2000 B 2093 MATHEMATICAL METHODS FOR PHYSICISTS AND ENGINEERS · sub gottingen 211 766 267 modern...
13
SUB Gottingen 211 766 267 MODERN MATHEMATICAL METHODS FOR PHYSICISTS AND ENGINEERS 2000 B 2093 C. D. CANTRELL CAMBRIDGE UNIVERSITY PRESS
Transcript of 2000 B 2093 MATHEMATICAL METHODS FOR PHYSICISTS AND ENGINEERS · sub gottingen 211 766 267 modern...
SUB Gottingen211 766 267
MODERNMATHEMATICALMETHODS FORPHYSICISTS ANDENGINEERS
2000 B 2093
C. D. CANTRELL
CAMBRIDGEUNIVERSITY PRESS
CONTENTS
Preface page xix
1 FOUNDATIONS OF COMPUTATION 11.1 Introduction 11.2 Representations of Numbers 2
1.2.1 Integers 31.2.2 Rational Numbers and Real Numbers 141.2.3 Representation's of Numbers as Text 171.2.4 Exercises for Section 1.2 20
1.3 Finite Floating-point Representations 211.3.1 Simple Cases 211.3.2 Practical Floating-point Representations 251.3.3 Approaching Zero or Infinity Gracefully 281.3.4 Exercises for Section 1.3 30
1.4 Floating-point Computation 311.4.1 Relative Error; Machine Epsilon 311.4.2 Rounding 32;1.4.3 Floating-point Addition and Subtraction 351.4.4 Exercises for Section 1.4 36
1.5 Propagation of Errors 371.5.1 General Formulas 371.5.2 Examples of Error Propagation 391.5.3 Estimates of the Mean and Variance 411.5.4 Exercises for Section 1.5 43
1.6 Bibliography and Endnotes 451.6.1 Bibliography 451.6.2 Endnotes 46
2 SETS AND MAPPINGS 472.1 Introduction 472.2 Basic Definitions 49
2.2.1 Sets 492.2.2 Mappings 532.2.3 Axiom of Choice 622.2.4 Cartesian Products 62
VII
viii CONTENTS
2.2.5 Equivalence and Equivalence Classes 65
2.2.6 Exercises for Section 2.2 672.3 Union, Intersection, and Complement 68
2.3.1 Unions of Sets 682.3.2 Intersections of Sets 692.3.3 Relative Complement 702.3.4 De Morgan's Laws 712.3.5 Exercises for Section 2.3 71
2.4 Infinite Sets 722.4.1 Basic Properties of Infinite Sets 722.4.2 Induction and Recursion 732.4.3 Countable Sets 762.4.4 Countable Unions and Intersections 772.4.5 Uncountable Sets 782.4.6 Exercises for Section 2.4 80
2.5 Ordered and Partially Ordered Sets 822.5.1 Partial Orderings 822.5.2 Orderings; Upper and Lower Bounds 832.5.3 Maximal Chains 842.5.4 Exercises for Section 2.5 84
2.6 Bibliography 85
3 EVALUATION OF FUNCTIONS 863.1 Introduction 863.2 Sensitivity and Condition Number 86
3.2.1 Definitions 863.2.2 Evaluation of Polynomials 873.2.3 Multiple Roots of Polynomials 893.2.4 Exercises for Section 3.2 91
3.3 Recursion and Iteration 923.3.1 Finding Roots by Bisection 923.3.2 Newton-Raphson Method 923.3.3 Evaluation of Series 953.3.4 Exercises for Section 3.3 97
3.4 Introduction to Numerical Integration 993.4.1 Rectangle Rules 1013.4.2 Trapezoidal Rule 1023.4.3 Local and Global Errors 1023.4.4 Exercises for Section 3.4 105
3.5 Solution of Differential Equations 1063.5.1 Euler's Method 1073.5.2 Truncation Error of Euler's Method 1093.5.3 Stability Analysis of Euler's Method 111
CONTENTS
112
118
120
121
121
122
122
127
130
136
141
143
143
147
149
152
152
158
162
164
165
165
167
169
171
173
175
175
179
180
181
184
184
184
186
189
189
VECTOR SPACES 1915.1 Introduction 1915.2 Basic Definitions and Examples 193
5.2.1 Axioms for a Vector Space 1935.2.2 Selected Realizations of the Vector-space Axioms 194
3.6
3.5.43.5.5
Selected Finite-difference MethodsExercises for Section 3.5
Bibliography
GROUPS,4.14.2
4.3
4.4
4.5
4.6
4.7
RINGS, AND FIELDSIntroductionGroup;4.2.14.2.24.2.34.2.44.2.54.2.64.2.74.2.84.2.9
3
AxiomsTwo-element GroupOrbits and CosetsCyclic GroupsDihedral GroupsCubic GroupsContinuous GroupsClasses of Conjugate ElementsExercises for Section 4.2
