Eringen Mathematics

1. Auflage 2013
ISBN: 978-1-4832-7716-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 694 Seiten, Web PDF

Continuum Physics: Volume 1 - Mathematics is a collection of papers that discusses certain selected mathematical methods used in the study of continuum physics. Papers in this collection deal with developments in mathematics in continuum physics and its applications such as, group theory functional analysis, theory of invariants, and stochastic processes. Part I explains tensor analysis, including the geometry of subspaces and the geometry of Finsler. Part II discusses group theory, which also covers lattices, morphisms, and crystallographic groups. Part III reviews the theory of invariants that includes isotrophy, transverse isotrophy, and nonpolynomial invariants. Part IV explains functional analysis that also includes set theory, vector spaces, topological spaces, and topological vector spaces. Part V deals with analytic function theory and covers topics, such as Cauchy's theorem, the residue theorem, and the Plemelj formulas. Part VI reviews the elements of stochastic processes and cites some examples where stochastic theory is applied. This book can be valuable for researchers and scientists involved in nuclear physicists, students, and academicians in the field of advanced physics.

Autoren/Hrsg.

Eringen, A. Cemal

Inhaltsverzeichnis

1;Front Cover;1
2;Mathematics;4
3;Copyright Page;5
4;Table of Contents;6
5;List of Contributors;14
6;Preface;16
7;Part I: Tensor Analysis;18
7.1;Introduction;19
7.2;Chapter 1. Tensor Algebra;21
7.2.1;1.1. SCOPE OF THE SECTION;21
7.2.2;1.2. CURVILINEAR COORDINATES;21
7.2.3;1.3. AFFINE GEOMETRY;23
7.2.4;1.4. VECTOR SPACE;28
7.2.5;1.5. GEOMETRIC OBJECTS;31
7.2.6;1.6. SCALARS, VECTORS, AND TENSORS;33
7.2.7;1.7. BASE VECTORS AND RECIPROCAL BASES;37
7.2.8;1.8. TENSOR ALGEBRA;44
7.2.9;1.9. MULTIVECTORS;49
7.2.10;1.10. SECOND-ORDER TENSORS;56
7.2.11;1.11. NORMAL FORM OF SYMMETRIC TENSORS;63
7.2.12;1.12. NORMAL FORM OF A BIVECTOR;69
7.3;Chapter 2. Tensor Analysis;72
7.3.1;2.1. SCOPE OF THE SECTION;72
7.3.2;2.2. METRIC TENSOR;73
7.3.3;2.3. ANHOLONOMIC COMPONENTS OF TENSORS;80
7.3.4;2.4. PHYSICAL COMPONENTS OF TENSORS;81
7.3.5;2.5. COVARIANT DIFFERENTIATION;85
7.3.6;2.6. INVARIANT DIFFERENTIAL OPERATORS;90
7.3.7;2.7. INTRINSIC DIFFERENTIATION;94
7.3.8;2.8. THE LIE DERIVATIVE;99
7.3.9;2.9. THE RIEMANN-CHRISTOFFEL TENSOR;104
7.4;Chapter 3. Geometry of Subspaces;107
7.4.1;3.1. SCOPE OF THE SECTION;107
7.4.2;3.2. CURVILINEAR COORDINATES FOR A SURFACE IN E3;108
7.4.3;3.3. SUBSPACE £m OF £n;113
