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.