Group Homomorphisms4.3.14.3.24.3.34.3.4
Definitions and Basic PropertiesNormal SubgroupsDirect Product GroupsExercises for Section 4.3
*Symmetric Groups4.4.14.4.24.4.34.4.44.4.5Rings4.5.14.5.24.5.34.5.44.5.5Fields4.6.14.6.24.6.3
PermutationsCayley's TheoremCyclic PermutationsEven and Odd PermutationsExercises for Section 4.4
and Integral Domains \Axioms and ExamplesBasic Properties of RingsRational Numbers*Ring HomomorphismsExercises for Section 4.5
Axioms and Examples* Galois FieldsExercises for Section 4.6
Bibliography
CONTENTS
5.2.3 Vector Subspaces 2015.2.4 ""Comments on Vector-space Axioms 2055.2.5 Exercises for Section 5.2 208
5.3 Linear Independence and Linear Dependence 2115.3.1 Definitions 2115.3.2 Basic Results on Linear Dependence 2125.3.3 Examples of Linear Independence 2165.3.4 Exercises for Section 5.3 219
5.4 Bases and Dimension 2215.4.1 Dimension of a Vector Space 2215.4.2 Selected Realizations of Vector-space Bases 2255.4.3 Vector-space Isomorphisms 2285.4.4 Gaussian Elimination and Linear Dependence 2325.4.5 Exercises for Section 5.4 237
5.5 Complementary Subspaces 2395.5.1 Vector Complements and Direct Sums 2395.5.2 Definition of Complementary Subspaces 2405.5.3 Dimensions of Complementary Subspaces 2415.5.4 Direct Sums of Vector Spaces 2425.5.5 Bases of Complementary Subspaces 2435.5.6 Examples of Direct Sums of Vector Spaces 2445.5.7 Exercises for Section 5.5 245
5.6 Bibliography and Endnotes 2465.6.1 Bibliography 2465.6.2 Endnotes 246
LINEAR MAPPINGS I 2486.1 Linear Mappings and their Matrices 248
6.1.1 Basic Properties 2486.1.2 Matrix of a Linear Mapping 2506.1.3 Computation of Matrix Products 2616.1.4 Invariant Subspaces and Direct Sums 2636.1.5 Other Examples of Linear Mappings 2656.1.6 Exercises for Section 6.1 268
6.2 Nonsingular Linear Mappings 2716.2.1 Definitions and Basic Properties 2716.2.2 Change of Basis 2756.2.3 Permutation Matrices 2776.2.4 General Linear Group of a Vector Space 2786.2.5 Exercises for Section 6.2 278
6.3 Singular Linear Mappings 2816.3.1 Singularity and Linear Dependence 2816.3.2 Visualization of Singular Linear Mappings 282
CONTENTS
6.3.3 Null Space of a Linear Mapping 2836.3.4 Other Examples of a Singular Linear Mappings 2856.3.5 Exercises for Section 6.3 287
6.4 Introduction to Digital Filters 2886.4.1 Definitions 2886.4.2 Noise Amplification by Digital Filters 2916.4.3 Difference Operators 2926.4.4 Exercises for Section 6.4 298
6.5 Trace and Determinant 2996.5.1 Trace of a Linear Mapping 2996.5.2 Determinants 3006.5.3 Exercises for Section 6.5 309
6.6 Solution of Linear Equations 3106.6.1 Basic Facts about Linear Equations 3106.6.2 Matrix Formulation of Gaussian Elimination 3126.6.3 Computational Aspects of Gaussian Elimination 3176.6.4 LU and LDMr Decompositions 3176.6.5 Bases of the Range and Null Space 3196.6.6 Rank-nullity Theorem 3216.6.7 Exercises for Section 6.6 322
6.7 Complements of Null Space 3246.7.1 Quotient Space V/null [A] 3246.7.2 Isomorphism of the Range to a Complement of the Null Space 3256.7.3 Rank-nullity Theorem (Again) 3276.7.4 Right Inverses of a Linear Mapping 3276.7.5 Examples of Right Inverses 3286.7.6 Exercise for Section 6.7 , 330
6.8 Bibliography 330
LINEAR FUNCTIONALS 3317.1 Motivation for Studying Functionals 3317.2 Dual Spaces 332
7.2.1 Definitions 3327.2.2 Range and Null Space of a Linear Functional 3347.2.3 Exercises for Section 7.2 335
