Higher mathematics for physics and engineering:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2010
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| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | XXI, 688 S. graph. Darst. 24 cm |
| ISBN: | 9783540878636 9783642425912 |
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MARC
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| 100 | 1 | |a Shima, Hiroyuki |e Verfasser |0 (DE-588)141263954 |4 aut | |
| 245 | 1 | 0 | |a Higher mathematics for physics and engineering |c Hiroyuki Shima ; Tsuneyoshi Nakayama |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2010 | |
| 300 | |a XXI, 688 S. |b graph. Darst. |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 700 | 1 | |a Nakayama, Tsuneyoshi |d 1945- |e Verfasser |0 (DE-588)124884199 |4 aut | |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-020489705 | |
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IMAGE 1
CONTENTS
1 PRELIMINARIES 1
1.1 BASIC NOTIONS OF A SET 1
1.1.1 SET AND ELEMENT 1
1.1.2 NUMBER SETS 3
1.1.3 BOUNDS 3
1.1.4 INTERVAL 4
1.1.5 NEIGHBORHOOD AND CONTACT POINT 5
1.1.6 CLOSED AND OPEN SETS 7
1.2 CONDITIONAL STATEMENTS 9
1.3 ORDER OF MAGNITUDE 10
1.3.1 SYMBOLS O, O, AND ~ 10
1.3.2 ASYMPTOTIC BEHAVIOR 11
1.4 VALUES OF INDETERMINATE FORMS 12
1.4.1 L'HOPITAL'S RULE 12
1.4.2 SEVERAL EXAMPLES 13
PART I REAL ANALYSIS
2 REAL SEQUENCES AND SERIES 17
2.1 SEQUENCES OF REAL NUMBERS 17
2.1.1 CONVERGENCE OF A SEQUENCE 17
2.1.2 BOUNDED SEQUENCES 18
2.1.3 MONOTONIE SEQUENCES 19
2.1.4 LIMIT SUPERIOR AND LIMIT INFERIOR 21
2.2 CAUCHY CRITERION FOR REAL SEQUENCES 25
2.2.1 CAUCHY SEQUENCE 25
2.2.2 CAUCHY CRITERION 26
2.3 INFINITE SERIES OF REAL NUMBERS 29
2.3.1 LIMITS OF INFINITE SERIES 29
2.3.2 CAUCHY CRITERION FOR INFINITE SERIES 31
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/993291163
DIGITALISIERT DURCH
IMAGE 2
X CONTENTS
2.3.3 ABSOLUTE AND CONDITIONAL CONVERGENCE 32
2.3.4 REARRANGEMENTS 34
2.4 CONVERGENCE TESTS FOR INFINITE REAL SERIES 38
2.4.1 LIMIT TESTS 38
2.4.2 RATIO TESTS 40
2.4.3 ROOT TESTS 41
2.4.4 ALTERNATING SERIES TEST 42
3 REAL FUNCTIONS 45
3.1 FUNDAMENTAL PROPERTIES 45
3.1.1 LIMIT OF A FUNCTION 45
3.1.2 CONTINUITY OF A FUNCTION 47
3.1.3 DERIVATIVE OF A FUNCTION 48
3.1.4 SMOOTH FUNCTIONS 50
