
11 Integrals and Series as Functions 317
11.1 Integrals as Functions . . . . . . . . . . . . . . . . . . . . . . . 317
11.1.1 GammaFunction......................318
11.1.2 The Beta Function . . . . . . . . . . . . . . . . . . . . . 320
11.1.3 TheErrorFunction ....................322
11.1.4 Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . 322
11.2PowerSeriesasFunctions .....................327
11.2.1 Hypergeometric Functions . . . . . . . . . . . . . . . . . 328
11.2.2 Confluent Hypergeometric Functions . . . . . . . . . . . 332
11.2.3 BesselFunctions ......................333
11.3Problems ..............................336
IV Analysis of Vectors 341
12 Vectors and Derivatives 343
12.1SolidAngle .............................344
12.1.1 Ordinary Angle Revisited . . . . . . . . . . . . . . . . . 344
12.1.2 SolidAngle .........................347
12.2 Time Derivative of Vectors . . . . . . . . . . . . . . . . . . . . 350
12.2.1 Equations of Motion in a Central Force Field . . . . . . 352
12.3TheGradient ............................355
12.3.1 Gradient and Extremum Problems . . . . . . . . . . . . 359
12.4Problems ..............................362
13 Flux and Divergence 365
13.1FluxofaVectorField .......................365
13.1.1 Flux Through an Arbitrary Surface . . . . . . . . . . . 370
13.2FluxDensity=Divergence ....................371
13.2.1 FluxDensity ........................371
13.2.2 Divergence Theorem . . . . . . . . . . . . . . . . . . . . 374
13.2.3 Continuity Equation . . . . . . . . . . . . . . . . . . . . 378
13.3Problems ..............................383
14 Line Integral and Curl 387
14.1TheLineIntegral..........................387
14.2 Curl of a Vector Field and Stokes’ Theorem . . . . . . . . . . . 391
14.3ConservativeVectorFields.....................398
14.4Problems ..............................404
CONTENTS xix
15 Applied Vector Analysis 407
15.1 Double Del Operations . . . . . . . . . . . . . . . . . . . . . . . 407
15.2MagneticMultipoles ........................409
15.3Laplacian ..............................411
15.3.1 A Primer of Fluid Dynamics . . . . . . . . . . . . . . . 413
15.4 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . 415
15.4.1 Maxwell’s Contribution . . . . . . . . . . . . . . . . . . 416
15.4.2 Electromagnetic Waves in Empty Space . . . . . . . . . 417
15.5Problems ..............................420
16 Curvilinear Vector Analysis 423
16.1ElementsofLength.........................423
16.2TheGradient ............................425
16.3TheDivergence...........................427
16.4TheCurl ..............................431
16.4.1 TheLaplacian .......................435
16.5Problems ..............................436
17 Tensor Analysis 439
17.1VectorsandIndices.........................439
17.1.1 Transformation Properties of Vectors . . . . . . . . . . . 441
17.1.2 Covariant and Contravariant Vectors . . . . . . . . . . . 445
17.2FromVectorstoTensors......................447
17.2.1 Algebraic Properties of Tensors . . . . . . . . . . . . . . 450
17.2.2 Numerical Tensors . . . . . . . . . . . . . . . . . . . . . 452
17.3MetricTensor............................454
17.3.1 Index Raising and Lowering . . . . . . . . . . . . . . . . 457
17.3.2 Tensors and Electrodynamics . . . . . . . . . . . . . . . 459
17.4DifferentiationofTensors .....................462
17.4.1 Covariant Differential and Affine Connection . . . . . . 462
17.4.2 Covariant Derivative . . . . . . . . . . . . . . . . . . . . 464
17.4.3 MetricConnection .....................465
17.5RiemannCurvatureTensor ....................468
17.6Problems ..............................471
V Complex Analysis 475
18 Complex Arithmetic 477
18.1 Cartesian Form of Complex Numbers . . . . . . . . . . . . . . . 477