7.3 Coordinate Functionals 3367.3.1 Definitions 3367.3.2 Coordinate Functionals on F" 3377.3.3 Isomorphism of V* to V 3387.3.4 Coordinate Functionals on Two-dimensional Euclidean Space 3397.3.5 Coordinate Functionals and the Reciprocal Lattice 3417.3.6 Isomorphism of V to V** 3457.3.7 Exercises for Section 7.3 346
CONTENTS
7.4 Annihilator of a Subspace 3477.4.1 Definitions 3477.4.2 Bases of the Annihilator 3477.4.3 Exercises for Section 7.4 - • 348
7.5 Other Realizations of Dual Spaces 3497.5.1 Dual Space of C 3497.5.2 DualofFz+ 3497.5.3 Boundary and Initial Conditions for Differential Equations 349
7.6 Polynomial Interpolation 3507.6.1 Lagrangian Interpolation 3507.6.2 Exercises for Section 7.6 352
7.7 Tensors 3527.7.1 Definitions and Basic Properties 3537.7.2 Components of Second-rank Tensors 3567.7.3 Tensor Products of Vectors 3587.7.4 Tensors of Rank m 3617.7.5 Linear Mappings of Tensors 3627.7.6 ^Exercises for Section 7.7 365
INNER PRODUCTS AND NORMS 3678.1 Inner-product Spaces 367
8.1.1 Definitions 3678.1.2 Canonical Inner Products 3698.1.3 Metric Tensor 3728.1.4 Indefinite Inner Products 3788.1.5 Orthogonality 3798.1.6 Exercises for Section 8.1 383
8.2 Geometry of Inner-product Space's 3848.2.1 Pythagoras's Theorem 3848.2.2 Orthonormal Bases 3928.2.3 Orthogonal Polynomials 3978.2.4 Exercises for Section 8.2 403
8.3 Projection Methods 4068.3.1 Projection of a Vector onto a Subspace: Definition 4068.3.2 Orthogonal Projectors 4078.3.3 Orthogonal Complement 4098.3.4 Exercises for Section 8.3 416
8.4 Least-squares Approximations 4178.4.1 Motivation 4178.4.2 Abstract Formulation 4188.4.3 Inequalities for Least-squares Approximations 4198.4.4 Approximation by Finite Fourier Sums 4208.4.5 Chebyshev Approximations 421
CONTENTS
8.4.6 Mapping a Function to its Fourier Coefficients 4228.4.7 Exercises for Section 8.4 423
8.5 Discrete Fourier Transform 4248.5.1 Approximation of Fourier Coefficients 4248.5.2 Discrete Fourier Basis 4258.5.3 Periodic Extension 4288.5.4 Aliasing 4308.5.5 Sampling Theorem and Alias Mapping 4338.5.6 Exercise for Section 8.5 437
8.6 Volume of an m-Parallelepiped 4378.6.1 Parallelepipeds 4378.6.2 Recursive Definition of Volume 4388.6.3 Volume as a Determinant 4388.6.4 Determinant as a Volume Ratio 4408.6.5 Jacobian Determinant 4418.6.6 Exercise for Section 8.6 443
8.7 Vector and Matrix Norms 4438.7.1 Vector Norms 4438.7.2 Norm of a Linear Mapping 4468.7.3 Matrix Norms 4508.7.4 Norm of an Integral 4538.7.5 Exercises for Section 8.7 453
8.8 Inner Products and Linear Functionals 4558.8.1 Introduction 4558.8.2 Inner-product Mapping 4568.8.3 Inverse Inner-product Mapping 4598.8.4 Exercises for Section 8.8 '. 464
8.9 Bibliography and Endnotes 4648.9.1 Bibliography 4658.9.2 Endnotes 465
LINEAR MAPPINGS II 4669.1 Dyads 466
9.1.1 Motivation 4669.1.2 Definition of a Dyad 4669.1.3 Dyadic Expansions 4699.1.4 Resolutions of the Identity Mapping 4719.1.5 Exercise for Section 9.1 473
9.2 Transpose and Adjoint 4739.2.1 Transpose 4739.2.2 Adjoint 4769.2.3 Other Realizations of the Adjoint 4809.2.4 Properties of the Adjoint 483
xiv CONTENTS
9.2.5 Hermitian and Self-adjoint Mappings 4859.2.6 Isometric and Unitary Mappings 4879.2.7 Exercises for Section 9.2 491
9.3 Eigenvalues and Eigenvectors 4939.3.1 Secular Equation 4939.3.2 Diagonalization of Hermitian Matrices 4959.3.3 Normal Linear Mappings 5029.3.4 Exercises for Section 9.3 504
9.4 Singular-value Decomposition 5079.4.1 Derivation of the Singular-value Decomposition 5079.4.2 Matrix Version of the Singular-value Decomposition 5099.4.3 The Fundamental Subspaces of a Linear Mapping 5119.4.4 Inverse and Pseudo-inverse in the SVD 5129.4.5 Data Compression Using the SVD 5149.4.6 Exercises for Section 9.4 514
