Description
Book SynopsisA comprehensive introduction to the multidisciplinary applications of mathematical methods, revised and updated
The second edition of Essentials of Mathematical Methods in Science and Engineering offers an introduction to the key mathematical concepts of advanced calculus, differential equations, complex analysis, and introductory mathematical physics for students in engineering and physics research. The book's approachable style is designed in a modular format with each chapter covering a subject thoroughly and thus can be read independently.
This updated second edition includes two new and extensive chapters that cover practical linear algebra and applications of linear algebra as well as a computer file that includes Matlab codes. To enhance understanding of the material presented, the text contains a collection of exercises at the end of each chapter. The author offers a coherent treatment of the topics with a style that makes the essential mathematic
Table of Contents
Preface xxiii
Acknowledgments xxix
1 Functional Analysis 1
1.1 Concept of Function 1
1.2 Continuity and Limits 3
1.3 Partial Differentiation 6
1.4 Total Differential 8
1.5 Taylor Series 9
1.6 Maxima and Minima of Functions 13
1.7 Extrema of Functions with Conditions 17
1.8 Derivatives and Differentials of Composite Functions 21
1.9 Implicit Function Theorem 23
1.10 Inverse Functions 28
1.11 Integral Calculus and the Definite Integral 30
1.12 Riemann Integral 32
1.13 Improper Integrals 35
1.14 Cauchy Principal Value Integrals 38
1.15 Integrals Involving a Parameter 40
1.16 Limits of Integration Depending on a Parameter 44
1.17 Double Integrals 45
1.18 Properties of Double Integrals 47
1.19 Triple and Multiple Integrals 48
References 49
Problems 49
2 Vector Analysis 55
2.1 Vector Algebra: Geometric Method 55
2.1.1 Multiplication of Vectors 57
2.2 Vector Algebra: Coordinate Representation 60
2.3 Lines and Planes 65
2.4 Vector Differential Calculus 67
2.4.1 Scalar Fields and Vector Fields 67
2.4.2 Vector Differentiation 69
2.5 Gradient Operator 70
2.5.1 Meaning of the Gradient 71
2.5.2 Directional Derivative 72
2.6 Divergence and Curl Operators 73
2.6.1 Meaning of Divergence and the Divergence Theorem 75
2.7 Vector Integral Calculus in Two Dimensions 79
2.7.1 Arc Length and Line Integrals 79
2.7.2 Surface Area and Surface Integrals 83
2.7.3 An Alternate Way to Write Line Integrals 84
2.7.4 Green’s Theorem 86
2.7.5 Interpretations of Green’s Theorem 88
2.7.6 Extension to Multiply Connected Domains 89
2.8 Curl Operator and Stokes’s Theorem 92
2.8.1 On the Plane 92
2.8.2 In Space 96
2.8.3 Geometric Interpretation of Curl 99
2.9 Mixed Operations with the Del Operator 99
2.10 Potential Theory 102
2.10.1 Gravitational Field of a Star 105
2.10.2 Work Done by Gravitational Force 106
2.10.3 Path Independence and Exact Differentials 108
2.10.4 Gravity and Conservative Forces 109
2.10.5 Gravitational Potential 111
2.10.6 Gravitational Potential Energy of a System 113
2.10.7 Helmholtz Theorem 115
2.10.8 Applications of the Helmholtz Theorem 116
2.10.9 Examples from Physics 120
References 123
Problems 123
3 Generalized Coordinates and Tensors 133
3.1 Transformations between Cartesian Coordinates 134
3.1.1 Basis Vectors and Direction Cosines 134
3.1.2 Transformation Matrix and Orthogonality 136
3.1.3 Inverse Transformation Matrix 137
3.2 Cartesian Tensors 139
3.2.1 Algebraic Properties of Tensors 141
3.2.2 Kronecker Delta and the Permutation Symbol 145
