Essentials of Mathematical Methods in Science and Engineering

Essentials of Mathematical Methods in Science and Engineering......Page 6
CONTENTS......Page 12
Preface......Page 24
Acknowledgments......Page 30
1.1 Concept of Function......Page 32
1.2 Continuity and Limits......Page 35
1.3 Partial Differentiation......Page 37
1.4 Total Differential......Page 39
1.5 Taylor Series......Page 41
1.6 Maxima and Minima of Functions......Page 45
1.7 Extrema of Functions with Conditions......Page 49
1.8 Derivatives and Differentials of Composite Functions......Page 53
1.9 Implicit Function Theorem......Page 55
1.10 Inverse Functions......Page 61
1.11 Integral Calculus and the Definite Integral......Page 63
1.12 Riemann Integral......Page 65
1.13 Improper Integrals......Page 68
1.14 Cauchy Principal Value Integrals......Page 71
1.15 Integrals Involving a Parameter......Page 73
1.16 Limits of Integration Depending on a Parameter......Page 77
1.17 Double Integrals......Page 78
1.18 Properties of Double Integrals......Page 80
1.19 Triple and Multiple Integrals......Page 81
Problems......Page 82
2.1 Vector Algebra: Geometric Method......Page 88
2.1.1 Multiplication of Vectors......Page 91
2.2 Vector Algebra: Coordinate Representation......Page 93
2.3 Lines and Planes......Page 99
2.4.1 Scalar Fields and Vector Fields......Page 101
2.4.2 Vector Differentiation......Page 103
2.5 Gradient Operator......Page 104
2.5.1 Meaning of the Gradient......Page 105
2.5.2 Directional Derivative......Page 106
2.6 Divergence and Curl Operators......Page 108
2.6.1 Meaning of Divergence and the Divergence Theorem......Page 109
2.7.1 Arc Length and Line Integrals......Page 114
2.7.2 Surface Area and Surface Integrals......Page 118
2.7.3 An Alternate Way to Write Line Integrals......Page 120
2.7.4 Green’s Theorem......Page 122
2.7.5 Interpretations of Green’s Theorem......Page 124
2.7.6 Extension to Multiply Connected Domains......Page 125
2.8.1 On the Plane......Page 128
2.8.2 In Space......Page 133
2.9 Mixed Operations with the Del Operator......Page 136
2.10 Potential Theory......Page 139
2.10.1 Gravitational Field of a Spherically Symmetric Star......Page 142
2.10.2 Work Done by Gravitational Force......Page 143
2.10.3 Path Independence and Exact Differentials......Page 145
2.10.4 Gravity and Conservative Forces......Page 147
2.10.5 Gravitational Potential......Page 149
2.10.6 Gravitational Potential Energy of a System......Page 151
2.10.7 Helmholtz Theorem......Page 153
2.10.8 Applications of the Helmholtz Theorem......Page 154
2.10.9 Examples from Physics......Page 158
Problems......Page 161
3 GENERALIZED COORDINATES and TENSORS......Page 170
3.1.1 Basis Vectors and Direction Cosines......Page 171
3.1.2 Transformation Matrix and the Orthogonality Relation......Page 173
3.1.3 Inverse Transformation Matrix......Page 175
3.2 Cartesian Tensors......Page 176
3.2.1 Algebraic Properties of Tensors......Page 179
3.2.2 Kronecker Delta and the Permutation Symbol......Page 182
3.3.1 Coordinate Curves and Surfaces......Page 185
3.3.2 Why Upper and Lower Indices......Page 190
3.4 General Tensors......Page 191
3.4.1 Einstein Summation Convention......Page 194
3.4.3 Metric Tensor......Page 195
3.4.4 How to Raise and Lower Indices......Page 196
3.4.5 Metric Tensor and the Basis Vectors......Page 197
3.4.6 Displacement Vector......Page 199
3.4.7 Transformation of Scalar Functions and Line Integrals......Page 200
3.4.8 Area Element in Generalized Coordinates......Page 202
3.4.9 Area of a Surface......Page 204
3.4.10 Volume Element in Generalized Coordinates......Page 208
3.4.11 Invariance and Covariance......Page 209
3.5.1 Gradient......Page 210
3.5.2 Divergence......Page 211
3.5.3 Curl......Page 213
3.6 Orthogonal Generalized Coordinates......Page 217
3.6.1 Cylindrical Coordinates......Page 218
3.6.2 Spherical Coordinates......Page 224
Problems......Page 229