7.4.4;3.4. SECTIONS AND REDUCTIONS OF TENSORS;116
7.4.5;3.5. TOTAL COVARIANT DIFFERENTIATION;122
7.4.6;3.6. CURVES IN SPACE;125
7.4.7;3.7. HYPERSURFACE Xn_1 IN Rn;130
7.4.8;3.8. THE VOLUME OF Xm IN Xn;135
7.4.9;3.9. THE STOKES THEOREM;138
7.5;Chapter 4. Nonriemannian Geometry;143
7.5.1;4.1. SCOPE OF THE SECTION;143
7.5.2;4.2. AFFINE CONNECTION;144
7.5.3;4.3. GEODESICS;150
7.5.4;4.4. CURVATURE;151
7.5.5;4.5. SOME IDENTITIES INVOLVING THE CURVATURE TENSOR;155
7.5.6;4.6. COVARIANT DIFFERENTIATION AND CURVATURE TENSORIN ANHOLONOMIC COORDINATES;156
7.6;Chapter 5. Geometry of Finsler;158
7.6.1;5.1. SCOPE OF THE SECTION;158
7.6.2;5.2. FINSLER SPACES;158
7.6.3;5.3. METRIC TENSOR DERIVABLE FROM A FUNCTION;160
7.6.4;5.4. COVARIANT DIFFERENTIATION;164
7.6.5;5.5. TORSION AND CURVATURE TENSORS;166
7.7;Acknowledgment;170
7.8;REFERENCES;170
8;Part II: Group Theory;174
8.1;Introduction;174
8.2;Chapter 1. Groups and Semigroups;175
8.3;Chapter 2. Lattices and Morphisms;193
8.4;Chapter 3. Lie Groups;201
8.5;Chapter 4. Linear Algebras, Frobenius and Lie;209
8.6;Chapter 5. Crystallographic Groups;214
8.7;BIBLIOGRAPHY;253
9;Part III Theory of Invariants;256
9.1;Chapter 1. Introduction;257
9.1.1;1.1. INVARIANTS OF VECTORS AND TENSORS;259
9.1.2;1.2. REDUCIBLE AND IRREDUCIBLE INVARIANTS; INTEGRITY BASES;262
9.1.3;1.3. RESULTS FROM CLASSICAL THEORY;263
9.1.4;1.4. THE ORTHOGONAL GROUPS AND CERTAIN SUBGROUPS;267
9.2;Chapter 2. Isotropy;273
9.2.1;2.1. INTEGRITY BASES FOR VECTORS;273
9.2.2;2.2. ISOTROPIC TENSORS;276
9.2.3;2.3. INVARIANTS OF VECTORS AND SECOND-ORDER TENSORS; GENERAL FORMS;278
9.2.4;2.4. RESULTS CONCERNING TRACES OF MATRIX POLYNOMIALS;279
9.2.5;2.5. INVARIANTS OF SYMMETRIC SECOND-ORDER TENSORS;285
9.2.6;2.6. INVARIANTS OF VECTORS AND SYMMETRIC SECOND-ORDER TENSORS; PROPER ORTHOGONAL GROUP;292
9.2.7;2.7. FULL ORTHOGONAL GROUP; INVARIANTS OF VECTORS AND SECOND-ORDER TENSORS;306
9.3;Chapter 3. Transverse Isotropy;309
9.3.1;3.1. INVARIANTS OF VECTORS AND TENSORS; GENERAL FORMS;311
9.3.2;3.2. RELATIONS FOR MATRIX POLYNOMIALS IN 2 x 2 MATRICES;313
9.3.3;3.3. INVARIANTS OF SYMMETRIC SECOND-ORDER TENSORS;314
9.3.4;3.4. INVARIANTS OF SYMMETRIC SECOND-ORDER TENSORS AND VECTORS;316
9.3.5;3.5. SYZYGIES FOR THE INVARIANTS;320
9.4;Chapter 4. The Crystal Classes;324
9.4.1;4.1. THEOREMS CONCERNING INTEGRITY BASES;325
9.4.2;4.2. INVARIANTS OF A SYMMETRIC TENSOR;327
9.4.3;4.4. INVARIANTS OF A SYMMETRIC TENSOR AND A VECTOR;333
9.4.4;4.5. INVARIANTS OF AN ARBITRARY NUMBER OF VECTORS;335
9.5;Chapter 5. Tensor Polynomial Functions of Vectors and Tensors;337
9.5.1;5.1. GENERAL STATEMENTS;337