3.2 SEQUENCES OF REAL FUNCTIONS 50
3.2.1 POINTWISE CONVERGENCE 50
3.2.2 UNIFORM CONVERGENCE 52
3.2.3 CAUCHY CRITERION FOR SERIES OF FUNCTIONS 53
3.2.4 CONTINUITY OF THE LIMIT FUNCTION 54
3.2.5 INTEGRABILITY OF THE LIMIT FUNCTION 56
3.2.6 DIFFERENTIABILITY OF THE LIMIT FUNCTION 57
3.3 SERIES OF REAL FUNCTIONS 61
3.3.1 SERIES OF FUNCTIONS 61
3.3.2 PROPERTIES OF UNIFORMLY CONVERGENT SERIES OF FUNCTIONS 62 3.3.3
WEIERSTRASS M-TEST 63
3.4 IMPROPER INTEGRALS 66
3.4.1 DEFINITIONS 66
3.4.2 CONVERGENCE OF AN IMPROPER INTEGRAL 67
3.4.3 PRINCIPAL VALUE INTEGRAL 67
3.4.4 CONDITIONS FOR CONVERGENCE 68
PART II FUNCTIONAL ANALYSIS
4 HUBERT SPACES 73
4.1 HUBERT SPACES 73
4.1.1 INTRODUCTION 73
4.1.2 ABSTRACT VECTOR SPACES 74
4.1.3 INNER PRODUCT 75
4.1.4 GEOMETRY OF INNER PRODUCT SPACES 76
4.1.5 ORTHOGONALITY 78
4.1.6 COMPLETENESS OF VECTOR SPACES 79
4.1.7 SEVERAL EXAMPLES OF HUBERT SPACES 80
4.2 HIERARCHICAL STRUCTURE OF VECTOR SPACES 83
4.2.1 PRECISE DEFINITIONS OF VECTOR SPACES 83
IMAGE 3
CONTENTS XI
4.2.2 METRIC SPACE 84
4.2.3 NORMED SPACES 85
4.2.4 SUBSPACES OF A NORMED SPACE 86
4.2.5 BASIS OF A VECTOR SPACE: REVISITED 87
4.2.6 ORTHOGONAL BASES IN HUBERT SPACES 88
4.3 HUBERT SPACES OF 2 AND L 2 91
4.3.1 COMPLETENESS OF THE 2 SPACES 91
4.3.2 COMPLETENESS OF THE L 2 SPACES 92
4.3.3 MEAN CONVERGENCE 95
4.3.4 GENERALIZED FOURIER COEFFICIENTS 95
4.3.5 RIESZ-FISHER THEOREM 96
4.3.6 ISOMORPHISM BETWEEN I 2 AND I? 98
ORTHONORMAL POLYNOMIALS 101
5.1 POLYNOMIAL APPROXIMATIONS 101
5.1.1 WEIERSTRASS THEOREM 101
5.1.2 EXISTENCE OF COMPLETE ORTHONORMAL SETS OF POLYNOMIALS 103 5.1.3
LEGENDRE POLYNOMIALS 105
5.1.4 FOURIER SERIES 108
5.1.5 SPHERICAL HARMONIC FUNCTIONS 109
5.2 CLASSIFICATION OF ORTHONORMAL FUNCTIONS 114
5.2.1 GENERAL RODRIGUES FORMULA 114
5.2.2 CLASSIFICATION OF THE POLYNOMIALS 116
5.2.3 THE RECURRENCE FORMULA 119
5.2.4 COEFFICIENTS OF THE RECURRENCE FORMULA 120
5.2.5 ROOTS OF ORTHOGONAL POLYNOMIALS 121
5.2.6 DIFFERENTIAL EQUATIONS SATISFIED BY THE POLYNOMIALS . . . 122
5.2.7 GENERATING FUNCTIONS (I) 124
5.2.8 GENERATING FUNCTIONS (II) 125
5.3 CHEBYSHEV POLYNOMIALS 128
5.3.1 MINIMAX PROPERTY 128
5.3.2 A CONCISE REPRESENTATION 131
5.3.3 DISCRETE ORTHOGONALITY RELATION 133
5.4 APPLICATIONS IN PHYSICS AND ENGINEERING 135
5.4.1 QUANTUM-MECHANICAL STATE IN AN HARMONIC POTENTIAL . 135 5.4.2