4.1 Basic Definitions......Page 238
4.2 Operations with Matrices......Page 239
4.3 Submatrix and Partitioned Matrices......Page 245
4.4 Systems of Linear Equations......Page 247
4.5 Gauss’s Method of Elimination......Page 248
4.6 Determinants......Page 252
4.7 Properties of Determinants......Page 254
4.8 Cramer's Rule......Page 257
4.9 Inverse of a Matrix......Page 261
4.10 Homogeneous Linear Equations......Page 264
Problems......Page 265
5.1 Fields and Vector Spaces......Page 272
5.2 Linear Combinations, Generators, and Bases......Page 275
5.3 Components......Page 277
5.4 Linear Transformations......Page 280
5.5 Matrix Representation of Transformations......Page 281
5.6 Algebra of Transformations......Page 283
5.7 Change of Basis......Page 285
5.9 Eigenvalues and Eigenvectors......Page 287
5.10 Moment of Inertia Tensor......Page 296
5.11 Inner product Spaces......Page 301
5.12 The Inner Product......Page 302
5.13 Orthogonality and Completeness......Page 305
5.14 Gram—Schmidt Orthogonalization......Page 307
5.15 Eigenvalue Problem for Real Symmetric Matrices......Page 308
5.16 Presence of Degenerate Eigenvalues......Page 309
5.17 Quadratic Forms......Page 316
5.18 Hermitian Matrices......Page 320
5.19 Matrix Representation of Linear Transformations......Page 324
5.20 Functions of Matrices......Page 325
5.21 Function Space and Hilbert Space......Page 327
5.22 Dirac's Bra and Ket vectors......Page 328
Problems......Page 329
6 SEQUENCES and SERIES......Page 334
6.1 Sequences......Page 335
6.2 Infinite Series......Page 339
6.3.3 Integral Test......Page 340
6.3.5 Root Test......Page 341
6.4 Operations with Series......Page 345
6.5 Sequences and Series of Functions......Page 347
6.6 M-Test for Uniform Convergence......Page 349
6.7 Properties of Uniformly Convergent Series......Page 350
6.8 Power Series......Page 352
6.10 Indeterminate Forms and Series......Page 355
Problems......Page 357
7 COMPLEX NUMBERS and FUNCTIONS......Page 362
7.1 The Algebra of Complex Numbers......Page 363
7.2 Roots of a Complex Number......Page 367
7.3 Infinity and the Extended Complex Plane......Page 370
7.4 Complex Functions......Page 373
7.5 Limits and Continuity......Page 375
7.6 Differentiation in the Complex Plane......Page 376
7.7 Analytic Functions......Page 380
7.8 Harmonic Functions......Page 381
7.9 Basic Differentiation Formulas......Page 383
7.10.1 Polynomials......Page 384
7.10.2 Exponential Function......Page 385
7.10.3 Trigonometric Functions......Page 387
7.10.4 Hyperbolic Functions......Page 388
7.10.5 Logarithmic Function......Page 389
7.10.6 Powers of Complex Numbers......Page 390
Problems......Page 393
8 COMPLEX ANALYSIS......Page 400
8.1 Contour Integrals......Page 401
8.2 Types of Contours......Page 403
8.3 The Cauchy–Goursat Theorem......Page 407
8.4 Indefinite Integrals......Page 410
8.6 The Cauchy Integral Formula......Page 412
8.7 Derivatives of Analytic Functions......Page 415
8.8.1 Taylor Series with the Remainder......Page 416
8.8.2 Laurent Series with the Remainder......Page 420
8.9 Convergence of Power Series......Page 424
8.10 Classification of Singular Points......Page 425
8.11 Residue Theorem......Page 428
Problems......Page 432
9 ORDINARY DIFFERENTIAL EQUATIONS......Page 438
9.1 Basic Definitions for Ordinary Differential Equations......Page 439
9.2 First-Order Differential Equations......Page 441
9.3.3 The Case of Separable f(x, y)......Page 443
9.3.5 Solution When f(x, y) Is a Rational Function......Page 444
9.3.6 Linear Equations of First-Order......Page 447
9.3.7 Exact Equations......Page 448
9.3.8 Integrating Factors......Page 450
9.3.9 Bernoulli Equation......Page 454
9.3.10 Riccati Equation......Page 455
9.3.11 Equations That Cannot Be Solved for y′......Page 457
9.4 Second-Order Differential Equations......Page 460
9.5 Second-Order Differential Equations: Methods of Solution......Page 461