3.3 Generalized Coordinates 148
3.3.1 Coordinate Curves and Surfaces 148
3.3.2 Why Upper and Lower Indices 152
3.4 General Tensors 153
3.4.1 Einstein Summation Convention 156
3.4.2 Line Element 157
3.4.3 Metric Tensor 157
3.4.4 How to Raise and Lower Indices 158
3.4.5 Metric Tensor and the Basis Vectors 160
3.4.6 Displacement Vector 161
3.4.7 Line Integrals 162
3.4.8 Area Element in Generalized Coordinates 164
3.4.9 Area of a Surface 165
3.4.10 Volume Element in Generalized Coordinates 169
3.4.11 Invariance and Covariance 171
3.5 Differential Operators in Generalized Coordinates 171
3.5.1 Gradient 171
3.5.2 Divergence 172
3.5.3 Curl 174
3.5.4 Laplacian 178
3.6 Orthogonal Generalized Coordinates 178
3.6.1 Cylindrical Coordinates 179
3.6.2 Spherical Coordinates 184
References 189
Problems 189
4 Determinants and Matrices 197
4.1 Basic Definitions 197
4.2 Operations with Matrices 198
4.3 Submatrix and Partitioned Matrices 204
4.4 Systems of Linear Equations 207
4.5 Gauss’s Method of Elimination 208
4.6 Determinants 211
4.7 Properties of Determinants 214
4.8 Cramer’s Rule 216
4.9 Inverse of a Matrix 221
4.10 Homogeneous Linear Equations 224
References 225
Problems 225
5 Linear Algebra 233
5.1 Fields and Vector Spaces 233
5.2 Linear Combinations, Generators, and Bases 236
5.3 Components 238
5.4 Linear Transformations 241
5.5 Matrix Representation of Transformations 242
5.6 Algebra of Transformations 244
5.7 Change of Basis 246
5.8 Invariants under Similarity Transformations 247
5.9 Eigenvalues and Eigenvectors 248
5.10 Moment of Inertia Tensor 257
5.11 Inner Product Spaces 262
5.12 The Inner Product 262
5.13 Orthogonality and Completeness 265
5.14 Gram–Schmidt Orthogonalization 267
5.15 Eigenvalue Problem for Real Symmetric Matrices 268
5.16 Presence of Degenerate Eigenvalues 270
5.17 Quadratic Forms 276
5.18 Hermitian Matrices 279
5.19 Matrix Representation of Hermitian Operators 283
5.20 Functions of Matrices 284
5.21 Function Space and Hilbert Space 286
5.22 Dirac’s Bra and Ket Vectors 287
References 288
Problems 289
6 Practical Linear Algebra 293
6.1 Systems of Linear Equations 294
6.1.1 Matrices and Elementary Row Operations 295
6.1.2 Gauss-Jordan Method 295
6.1.3 Information From the Row-Echelon Form 300
6.1.4 Elementary Matrices 301
6.1.5 Inverse by Gauss-Jordan Row-Reduction 302
6.1.6 Row Space, Column Space, and Null Space 303
6.1.7 Bases for Row, Column, and Null Spaces 307
6.1.8 Vector Spaces Spanned by a Set of Vectors 310
6.1.9 Rank and Nullity 312
6.1.10 Linear Transformations 315
6.2 Numerical Methods of Linear Algebra 317
6.2.1 Gauss-Jordan Row-Reduction and Partial Pivoting 317
6.2.2 LU-Factorization 321
6.2.3 Solutions of Linear Systems by Iteration 325
6.2.4 Interpolation 328
6.2.5 Power Method for Eigenvalues 331
6.2.6 Solution of Equations 333
6.2.7 Numerical Integration 343
References 349
Problems 350
7 Applications of Linear Algebra 355
7.1 Chemistry and Chemical Engineering 355
7.1.1 Independent Reactions and Stoichiometric Matrix 356
7.1.2 Independent Reactions from a Set of Species 359
7.2 Linear Programming 362
7.2.1 The Geometric Method 363
7.2.2 The Simplex Method 367
7.3 Leontief Input–Output Model of Economy 375
7.3.1 Leontief Closed Model 375
7.3.2 Leontief Open Model 378
7.4 Applications to Geometry 381
7.4.1 Orbit Calculations 382
7.5 Elimination Theory 383
7.5.1 Quadratic Equations and the Resultant 384
7.6 Coding Theory 388
7.6.1 Fields and Vector Spaces 388
7.6.2 Hamming (7,4) Code 390