ELECTROSTATIC POTENTIAL GENERATED BY A MULTIPOLE 136
LEBESGUE INTEGRALS 139
6.1 MEASURE AND SUMMABILITY 139
6.1.1 RIEMANN INTEGRAL REVISITED 139
6.1.2 MEASURE 141
6.1.3 THE PROBABILITY MEASURE 142
6.1.4 SUPPORT AND AREA OF A STEP FUNCTION 144
6.1.5 A-SUMMABILITY 146
6.1.6 PROPERTIES OF A-SUMMABLE FUNCTIONS 147
IMAGE 4
XII CONTENTS
6.2 LEBESGUE INTEGRAL 149
6.2.1 LEBESGUE MEASURE 149
6.2.2 DEFINITION OF THE LEBESGUE INTEGRAL 151
6.2.3 RIEMANN INTEGRALS VS. LEBESGUE INTEGRALS 152
6.2.4 PROPERTIES OF THE LEBESGUE INTEGRALS 153
6.2.5 NULL-MEASURE PROPERTY OF COUNTABLE SETS 154
6.2.6 THE CONCEPT OF ALMOST EVERYWHERE 155
6.3 IMPORTANT THEOREMS FOR LEBESGUE INTEGRALS 158
6.3.1 MONOTONE CONVERGENCE THEOREM 158
6.3.2 DOMINATED CONVERGENCE THEOREM (I) 160
6.3.3 FATOU LEMMA 160
6.3.4 DOMINATED CONVERGENCE THEOREM (II) 161
6.3.5 FUBINI THEOREM 162
6.4 THE LEBESGUE SPACES L P 167
6.4.1 THE SPACES OF L P 167
6.4.2 HOLDER INEQUALITY 168
6.4.3 MINKOWSKI INEQUALITY 169
6.4.4 COMPLETENESS OF LP SPACES 170
6.5 APPLICATIONS IN PHYSICS AND ENGINEERING 172
6.5.1 PRACTICAL SIGNIFICANCE OF LEBESGUE INTEGRALS 172 6.5.2 CONTRACTION
MAPPING 173
6.5.3 PRELIMINARIES FOR THE CENTRAL LIMIT THEOREM 175 6.5.4 CENTRAL
LIMIT THEOREM 177
6.5.5 PROOF OF THE CENTRAL LIMIT THEOREM 178
PART III COMPLEX ANALYSIS
7 COMPLEX FUNCTIONS 185
7.1 ANALYTIC FUNCTIONS 185
7.1.1 CONTINUITY AND DIFFERENTIABILITY 185
7.1.2 DEFINITION OF AN ANALYTIC FUNCTION 187
7.1.3 CAUCHY-RIEMANN EQUATIONS 189
7.1.4 HARMONIC FUNCTIONS 191
7.1.5 GEOMETRIC INTERPRETATION OF ANALYTICITY 192
7.2 COMPLEX INTEGRATIONS 195
7.2.1 INTEGRATION OF COMPLEX FUNCTIONS 195
7.2.2 CAUCHY THEOREM 197
7.2.3 INTEGRATIONS ON A MULTIPLY CONNECTED REGION 199 7.2.4 PRIMITIVE
FUNCTIONS 201
7.3 CAUCHY INTEGRAL FORMULA AND RELATED THEOREM 204
7.3.1 CAUCHY INTEGRAL FORMULA 204
7.3.2 GOURSAT FORMULA 206
7.3.3 ABSENCE OF EXTREMA IN ANALYTIC REGIONS 207
7.3.4 LIOUVILLE THEOREM 208
IMAGE 5
CONTENTS XIII
7.3.5 FUNDAMENTAL THEOREM OF ALGEBRA 209
7.3.6 MORERA THEOREM 210
7.4 SERIES REPRESENTATIONS 213
7.4.1 CIRCLE OF CONVERGENCE 213
7.4.2 SINGULARITY ON THE RADIUS OF CONVERGENCE 215 7.4.3 TAYLOR SERIES
217
7.4.4 APPARENT PARADOXES 218
7.4.5 LAURENT SERIES 219
7.4.6 REGULAR AND PRINCIPAL PARTS 221
7.4.7 UNIQUENESS OF LAURENT SERIES 222