9.5 Linear Equations II 5159.5.1 Numerical Versus Analytical Methods 5159.5.2 Diagonal Dominance 5169.5.3 Condition Number of the Linear-equation Problem 5179.5.4 The LDLf and Cholesky Decompositions 520
9.6 Selected Applications of Linear Equations 5219.6.1 The Linear Least-squares Problem 5219.6.2 Linear Difference Equations 5239.6.3 Solution of Tridiagonal Systems 5299.6.4 Exercises for Section 9.6 530
9.7 Bibliography 531
10 CONVERGENCE IN NORMED VECTOR SPACES 53210.1 Metrics and Norms 532
10.1.1 Metric Spaces 53210.1.2 Normed Vector Spaces 53410.1.3 Examples of Metric and Normed Vector Spaces 53610.1.4 Open Sets 53910.1.5 Exercises for Section 10.1 541
10.2 Limit Points 54310.2.1 Limit Points and Closed Sets 54310.2.2 Dense Sets and Separable Spaces 54810.2.3 Exercises for Section 10.2 552
10.3 Convergence of Sequences and Series 55310.3.1 Convergence of Sequences 55310.3.2 Numerical Sequences 55810.3.3 Numerical Series 56010.3.4 Exercises for Section 10.3 565
CONTENTS xv
10.4 Strong and Pointwise Convergence 56610.4.1 Strong Convergence 56610.4.2 Operators 57010.4.3 Sequences of Real-valued Functions 57210.4.4 Series of Real-valued Functions 57510.4.5 Exercises for Section 10.4 576
10.5 Continuity 57710.5.1 Pointwise Continuity 57710.5.2 Uniform Continuity 580
10.6 Best Approximations in the Maximum and SupremumNorms 58110.6.1 Best Approximations in the Maximum Norm 58310.6.2 Best Approximations in the Supremum Norm 58910.6.3 Exercises for Section 10.6 592
10.7 Hilbert and Banach Spaces 59410.7.1 Survey of Complete Metric Vector Spaces 59410.7.2 Complete Orthonormal Sets 59710.7.3 Orthogonal Series 59910.7.4 Practical Aspects of Fourier Series 60310.7.5 Orthogonal-polynomial Expansions 61010.7.6 Exercises for Section 10.7 613
10.8 Bibliography 615
11 GROUP REPRESENTATIONS 61611.1 Preliminaries 616
11.1.1 Background 61611.1.2 Symmetry-adapted Functions 61811.1.3 Partner Functions : 61911.1.4 Exercises for Section 11.1 621
11.2 Reducibility of Representations 62111.2.1 Invariant Subspaces and Irreducibility 62211.2.2 Schur's Lemma 62311.2.3 Eigenvectors of Invariant Operators 62711.2.4 Exercises for Section 11.2 629
11.3 Unitarity and Orthogonality 63011.3.1 Consequences of the Rearrangement Theorem 63011.3.2 Unitary Representations 63111.3.3 Orthogonality Theorems 63311.3.4 Product Relation for Characters 63811.3.5 Reduction of Unitary Representations 64011.3.6 Construction of Character Tables 64311.3.7 Characters of Kronecker Products 64411.3.8 Exercises for Section 11.3 646
CONTENTS
11.4 Two-dimensional Rotation Group 64711.4.1 Representation Space for S 0 (2) 64811.4.2 Representations of S O (2) 64911.4.3 Completeness Relation for {e~imB} 65011.4.4 Exercise for Section 11.4 652
11.5 Symmetry and the One-dimensional Wave Equation 65211.5.1 Boundary Conditions and Symmetry 65211.5.2 Wave Equation for a Vibrating String 65311.5.3 Boundary Conditions for the One-dimensional Wave
Equation 65311.5.4 Form Invariance of the Wave Equation 65311.5.5 Invariance of the Wave Equation Under Translations 65611.5.6 Invariance of the Wave Equation Under Lorentz
Transformations 65611.5.7 D' Alembert's Solution of the Wave Equation 65711.5.8 Solution for a String of Infinite Length 65811.5.9 Solution for a String of Finite Length 65911.5.10 Exercises for Section 11.5 661