9.5.1 Linear Homogeneous Equations with Constant Coefficients......Page 462
9.5.2 Operator Approach......Page 468
9.5.3 Linear Homogeneous Equations with Variable Coefficients......Page 469
9.5.4 Cauchy—Euler Equation......Page 472
9.5.5 Exact Equations and Integrating Factors......Page 473
9.5.6 Linear Nonhomogeneous Equations......Page 475
9.5.7 Variation of Parameters......Page 476
9.5.8 Method of Undetermined Coefficients......Page 477
9.6.1 With Constant Coefficients......Page 481
9.6.3 Nonhomogeneous Equations......Page 482
9.8 Series Solutions: Frobenius Method......Page 483
9.8.1 Frobenius Method and First-Order Equations......Page 493
Problems......Page 494
10 SECOND-ORDER DIFFERENTIAL EQUATIONS and SPECIAL FUNCTIONS......Page 500
10.1.1 Series Solution......Page 501
10.1.2 Effect of Boundary Conditions......Page 504
10.1.3 Legendre Polynomials......Page 505
10.1.4 Rodriguez Formula......Page 508
10.1.5 Generating Function......Page 509
10.1.6 Special Values......Page 511
10.1.7 Recursion Relations......Page 512
10.1.8 Orthogonality......Page 513
10.1.9 Legendre Series......Page 515
10.2.1 Series Solution......Page 518
10.2.2 Hermite Polynomials......Page 522
10.2.3 Contour Integral Representation......Page 523
10.2.4 Rodriguez Formula......Page 524
10.2.5 Generating Function......Page 525
10.2.7 Recursion Relations......Page 526
10.2.8 Orthogonality......Page 527
10.2.9 Series Expansions in Hermite Polynomials......Page 530
10.3.1 Series Solution......Page 531
10.3.3 Contour Integral Representation......Page 533
10.3.4 Rodriguez Formula......Page 534
10.3.6 Special Values and Recursion Relations......Page 535
10.3.7 Orthogonality......Page 536
10.3.8 Series Expansions in Laguerre Polynomials......Page 537
Problems......Page 538
11 BESSEL’S EQUATION and BESSEL FUNCTIONS......Page 540
11.1 Bessel‘s Equation and Its Series Solution......Page 541
11.1.1 Bessel Functions J±m(x), Nm(x), and Hm(1,2)(x)......Page 545
11.1.2 Recursion Relations......Page 549
11.1.3 Generating Function......Page 550
11.1.4 Integral Definitions......Page 552
11.1.5 Linear Independence of Bessel Functions......Page 553
11.1.6 Modified Bessel Functions Im(x) and Km(x)......Page 554
11.1.7 Spherical Bessel Functions ji (x) , ni(x) , and hi(1,2)(x)......Page 556
11.2 Orthogonality and the Roots of Bessel Functions......Page 558
11.2.2 Boundary Conditions for the Bessel Functions......Page 562
Problems......Page 566
12 PARTIAL DIFFERENTIAL EQUATIONS and SEPARATION of VARIABLES......Page 572
12.1 Separation of Variables in Cartesian Coordinates......Page 573
12.1.1 Wave Equation......Page 575
12.1.2 Laplace Equation......Page 577
12.1.3 Diffusion and Heat Flow Equations......Page 581
12.2 Separation of Variables in Spherical Coordinates......Page 584
12.2.1 Laplace Equation......Page 588
12.2.2 Boundary Conditions for a Spherical Boundary......Page 589
12.2.4 Wave Equation......Page 594
12.2.5 Diffusion and Heat Flow Equations......Page 595
12.2.6 Time-Independent Schrödinger Equation......Page 596
12.2.7 Time-Dependent Schrödinger Equation......Page 597
12.3 Separation of Variables in Cylindrical Coordinates......Page 598
12.3.1 Laplace Equation......Page 600
12.3.3 Wave Equation......Page 601
12.3.4 Diffusion and Heat Flow Equations......Page 603
Problems......Page 611
13.1 Orthogonal Systems of Functions......Page 616
13.2 Fourier Series......Page 622
13.3 Exponential Form of the Fourier Series......Page 623
13.4 Convergence of Fourier Series......Page 624
13.5 Sufficient Conditions for Convergence......Page 626
13.6 The Fundamental Theorem......Page 627
13.8.1 Square Wave......Page 628
13.8.2 Triangular Wave......Page 630
13.8.3 Periodic Extension......Page 631
13.9 Fourier Sine and Cosine Series......Page 632
13.10 Change of Interval......Page 633
13.11 Integration and Differentiation of Fourier Series......Page 634
Problems......Page 635