7.4.8 TECHNIQUES FOR LAURENT EXPANSION 223
7.5 APPLICATIONS IN PHYSICS AND ENGINEERING 228
7.5.1 FLUID DYNAMICS 228
7.5.2 KUTTA-JOUKOWSKI THEOREM 229
7.5.3 BLASIUS FORMULA 231
SINGULARITY AND CONTINUATION 233
8.1 SINGULARITY 233
8.1.1 ISOLATED SINGULARITIES 233
8.1.2 NONISOLATED SINGULARITIES 235
8.1.3 WEIERSTRASS THEOREM FOR ESSENTIAL SINGULARITIES 236 8.1.4 RATIONAL
FUNCTIONS 237
8.2 MULTIVALUEDNESS 240
8.2.1 MULTIVALUED FUNCTIONS 240
8.2.2 RIEMANN SURFACES 241
8.2.3 BRANCH POINT AND BRANCH CUT 243
8.3 ANALYTIC CONTINUATION 245
8.3.1 CONTINUATION BY TAYLOR SERIES 245
8.3.2 FUNCTION ELEMENTS 246
8.3.3 UNIQUENESS THEOREM 250
8.3.4 CONSERVATION OF FUNCTIONAL EQUATIONS 250
8.3.5 CONTINUATION AROUND A BRANCH POINT 252
8.3.6 NATURAL BOUNDARIES 252
8.3.7 TECHNIQUE OF ANALYTIC CONTINUATIONS 254
8.3.8 THE METHOD OF MOMENT 255
CONTOUR INTEGRALS 259
9.1 CALCULUS OF RESIDUES 259
9.1.1 RESIDUE THEOREM 259
9.1.2 REMARKS ON RESIDUES 261
9.1.3 WINDING NUMBER 262
9.1.4 RATIO METHOD 263
9.1.5 EVALUATING THE RESIDUES 264
9.2 APPLICATIONS TO REAL INTEGRALS 267
9.2.1 CLASSIFICATION OF EVALUABLE REAL INTEGRALS 267
IMAGE 6
XIV CONTENTS
9.2.2 TYPE 1: INTEGRALS OF /(COSO,SINO) 268
9.2.3 TYPE 2: INTEGRALS OF RATIONAL FUNCTION 268
9.2.4 TYPE 3: INTEGRALS OF F(X)E IX 270
9.2.5 TYPE 4: INTEGRALS OF F(X)/X A 271
9.2.6 TYPE 5: INTEGRALS OF F(X) LOG X 273
9.3 MORE APPLICATIONS OF RESIDUE CALCULUS 277
9.3.1 INTEGRALS ON RECTANGULAR CONTOURS 277
9.3.2 FRESNEL INTEGRALS 279
9.3.3 SUMMATION OF SERIES 281
9.3.4 LANGEVIN AND RIEMANN ZETA FUNCTIONS 283
9.4 ARGUMENT PRINCIPLE 285
9.4.1 THE PRINCIPLE 285
9.4.2 VARIATION OF THE ARGUMENT 288
9.4.3 EXTENTSON OF THE ARGUMENT PRINCIPLE 289
9.4.4 ROUCHE THEOREM 290
9.5 DISPERSION RELATIONS 293
9.5.1 PRINCIPAL VALUE INTEGRALS 293
9.5.2 SEVERAL REMARKS 295
9.5.3 DISPERSION RELATIONS 297
9.5.4 KRAMERS-KRONIG RELATIONS 298
9.5.5 SUBTRACTED DISPERSION RELATION 299
9.5.6 DERIVATION OF DISPERSION RELATIONS 300
10 CONFORMAI MAPPING 305
10.1 FUNDAMENTALS 305
10.1.1 CONFORMAI PROPERTY OF ANALYTIC FUNCTIONS 305 10.1.2 SCALE FACTOR
307
10.1.3 MAPPING OF A DIFFERENTIAL AREA 308
10.1.4 MAPPING OF A TANGENT LINE 309
10.1.5 THE POINT AT INFINITY 311
10.1.6 SINGULAR POINT AT INFINITY 312
10.2 ELEMENTARY TRANSFORMATIONS 315
10.2.1 LINEAR TRANSFORMATIONS 315