18.2 Polar Form of Complex Numbers . . . . . . . . . . . . . . . . . 482
18.3FourierSeriesRevisited ......................488
18.4 A Representation of Delta Function . . . . . . . . . . . . . . . 491
18.5Problems ..............................493
xx CONTENTS
19 Complex Derivative and Integral 497
19.1 Complex Functions . . . . . . . . . . . . . . . . . . . . . . . . . 497
19.1.1 Derivatives of Complex Functions . . . . . . . . . . . . . 499
19.1.2 Integration of Complex Functions . . . . . . . . . . . . . 503
19.1.3 Cauchy Integral Formula . . . . . . . . . . . . . . . . . 508
19.1.4 Derivatives as Integrals . . . . . . . . . . . . . . . . . . 509
19.2Problems ..............................511
20 Complex Series 515
20.1PowerSeries.............................516
20.2TaylorandLaurentSeries .....................518
20.3Problems ..............................522
21 Calculus of Residues 525
21.1TheResidue.............................525
21.2IntegralsofRationalFunctions ..................529
21.3 Products of Rational and Trigonometric Functions . . . . . . . 532
21.4 Functions of Trigonometric Functions . . . . . . . . . . . . . . 534
21.5Problems ..............................536
VI Differential Equations 539
22 From PDEs to ODEs 541
22.1 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . 542
22.2 Separation in Cartesian Coordinates . . . . . . . . . . . . . . . 544
22.3 Separation in Cylindrical Coordinates . . . . . . . . . . . . . . 547
22.4 Separation in Spherical Coordinates . . . . . . . . . . . . . . . 548
22.5Problems ..............................550
23 First-Order Differential Equations 551
23.1NormalFormofaFODE .....................551
23.2IntegratingFactors.........................553
23.3First-OrderLinearDifferentialEquations ............556
23.4Problems ..............................561
24 Second-Order Linear Differential Equations 563
24.1Linearity,Superposition,andUniqueness.............564
24.2TheWronskian...........................566
24.3ASecondSolutiontotheHSOLDE................567
24.4TheGeneralSolutiontoanISOLDE...............569
24.5 Sturm–Liouville Theory . . . . . . . . . . . . . . . . . . . . . . 570
24.5.1 Adjoint Differential Operators . . . . . . . . . . . . . . 571
24.5.2 Sturm–Liouville System . . . . . . . . . . . . . . . . . . 574
24.6 SOLDEs with Constant Coefficients . . . . . . . . . . . . . . . 575
24.6.1 TheHomogeneousCase ..................576
24.6.2 Central Force Problem . . . . . . . . . . . . . . . . . . . 579
CONTENTS xxi
24.6.3 The Inhomogeneous Case . . . . . . . . . . . . . . . . . 583
24.7Problems ..............................587
25 Laplace’s Equation: Cartesian Coordinates 591
25.1UniquenessofSolutions ......................592
25.2CartesianCoordinates .......................594
25.3Problems ..............................603
26 Laplace’s Equation: Spherical Coordinates 607
26.1FrobeniusMethod .........................608
26.2LegendrePolynomials .......................610
26.3 Second Solution of the Legendre DE . . . . . . . . . . . . . . . 617
26.4CompleteSolution .........................619
26.5 Properties of Legendre Polynomials . . . . . . . . . . . . . . . . 622
26.5.1 Parity............................622
26.5.2 Recurrence Relation . . . . . . . . . . . . . . . . . . . . 622
26.5.3 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . 624
26.5.4 RodriguesFormula.....................626
26.6 Expansions in Legendre Polynomials . . . . . . . . . . . . . . . 628
26.7PhysicalExamples .........................631
26.8Problems ..............................635
27 Laplace’s Equation: Cylindrical Coordinates 639
27.1TheODEs..............................639