7.6.3 Hamming Algorithm for Error Correction 393
7.7 Cryptography 396
7.7.1 Single-Key Cryptography 396
7.8 Graph Theory 399
7.8.1 Basic Definition 399
7.8.2 Terminology 400
7.8.3 Walks, Trails, Paths and Circuits 402
7.8.4 Trees and Fundamental Circuits 404
7.8.5 Graph Operations 404
7.8.6 Cut Sets and Fundamental Cut Sets 405
7.8.7 Vector Space Associated with a Graph 407
7.8.8 Rank and Nullity 409
7.8.9 Subspaces in WG 410
7.8.10 Dot Product and Orthogonal vectors 411
7.8.11 Matrix Representation of Graphs 413
7.8.12 Dominance Directed Graphs 417
7.8.13 Gray Codes in Coding Theory 419
References 419
Problems 420
8 Sequences and Series 425
8.1 Sequences 426
8.2 Infinite Series 430
8.3 Absolute and Conditional Convergence 431
8.3.1 Comparison Test 431
8.3.2 Limit Comparison Test 431
8.3.3 Integral Test 431
8.3.4 Ratio Test 432
8.3.5 Root Test 432
8.4 Operations with Series 436
8.5 Sequences and Series of Functions 438
8.6 M-Test for Uniform Convergence 441
8.7 Properties of Uniformly Convergent Series 441
8.8 Power Series 443
8.9 Taylor Series and Maclaurin Series 446
8.10 Indeterminate Forms and Series 447
References 448
Problems 448
9 Complex Numbers and Functions 453
9.1 The Algebra of Complex Numbers 454
9.2 Roots of a Complex Number 458
9.3 Infinity and the Extended Complex Plane 460
9.4 Complex Functions 463
9.5 Limits and Continuity 465
9.6 Differentiation in the Complex Plane 467
9.7 Analytic Functions 470
9.8 Harmonic Functions 471
9.9 Basic Differentiation Formulas 474
9.10 Elementary Functions 475
9.10.1 Polynomials 475
9.10.2 Exponential Function 476
9.10.3 Trigonometric Functions 477
9.10.4 Hyperbolic Functions 478
9.10.5 Logarithmic Function 479
9.10.6 Powers of Complex Numbers 481
9.10.7 Inverse Trigonometric Functions 483
References 483
Problems 484
10 Complex Analysis 491
10.1 Contour Integrals 492
10.2 Types of Contours 494
10.3 The Cauchy–Goursat Theorem 497
10.4 Indefinite Integrals 500
10.5 Simply and Multiply Connected Domains 502
10.6 The Cauchy Integral Formula 503
10.7 Derivatives of Analytic Functions 505
10.8 Complex Power Series 506
10.8.1 Taylor Series with the Remainder 506
10.8.2 Laurent Series with the Remainder 510
10.9 Convergence of Power Series 514
10.10 Classification of Singular Points 514
10.11 Residue Theorem 517
References 522
Problems 522
11 Ordinary Differential Equations 527
11.1 Basic Definitions for Ordinary Differential Equations 528
11.2 First-Order Differential Equations 530
11.2.1 Uniqueness of Solution 530
11.2.2 Methods of Solution 532
11.2.3 Dependent Variable is Missing 532
11.2.4 Independent Variable is Missing 532
11.2.5 The Case of Separable f(x, y) 532
11.2.6 Homogeneous f(x, y) of Zeroth Degree 533
11.2.7 Solution When f(x, y) is a Rational Function 533
11.2.8 Linear Equations of First-order 535
11.2.9 Exact Equations 537
11.2.10 Integrating Factors 539
11.2.11 Bernoulli Equation 542
11.2.12 Riccati Equation 543
11.2.13 Equations that Cannot Be Solved for y' 546
11.3 Second-Order Differential Equations 548
11.3.1 The General Case 549
11.3.2 Linear Homogeneous Equations with Constant Coefficients 551
11.3.3 Operator Approach 556
11.3.4 Linear Homogeneous Equations with Variable Coefficients 557
11.3.5 Cauchy–Euler Equation 560
11.3.6 Exact Equations and Integrating Factors 561
11.3.7 Linear Nonhomogeneous Equations 564
11.3.8 Variation of Parameters 564
11.3.9 Method of Undetermined Coefficients 566
11.4 Linear Differential Equations of Higher Order 569