10.2.2 BILINEAR TRANSFORMATIONS 316
10.2.3 MISCELLANEOUS TRANSFORMATIONS 317
10.2.4 MAPPING OF FINITE-RADIUS CIRCLE 321
10.2.5 INVARIANCE OF THE CROSS RATIO 322
10.3 APPLICATIONS TO BOUNDARY-VALUE PROBLEMS 325
10.3.1 SCHWARZ-CHRISTOFFEL TRANSFORMATION 325
10.3.2 DERIVATION OF THE SCHWARTZ-CHRISTOFFEL TRANSFORMATION . 326
10.3.3 THE METHOD OF INVERSION 327
10.4 APPLICATIONS IN PHYSICS AND ENGINEERING 332
10.4.1 ELECTRIC POTENTIAL FIELD IN A COMPLICATED GEOMETRY. 332 10.4.2
JOUKOWSKY AIRFOIL 335
IMAGE 7
CONTENTS XV
PART IV FOURIER ANALYSIS
11 FOURIER SERIES 339
11.1 BASIC PROPERTIES 339
11.1.1 DEFINITION 339
11.1.2 DIRICHLET THEOREM 340
11.1.3 FOURIER SERIES OF PERIODIC FUNCTIONS 342
11.1.4 HALF-RANGE FOURIER SERIES 343
11.1.5 FOURIER SERIES OF NONPERIODIC FUNCTIONS 344 11.1.6 THE RATE OF
CONVERGENCE 346
11.1.7 FOURIER SERIES IN HIGHER DIMENSIONS 347
11.2 MEAN CONVERGENCE OF FOURIER SERIES 351
11.2.1 MEAN CONVERGENCE PROPERTY 351
11.2.2 DIRICHLET AND FEJER INTEGRALS 353
11.2.3 PROOF OF THE MEAN CONVERGENCE OF FOURIER SERIES 355 11.2.4
PARSEVAL IDENTITY 356
11.2.5 RIEMANN-LEBESGUE THEOREM 357
11.3 UNIFORM CONVERGENCE OF FOURIER SERIES 360
11.3.1 CRITERION FOR UNIFORM AND POINTWISE CONVERGENCE 360 11.3.2 FEJER
THEOREM 360
11.3.3 PROOF OF UNIFORM CONVERGENCE 361
11.3.4 POINTWISE CONVERGENCE AT DISCONTINUOUS POINTS 363 11.3.5 GIBBS
PHENOMENON 365
11.3.6 OVERSHOOT AT A DISCONTINUOUS POINT 366
11.4 APPLICATIONS IN PHYSICS AND ENGINEERING 371
11.4.1 TEMPERATURE VARIATION OF THE GROUND 371
11.4.2 STRING VIBRATION UNDER IMPACT 373
12 FOURIER TRANSFORMATION 377
12.1 FOURIER TRANSFORM 377
12.1.1 DERIVATION OF FOURIER TRANSFORM 377
12.1.2 FOURIER INTEGRAL THEOREM 379
12.1.3 PROOF OF THE FOURIER INTEGRAL THEOREM 380
12.1.4 INVERSE RELATIONS OF THE HALF-WIDTH 381
12.1.5 PARSEVAL IDENTITY FOR FOURIER TRANSFORMS 382 12.1.6 FOURIER
TRANSFORMS IN HIGHER DIMENSIONS 384 12.2 CONVOLUTION AND CORRELATIONS
387
12.2.1 CONVOLUTION THEOREM 387
12.2.2 CROSS-CORRELATION FUNCTIONS 388
12.2.3 AUTOCORRELATION FUNCTIONS 390
12.3 DISCRETE FOURIER TRANSFORM 391
12.3.1 DEFINITIONS 391
12.3.2 INVERSE TRANSFORM 392
12.3.3 NYQUEST FREQUENCY AND ALIASING 393
IMAGE 8
XVI CONTENTS
12.3.4 SAMPLING THEOREM 394
12.3.5 FAST FOURIER TRANSFORM 396
12.3.6 MATRIX REPRESENTATION OF FFT ALGORITHM 398 12.3.7 DECOMPOSITION