27.2 Solutions of the Bessel DE . . . . . . . . . . . . . . . . . . . . . 642
27.3 Second Solution of the Bessel DE . . . . . . . . . . . . . . . . . 645
27.4 Properties of the Bessel Functions . . . . . . . . . . . . . . . . 646
27.4.1 Negative Integer Order . . . . . . . . . . . . . . . . . . . 646
27.4.2 Recurrence Relations . . . . . . . . . . . . . . . . . . . . 646
27.4.3 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . 647
27.4.4 Generating Function . . . . . . . . . . . . . . . . . . . . 649
27.5 Expansions in Bessel Functions . . . . . . . . . . . . . . . . . . 653
27.6PhysicalExamples .........................654
27.7Problems ..............................657
28 Other PDEs of Mathematical Physics 661
28.1TheHeatEquation.........................661
28.1.1 Heat-Conducting Rod . . . . . . . . . . . . . . . . . . . 662
28.1.2 Heat Conduction in a Rectangular Plate . . . . . . . . . 663
28.1.3 Heat Conduction in a Circular Plate . . . . . . . . . . . 664
28.2 The Schr¨odingerEquation.....................666
28.2.1 Quantum Harmonic Oscillator . . . . . . . . . . . . . . 667
28.2.2 QuantumParticleinaBox ................675
28.2.3 Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . 677
28.3TheWaveEquation ........................680
28.3.1 GuidedWaves .......................682
xxii CONTENTS
28.3.2 VibratingMembrane....................686
28.4Problems ..............................687
VII Special Topics 691
29 Integral Transforms 693
29.1TheFourierTransform.......................693
29.1.1 Properties of Fourier Transform . . . . . . . . . . . . . . 696
29.1.2 SineandCosineTransforms................697
29.1.3 Examples of Fourier Transform . . . . . . . . . . . . . . 698
29.1.4 Application to Differential Equations . . . . . . . . . . . 702
29.2 Fourier Transform and Green’s Functions . . . . . . . . . . . . 705
29.2.1 Green’s Function for the Laplacian . . . . . . . . . . . . 708
29.2.2 Green’s Function for the Heat Equation . . . . . . . . . 709
29.2.3 Green’s Function for the Wave Equation . . . . . . . . . 711
29.3TheLaplaceTransform ......................712
29.3.1 Properties of Laplace Transform . . . . . . . . . . . . . 713
29.3.2 Derivative and Integral of the Laplace Transform . . . . 717
29.3.3 Laplace Transform and Differential Equations . . . . . . 718
29.3.4 Inverse of Laplace Transform . . . . . . . . . . . . . . . 721
29.4Problems ..............................723
30 Calculus of Variations 727
30.1 Variational Problem . . . . . . . . . . . . . . . . . . . . . . . . 728
30.1.1 Euler-Lagrange Equation . . . . . . . . . . . . . . . . . 729
30.1.2 Beltramiidentity......................731
30.1.3 Several Dependent Variables . . . . . . . . . . . . . . . 734
30.1.4 Several Independent Variables . . . . . . . . . . . . . . . 734
30.1.5 Second Variation . . . . . . . . . . . . . . . . . . . . . . 735
30.1.6 Variational Problems with Constraints . . . . . . . . . . 738
30.2LagrangianDynamics .......................740
30.2.1 From Newton to Lagrange . . . . . . . . . . . . . . . . . 740
30.2.2 Lagrangian Densities . . . . . . . . . . . . . . . . . . . . 744
30.3 Hamiltonian Dynamics . . . . . . . . . . . . . . . . . . . . . . . 747
30.4Problems ..............................750
31 Nonlinear Dynamics and Chaos 753
31.1SystemsObeyingIteratedMaps..................754
31.1.1 Stable and Unstable Fixed Points . . . . . . . . . . . . . 755
31.1.2 Bifurcation .........................757
31.1.3 OnsetofChaos.......................761
31.2SystemsObeyingDEs .......................763
31.2.1 The Phase Space . . . . . . . . . . . . . . . . . . . . . . 764
31.2.2 Autonomous Systems . . . . . . . . . . . . . . . . . . . 766
31.2.3 OnsetofChaos.......................770