9.5.2;5.2. EXAMPLES OF TENSOR AND VECTOR POLYNOMIAL FUNCTIONS; PROPER ORTHOGONAL TRANSFORMATION GROUP;340
9.5.3;5.3. EXAMPLES OF TENSOR AND VECTOR POLYNOMIAL FUNCTIONS; FULL ORTHOGONAL TRANSFORMATION GROUP;346
9.6;Chapter 6. Invariant Functionals; Vector and Tensor Functionals;351
9.6.1;6.1. GENERAL STATEMENTS;351
9.6.2;6.2. DIFFERENTIAL APPROXIMATIONS TO FUNCTIONALS;353
9.6.3;6.3. INTEGRAL APPROXIMATIONS TO FUNCTIONALS;353
9.7;Chapter 7. Minimality of the Integrity Bases;354
9.7.1;7.1. THE NUMBER OF LINEARLY INDEPENDENT INVARIANTS;355
9.7.2;7.2. METHOD OF DEMONSTRATING MINIMALITY OF AN INTEGRITY BASIS;364
9.7.3;7.3. MINIMALITY OF INTEGRITY BASES. THE CRYSTAL CLASSES;366
9.7.4;7.4. MINIMALITY OF INTEGRITY BASES. FULL AND PROPER ORTHOGONAL GROUPS;366
9.8;Chapter 8. Nonpolynomial Invariants;367
9.8.1;8.1. NONPOLYNOMIAL INVARIANTS;367
9.9;REFERENCES;369
10;Part IV: Functional Analysis;372
10.1;Introduction;373
10.2;Chapter 1. Set Theory;374
10.2.1;1.1. NOTATION;374
10.2.2;1.2. THE ALGEBRA OF SETS;375
10.2.3;1.3. MAPPINGS;377
10.2.4;1.4. COUNTABLE SETS;382
10.2.5;1.5. CLASSES OF SUBSETS;383
10.2.6;1.6. SET FUNCTIONS;386
10.3;Chapter 2. Vector Spaces;389
10.3.1;2.1. DEFINITION OF A VECTOR SPACE;390
10.3.2;2.2. LINEAR MAPPINGS;391
10.3.3;2.3. THE ALGEBRAIC DUAL OF A VECTOR SPACE;392
10.3.4;2.4. CONVEX SETS IN A VECTOR SPACE;392
10.3.5;2.5. MAXIMAL SUBSPACES AND HYPERPLANES;395
10.3.6;2.6. LINEAR TRANSFORMATIONS FROM Rn INTO Rm IN TERMS OF COORDINATES;395
10.4;Chapter 3. Topological Spaces;397
10.4.1;3.1. OPEN SETS;397
10.4.2;3.2. CLOSED SETS;400
10.4.3;3.3. METRIC SPACES;402
10.4.4;3.4. CONTINUOUS MAPPINGS;404
10.4.5;3.5. HAUSDORFF SPACES;405
10.4.6;3.6. COMPACT SETS;407
10.4.7;3.7. COMPLETE METRIC SPACES;410
10.4.8;3.8. HOMEOMORPHISMS;415
10.5;Chapter 4. Topological Vector Spaces;417
10.5.1;4.1. DEFINITION OF A TOPOLOGICAL VECTOR SPACE;418
10.5.2;4.2. NORMED VECTOR SPACES;419
10.5.3;4.3. VECTOR SUBSPACES;425
10.5.4;4.4. BANACH SPACES;426
10.5.5;4.5. FlNITE-DlMENSIONAL NORMED SPACES;430
10.5.6;4.6. HILBERT SPACES;433
10.5.7;4.7. LOCALLY CONVEX SPACES;454
10.6;Chapter 5. Spectral Theory of Linear Operators;457
10.6.1;5.1. THE SPECTRUM OF AN OPERATOR;457
10.6.2;5.2. NORMAL OPERATORS IN A HILBERT SPACE;458
10.6.3;5.3. SELF-ADJOINT OPERATORS IN A HILBERT SPACE;458
10.6.4;5.4. COMPACT SYMMETRIC OPERATORS IN A HILBERT SPACE;461
10.7;Chapter 6. Differential Calculus;462
10.7.1;6.1. THE GATEAUX DERIVATIVE;463
10.7.2;6.2. THE FRÉCHET DERIVATIVE;464
10.7.3;6.3. THE CHAIN RULE;465
10.7.4;6.4. NEWTON'S METHOD;467
10.7.5;6.5. HIGHER DERIVATIVES;468