14.1 Types of Signals......Page 638
14.2 Spectral Analysis and Fourier Transforms......Page 641
14.3 Correlation with Cosines and Sines......Page 642
14.5 Inverse Fourier Transform......Page 646
14.6 Frequency Spectrums......Page 648
14.7 Dirac-Delta Function......Page 649
14.8 A Case with Two Cosines......Page 650
14.9 General Fourier Transforms and Their Properties......Page 651
14.10 Basic Definition of Laplace Transform......Page 653
14.11 Differential Equations and Laplace Transforms......Page 656
14.12 Transfer Functions and Signal Processors......Page 658
14.13 Connection of Signal Processors......Page 660
Problems......Page 663
15 CALCULUS of VARIATIONS......Page 668
15.1 A Simple Case......Page 669
15.2 Variational Analysis......Page 670
15.3 Alternate Form of Euler Equation......Page 673
15.4 Variational Notation......Page 676
15.5 A More General Case......Page 678
15.6 Hamilton's Principle......Page 682
15.7 Lagrange's Equations of Motion......Page 684
15.8 Definition of Lagrangian......Page 688
15.9 Presence of Constraints in Dynamical Systems......Page 690
15.10 Conservation Laws......Page 693
Problems......Page 694
16 PROBABILITY THEORY and DISTRIBUTIONS......Page 698
16.1.1 Fundamental Concepts......Page 699
16.1.3 Basic Theorems of Probability......Page 700
16.1.4 Statistical Definition of Probability......Page 703
16.1.5 Conditional Probability and Multiplication Theorem......Page 704
16.1.6 Bayes' Theorem......Page 705
16.1.7 Geometric Probability and Buffon's Needle Problem......Page 708
16.2.1 The Case of Distinguishable Balls with Replacement......Page 709
16.2.2 The Case of Distinguishable Balls Without Replacement......Page 710
16.2.3 The Case of Indistinguishable Balls......Page 711
16.2.4 Binomial and Multinomial Coefficients......Page 712
16.3 Applications to Statistical Mechanics......Page 713
16.3.1 Boltzmann Distribution for Solids......Page 715
16.3.2 Boltzmann Distribution for Gases......Page 717
16.3.3 Bose—Einstein Distribution for Perfect Gases......Page 718
16.3.4 Fermi—Dirac Distribution......Page 719
16.4.1 Probability and Entropy......Page 720
16.4.2 Derivation of β......Page 722
16.5 Random Variables and Distributions......Page 724
16.6 Distribution Functions and Probability......Page 727
16.7.1 Uniform Distribution......Page 729
16.7.3 Gamma Distribution......Page 730
16.8 Discrete Probability Distributions......Page 731
16.8.2 Binomial Distribution......Page 732
16.8.3 Poisson Distribution......Page 734
16.9 Fundamental Theorem of Averages......Page 735
16.10 Moments of Distribution Functions......Page 736
16.10.1 Moments of the Gaussian Distribution......Page 737
16.10.2 Moments of the Binomial Distribution......Page 738
16.10.3 Moments of the Poisson Distribution......Page 739
16.11 Chebyshev’s Theorem......Page 741
16.12 Law of Large Numbers......Page 743
Problems......Page 744
17 INFORMATION THEORY......Page 752
17.1 Elements of Information Processing Mechanisms......Page 755
17.2 Classical Information Theory......Page 757
17.2.1 Prior Uncertainty and Entropy of Information......Page 760
17.2.2 Joint and Conditional Entropies of Information......Page 762
17.2.3 Decision Theory......Page 766
17.2.4 Decision Theory and Game Theory......Page 767
17.2.5 Traveler’s Dilemma and Nash Equilibrium......Page 773
17.2.6 Classical Bit or Cbit......Page 777
17.2.7 Operations on Cbits......Page 781
17.3.1 Basic Quantum Theory......Page 783
17.3.2 Single-Particle Systems and Quantum Information......Page 789
17.3.3 Mach—Zehnder Interferometer......Page 791
17.3.4 Mathematics of the Mach—Zehnder Interferometer......Page 794
17.3.5 Quantum Bit or Qbit......Page 798
17.3.6 The No-Cloning Theorem......Page 801
17.3.7 Entanglement and Bell States......Page 802
17.3.8 Quantum Dense Coding......Page 807
17.3.9 Quantum Teleportation......Page 808
Problems......Page 811
References......Page 818
Index......Page 824