11.4.1 With Constant Coefficients 569
11.4.2 With Variable Coefficients 570
11.4.3 Nonhomogeneous Equations 570
11.5 Initial Value Problem and Uniqueness of the Solution 571
11.6 Series Solutions: Frobenius Method 571
11.6.1 Frobenius Method and First-order Equations 581
References 582
Problems 582
12 Second-Order Differential Equations and Special Functions 589
12.1 Legendre Equation 590
12.1.1 Series Solution 590
12.1.2 Effect of Boundary Conditions 593
12.1.3 Legendre Polynomials 594
12.1.4 Rodriguez Formula 596
12.1.5 Generating Function 597
12.1.6 Special Values 599
12.1.7 Recursion Relations 600
12.1.8 Orthogonality 601
12.1.9 Legendre Series 603
12.2 Hermite Equation 606
12.2.1 Series Solution 606
12.2.2 Hermite Polynomials 610
12.2.3 Contour Integral Representation 611
12.2.4 Rodriguez Formula 612
12.2.5 Generating Function 613
12.2.6 Special Values 614
12.2.7 Recursion Relations 614
12.2.8 Orthogonality 616
12.2.9 Series Expansions in Hermite Polynomials 618
12.3 Laguerre Equation 619
12.3.1 Series Solution 620
12.3.2 Laguerre Polynomials 621
12.3.3 Contour Integral Representation 622
12.3.4 Rodriguez Formula 623
12.3.5 Generating Function 623
12.3.6 Special Values and Recursion Relations 624
12.3.7 Orthogonality 624
12.3.8 Series Expansions in Laguerre Polynomials 625
References 626
Problems 626
13 Bessel’s Equation and Bessel Functions 629
13.1 Bessel’s Equation and Its Series Solution 630
13.1.1 Bessel Functions J±m(x), Nm(x), and H(1,2)m (x) 634
13.1.2 Recursion Relations 639
13.1.3 Generating Function 639
13.1.4 Integral Definitions 641
13.1.5 Linear Independence of Bessel Functions 642
13.1.6 Modified Bessel Functions Im(x) and Km(x) 644
13.1.7 Spherical Bessel Functions jl(x), nl(x), and h(1,2)l (x) 645
13.2 Orthogonality and the Roots of Bessel Functions 648
13.2.1 Expansion Theorem 652
13.2.2 Boundary Conditions for the Bessel Functions 652
References 656
Problems 656
14 Partial Differential Equations and Separation of Variables 661
14.1 Separation of Variables in Cartesian Coordinates 662
14.1.1 Wave Equation 665
14.1.2 Laplace Equation 666
14.1.3 Diffusion and Heat Flow Equations 671
14.2 Separation of Variables in Spherical Coordinates 673
14.2.1 Laplace Equation 677
14.2.2 Boundary Conditions for a Spherical Boundary 678
14.2.3 Helmholtz Equation 682
14.2.4 Wave Equation 683
14.2.5 Diffusion and Heat Flow Equations 684
14.2.6 Time-Independent Schrödinger Equation 685
14.2.7 Time-Dependent Schrödinger Equation 685
14.3 Separation of Variables in Cylindrical Coordinates 686
14.3.1 Laplace Equation 688
14.3.2 Helmholtz Equation 689
14.3.3 Wave Equation 690
14.3.4 Diffusion and Heat Flow Equations 691
References 701
Problems 701
15 Fourier Series 705
15.1 Orthogonal Systems of Functions 705
15.2 Fourier Series 711
15.3 Exponential Form of the Fourier Series 712
15.4 Convergence of Fourier Series 713
15.5 Sufficient Conditions for Convergence 715
15.6 The Fundamental Theorem 716
15.7 Uniqueness of Fourier Series 717
15.8 Examples of Fourier Series 717
15.8.1 Square Wave 717
15.8.2 Triangular Wave 719
15.8.3 Periodic Extension 720
15.9 Fourier Sine and Cosine Series 721
15.10 Change of Interval 722
15.11 Integration and Differentiation of Fourier Series 723
References 724
Problems 724
16 Fourier and Laplace Transforms 727
16.1 Types of Signals 727
16.2 Spectral Analysis and Fourier Transforms 730
16.3 Correlation with Cosines and Sines 731
16.4 Correlation Functions and Fourier Transforms 735