10.7.6;6.6. DIFFERENTIABLE MANIFOLDS;468
10.8;Chapter 7. Distributions;470
10.8.1;7.1. THE SPACE OF TEST FUNCTIONS;472
10.8.2;7.2. DISTRIBUTIONS;475
10.8.3;7.3. EXAMPLES OF DISTRIBUTIONS;476
10.8.4;7.4. DIFFERENTIATION OF DISTRIBUTIONS;478
10.8.5;7.5. MULTIPLICATION OF DISTRIBUTIONS;485
10.8.6;7.6. DISTRIBUTIONS WITH COMPACT SUPPORT;486
10.8.7;7.7. TENSOR PRODUCT OF DISTRIBUTIONS;486
10.8.8;7.8. CONVOLUTION OF DISTRIBUTIONS;487
10.8.9;7.9. FOURIER TRANSFORMS;489
10.8.10;7.10. SOBOLEV SPACES;492
10.9;Appendix: The Lebesgue Integral;493
10.9.1;A.l. LEBESGUE MEASURE;493
10.9.2;A.2. THE LEBESGUE INTEGRAL;496
10.9.3;A.3. PROPERTIES OF THE LEBESQUE INTEGRAL;498
10.9.4;A.4. THE Lp-SPACES;502
10.10;BIBLIOGRAPHY;504
11;Part V: Analytic Function Theory;508
11.1;Chapter 1. Cauchy Integrals;509
11.1.1;1.1. INTRODUCTION;509
11.1.2;1.2. CAUCHY'S THEOREM AND INTEGRAL FORMULA;512
11.1.3;1.3. SINGULARITIES: THE RESIDUE THEOREM;515
11.1.4;1.4. THE POINT AT INFINITY;518
11.1.5;1.5. CAUCHY PRINCIPAL VALUES;521
11.1.6;1.6. THE RIEMANN-STIELTJES INTEGRAL: THE DELTA FUNCTION;525
11.1.7;1.7. THE PLEMELJ FORMULAS;527
11.1.8;1.8. CONTOUR PASSING THROUGH THE POINT AT INFINITY;529
11.1.9;1.9. INVERSION FORMULAS: THE POINCARÉ-BERTRAND FORMULA;531
11.1.10;1.10. SINGULARITIES ON THE CONTOUR C;534
11.2;Chapter 2. The Fundamental Problems of Potential Theory;536
11.2.1;2.1. THE DIRICHLET AND NEUMANN PROBLEMS;536
11.2.2;2.2. SOLUTIONS OF THE FUNDAMENTAL PROBLEMS FOR A CIRCLE;538
11.2.3;2.3. THE REFLECTION PRINCIPLE;542
11.2.4;2.4. SOLUTIONS OF THE FUNDAMENTAL PROBLEMS FOR AN INFINITE STRIP;544
11.2.5;2.5. SIMPLY-MIXED BOUNDARY CONDITIONS FOR A STRIP;547
11.2.6;2.6. PERIODIC BOUNDARY CONDITIONS;549
11.2.7;2.7. PERIODIC BOUNDARY CONDITIONS FOR THE INFINITE STRIP;553
11.2.8;2.8. THE SOLUTION OF THE FUNDAMENTAL PROBLEMS FOR THE RECTANGLE;556
11.2.9;2.9. UNIQUENESS OF SOLUTIONS TO THE DIRICHLET PROBLEM;558
11.2.10;2.10. REDUCTION OF THE DIRICHLET PROBLEM TO AN INTEGRAL EQUATION;561
11.2.11;2.11. GREEN'S FUNCTION;565
11.3;Chapter 3. Conformal Mapping;569
11.3.1;3.1. GENERAL PRINCIPLES;569
11.3.2;3.2. THE SCHWARZ-CHRISTOFFEL MAPPING THEOREM;573
11.3.3;3.3. AN INTEGRAL EQUATION FOR MAPPING ON THE UPPER HALF PLANE;578
11.3.4;3.4. GENERALIZATIONS OF THE S-C MAPPING THEOREM;580
11.3.5;3.5. THE MAPPING OF NEARLY CIRCULAR DOMAINS;584
11.3.6;3.6. THE MAPPING OF DOUBLY CONNECTED REGIONS;586
11.3.7;3.7. GREEN'S FUNCTION AND CONFORMAL MAPPING;588
11.3.8;3.8. THE KERNEL FUNCTION;590
11.3.9;3.9. MAPPING AND FLUID DYNAMICS;592
11.3.10;3.10. PRACTICAL METHODS OF CONSTRUCTING CONFORMAL MAPS;594