11.6 Discrete Translation Groups 66211.6.1 Motivation 66211.6.2 Invariance Under the Discrete Translation Group 66211.6.3 Discrete-shift-invariant Digital Filters 66311.6.4 Representations of the Discrete Translation Group 66411.6.5 Discrete-time Transfer Function 66511.6.6 Exercises for Section 11.6 668
11.7 Continuous Translation Groups 66911.7.1 Translation Group of the Real Line 66911.7.2 Irreducible Representations of T(R) 66911.7.3 {VOc} as Momentum Eigenfunctions 67111.7.4 Representation of T(E2) Carried by fk 67211.7.5 Translation Group of Euclidean n-space 67311.7.6 Exercises for Section 11.7 676
11.8 Fourier Transforms 67611.8.1 Fourier Transform in One Dimension 67611.8.2 Completeness Relation for the {fk} 61111.8.3 Fourier Transforms in n-dimensional Euclidean
Space 67811.8.4 Poisson Sum Formula 67911.8.5 Exercises for Section 11.8 679
11.9 Linear, Shift-invariant Systems 68011.9.1 Continuous-time-shift Invariance 68011.9.2 Continuous-time Transfer Function 68111.9.3 Exercises for Section 11.9 682
CONTENTS XVII
11.10 Two-dimensional Euclidean Group E(2) 68211.10.1 Representation of E(2) Carried by {x/fk} 68411.10.2 Discussion 68411.10.3 Exercises for Section 11.10 685
11.11 Bibliography 685
12 SPECIAL FUNCTIONS 68612.1 Group Theory and Special Functions 686
12.1.1 Separation of Variables 68612.1.2 Special Functions as Matrix Elements 68712.1.3 Symmetries of the Helmholtz Equation 68712.1.4 Exercise for Section 12.1 690
12.2 Definition of the Bessel Functions 69012.2.1 Fourier Expansion of a Plane Wave 69012.2.2 Definition of the Bessel Function Jm of Integer
Order 69212.2.3 Jacobi-Anger Expansion 69212.2.4 Frequency Content of an FM Signal 69412.2.5 Exercises for Section 12.2 694
12.3 Bessel-function Addition Formulas 69512.3.1 Exercises for Section 12.3 698
12.4 Bessel Raising and Lowering Operators 69812.4.1 Recurrence Relations 69812.4.2 Raising and Lowering Operators for Jm 70012.4.3 Raising and Lowering Operators for the Helmholtz
Functions 70012.4.4 Exercises for Section 12.4 701
12.5 Bessel Differential Equations ' 70112.5.1 Bessel's Differential Equation 70112.5.2 Helmholtz Equation in Two Dimensions 70212.5.3 Qualitative Behavior of Jm 70212.5.4 Exercises for Section 12.5 704
12.6 Orthogonal Series in /„ 70412.6.1 Boundary Conditions that Ensure Self-adjointness 70412.6.2 Orthogonality Relations 70812.6.3 Fourier-Bessel Series 70912.6.4 Exercise for Section 12.6 711
12.7 Vibrations of a Drumhead 71112.7.1 Exercise for Section 12.7 713
12.8 Power Series for Jm 71312.8.1 Derivation Using Raising and Lowering Operators 71312.8.2 Properties Deduced from the Power Series 71512.8.3 Exercises for Section 12.8 715
xviii CONTENTS
12.9 Completeness Relations Using Jn 71612.10 Bibliography 718
APPENDIX A INDEX OF NOTATION 719A.I Quantifiers and Other Logical Symbols 719A.2 Sets and Mappings 719A.3 Vector Spaces, Linear Mappings and Matrices 720A.4 Norms and Inner Products 721A.5 Functions 721
A.5.1 General Notation for Functions 721A.5.2 Special Functions 722
A.6 Probability 722
APPENDIX B AFFINE MAPPINGS 723B.I Affine Group of a Vector Space 723
r B.2 Coordinate Transformations 726B.2.1 Active Transformations 727B.2.2 Passive Transformations 727B.2.3 Relation between Active and Passive Transformations 728
B.3 Exercises 728
APPENDIX C PSEUDO-UNITARY SPACES 730
APPENDIX D REMAINDER TERM 733
APPENDIX E BOLZANO-WEIERSTRAB THEOREM 735E.I The Real Numbers ' 735
E.2 Finite-dimensional Hilbert Spaces 736
APPENDIX F WEIERSTRAB APPROXIMATION THEOREM 738
Index 745