16.5 Inverse Fourier Transform 736
16.6 Frequency Spectrums 736
16.7 Dirac-Delta Function 738
16.8 A Case with Two Cosines 739
16.9 General Fourier Transforms and Their Properties 740
16.10 Basic Definition of Laplace Transform 743
16.11 Differential Equations and Laplace Transforms 746
16.12 Transfer Functions and Signal Processors 748
16.13 Connection of Signal Processors 750
References 753
Problems 753
17 Calculus of Variations 757
17.1 A Simple Case 758
17.2 Variational Analysis 759
17.2.1 Case I: The Desired Function is Prescribed at the End Points 761
17.2.2 Case II: Natural Boundary Conditions 762
17.3 Alternate Form of Euler Equation 763
17.4 Variational Notation 765
17.5 A More General Case 767
17.6 Hamilton’s Principle 772
17.7 Lagrange’s Equations of Motion 773
17.8 Definition of Lagrangian 777
17.9 Presence of Constraints in Dynamical Systems 779
17.10 Conservation Laws 783
References 784
Problems 784
18 Probability Theory and Distributions 789
18.1 Introduction to Probability Theory 790
18.1.1 Fundamental Concepts 790
18.1.2 Basic Axioms of Probability 791
18.1.3 Basic Theorems of Probability 791
18.1.4 Statistical Definition of Probability 794
18.1.5 Conditional Probability and Multiplication Theorem 795
18.1.6 Bayes’ Theorem 796
18.1.7 Geometric Probability and Buffon’s Needle Problem 798
18.2 Permutations and Combinations 800
18.2.1 The Case of Distinguishable Balls with Replacement 800
18.2.2 The Case of Distinguishable Balls without Replacement 801
18.2.3 The Case of Indistinguishable Balls 802
18.2.4 Binomial and Multinomial Coefficients 803
18.3 Applications to Statistical Mechanics 804
18.3.1 Boltzmann Distribution for Solids 805
18.3.2 Boltzmann Distribution for Gases 807
18.3.3 Bose–Einstein Distribution for Perfect Gases 808
18.3.4 Fermi–Dirac Distribution 810
18.4 Statistical Mechanics and Thermodynamics 811
18.4.1 Probability and Entropy 811
18.4.2 Derivation of β 812
18.5 Random Variables and Distributions 814
18.6 Distribution Functions and Probability 817
18.7 Examples of Continuous Distributions 819
18.7.1 Uniform Distribution 819
18.7.2 Gaussian or Normal Distribution 820
18.7.3 Gamma Distribution 821
18.8 Discrete Probability Distributions 821
18.8.1 Uniform Distribution 822
18.8.2 Binomial Distribution 822
18.8.3 Poisson Distribution 824
18.9 Fundamental Theorem of Averages 825
18.10 Moments of Distribution Functions 826
18.10.1 Moments of the Gaussian Distribution 827
18.10.2 Moments of the Binomial Distribution 827
18.10.3 Moments of the Poisson Distribution 829
18.11 Chebyshev’s Theorem 831
18.12 Law of Large Numbers 832
References 833
Problems 834
19 Information Theory 841
19.1 Elements of Information Processing Mechanisms 844
19.2 Classical Information Theory 846
19.2.1 Prior Uncertainty and Entropy of Information 848
19.2.2 Joint and Conditional Entropies of Information 851
19.2.3 Decision Theory 854
19.2.4 Decision Theory and Game Theory 856
19.2.5 Traveler’s Dilemma and Nash Equilibrium 862
19.2.6 Classical Bit or Cbit 866
19.2.7 Operations on Cbits 869
19.3 Quantum Information Theory 871
19.3.1 Basic Quantum Theory 872
19.3.2 Single-Particle Systems and Quantum Information 878
19.3.3 Mach–Zehnder Interferometer 880
19.3.4 Mathematics of the Mach–Zehnder Interferometer 882
19.3.5 Quantum Bit or Qbit 886
19.3.6 The No-Cloning Theorem 889
19.3.7 Entanglement and Bell States 890
19.3.8 Quantum Dense Coding 895
19.3.9 Quantum Teleportation 896
References 900
Problems 901
Further Reading 907
Index 915