11.4;Chapter 4. The Hubert Problem and Applications;595
11.4.1;4.1. THE HILBERT PROBLEM;595
11.4.2;4.2. THE RIEMANN-HILBERT PROBLEM FOR SINGLY CONNECTED DOMAINS;597
11.4.3;4.3. AN ALTERNATIVE TREATMENT OF THE RIEMANN-HILBERT PROBLEM;601
11.4.4;4.4. THE RIEMANN-HILBERT PROBLEM WITH DISCONTINUOUS COEFFICIENTS;603
11.4.5;4.5. INVERSION FORMULAS FOR ARCS;607
11.4.6;4.6. SINGULAR INTEGRAL EQUATIONS;608
11.4.7;4.7. THE POINCARÉ PROBLEM;613
11.4.8;4.8. THE WIENER–HOPF TECHNIQUE;615
11.5;ACKNOWLEDGMENT;619
11.6;REFERENCES;619
12;Part VI: Elements of Stochastic Processes;622
12.1;Chapter 1. Introduction;623
12.1.1;1.1. SOME APPLICATIONS OF THE THEORY OF STOCHASTIC PROCESSES;623
12.1.2;1.2. HISTORICAL DEVELOPMENT OF THE MATHEMATICAL THEORY OF STOCHASTIC PROCESSES;624
12.1.3;1.3. SCOPE OF THIS WORK;625
12.1.4;1.4. PHYSICAL DESCRIPTION OF RANDOM PROCESSES;625
12.1.5;1.5. MATHEMATICAL DESCRIPTION OF RANDOM PROCESSES;627
12.1.6;1.6. RANDOM FUNCTIONS OF SEVERAL INDEPENDENT VARIABLES;629
12.1.7;1.7. RANDOM TENSORS;629
12.1.8;1.8. CLASSIFICATION OF RANDOM FUNCTIONS;630
12.2;Chapter 2. Calculus and Second-Order Properties of Random Functions;630
12.2.1;2.1. STOCHASTIC LIMITS AND MODES OF CONVERGENCE;630
12.2.2;2.2. REGULAR RANDOM FUNCTIONS;632
12.2.3;2.3. CONTINUITY OF RANDOM FUNCTIONS;632
12.2.4;2.4. DIFFERENTIABILITY OF RANDOM FUNCTIONS;633
12.2.5;2.5. INTEGRATION OF RANDOM FUNCTIONS;634
12.2.6;2.6. THEOREMS ON STOCHASTIC DIFFERENTIATION AND INTEGRATION;635
12.2.7;2.7. AUTOCOVARIANCE AND AUTOCORRELATION FUNCTIONS;636
12.2.8;2.8. SPECTRAL DENSITY FUNCTION;643
12.2.9;2.9. ORTHOGONAL EXPANSIONS OF RANDOM FUNCTIONS;650
12.2.10;2.10. FOURIER ANALYSIS OF STATIONARY PROCESSES;654
12.2.11;2.11. THE GAUSSIAN RANDOM PROCESS, DEFINITIONS;655
12.2.12;2.12. THEOREMS AND RESULTS;656
12.3;Chapter 3. Differential Equations for Distribution Functions of Stochastic Processes and Some Special Properties of Stochastic Processes;658
12.3.1;3.1. DIFFERENTIAL EQUATION FOR THE CHARACTERISTIC AND DISTRIBUTION FUNCTIONS OF PHYSICAL PROCESSES (Sufficiently Smooth Processes);658
12.3.2;3.2. MARKOFF PROCESSES;664
12.3.3;3.3. EXPECTED NUMBER OF ZEROS PER UNIT OF TIME;667
12.3.4;3.4. EXPECTED NUMBER OF MAXIMA PER UNIT OF TIME;670
12.3.5;3.5. OTHER PROPERTIES OF INTEREST;671
12.4;Chapter 4. Stochastic Boundary and Initial Value Problems;671
12.4.1;4.1. FORMULATION OF BOUNDARY AND INITIAL VALUE PROBLEMS UNDER STOCHASTIC CONDITIONS;672
12.4.2;4.2. STOCHASTIC ORDINARY DIFFERENTIAL EQUATIONS;675
12.4.3;4.3. STOCHASTIC STABILITY THEORY;677
12.5;REFERENCES;678
13;Author Index;682
14;Subject Index;687