METHOD FOR FFT 400
12.4 APPLICATIONS IN PHYSICS AND ENGINEERING 401
12.4.1 FRAUNHOFER DIFFRACTION I 401
12.4.2 FRAUNHOFER DIFFRACTION II 403
12.4.3 AMPLITUDE MODULATION TECHNIQUE 404
13 LAPLACE TRANSFORMATION 407
13.1 BASIC OPERATIONS 407
13.1.1 DEFINITIONS 407
13.1.2 SEVERAL REMARKS 408
13.1.3 SIGNIFICANCE OF ANALYTIC CONTINUATION 409
13.1.4 CONVERGENCE OF LAPLACE INTEGRALS 410
13.1.5 ABSCISSA OF ABSOLUTE CONVERGENCE 411
13.1.6 LAPLACE TRANSFORMS OF ELEMENTARY FUNCTIONS 412 13.2 PROPERTIES
OF LAPLACE TRANSFORMS 415
13.2.1 FIRST SHIFTING THEOREM 415
13.2.2 SECOND SHIFTING THEOREM 416
13.2.3 LAPLACE TRANSFORM OF PERIODIC FUNCTIONS 417 13.2.4 LAPLACE
TRANSFORM OF DERIVATIVES AND INTEGRALS 418 13.2.5 LAPLACE TRANSFORMS
LEADING TO MULTIVALUED FUNCTIONS . 420 13.3 CONVERGENCE THEOREMS FOR
LAPLACE INTEGRALS 422
13.3.1 FUNCTIONS OF EXPONENTIAL ORDER 422
13.3.2 CONVERGENCE FOR EXPONENTIAL-ORDER CASES 424 13.3.3 UNIFORM
CONVERGENCE FOR EXPONENTIAL-ORDER CASES . . . 425 13.3.4 CONVERGENCE
FOR GENERAL CASES 427
13.3.5 UNIFORM CONVERGENCE FOR GENERAL CASES 429 13.3.6 DISTINCTION
BETWEEN EXPONENTIAL-ORDER CASES AND GENERAL CASES 431
13.3.7 ANALYTIC PROPERTY OF LAPLACE TRANSFORMS 432 13.4 INVERSE LAPLACE
TRANSFORM 432
13.4.1 THE TWO-SIDED LAPLACE TRANSFORM 432
13.4.2 INVERSE OF THE TWO-SIDED LAPLACE TRANSFORM 434 13.4.3 INVERSE OF
THE ONE-SIDED LAPLACE TRANSFORM 436 13.4.4 USEFUL FORMULA FOR INVERSE
LAPLACE TRANSFORMATION . . . 436 13.4.5 EVALUATING INVERSE
TRANSFORMATIONS 439
13.4.6 INVERSE TRANSFORM OF MULTIVALUED FUNCTIONS 441 13.5 APPLICATIONS
IN PHYSICS AND ENGINEERING 445
13.5.1 ELECTRIC CIRCUITS I 445
13.5.2 ELECTRIC CIRCUITS II 447
IMAGE 9
CONTENTS XVII
14 WAVELET TRANSFORMATION 449
14.1 CONTINUOUS WAVELET ANALYSES 449
14.1.1 DEFINITION OF WAVELET 449
14.1.2 THE WAVELET TRANSFORM 451
14.1.3 CORRELATION BETWEEN WAVELET AND SIGNAL 452 14.1.4 ACTUAL
APPLICATION OF THE WAVELET TRANSFORM 455 14.1.5 INVERSE WAVELET
TRANSFORM 456
14.1.6 NOISE REDUCTION TECHNIQUE 457
14.2 DISCRETE WAVELET ANALYSIS 460
14.2.1 DISCRETE WAVELET TRANSFORMS 460
14.2.2 COMPLETE ORTHONORMAL WAVELETS 462
14.2.3 MULTIRESOLUTION ANALYSIS 463
14.2.4 ORTHOGONAL DECOMPOSITION 464
14.2.5 CONSTRUCTING AN ORTHONORMAL BASIS 466
14.2.6 TWO-SCALE RELATIONS 467
14.2.7 CONSTRUCTING THE MOTHER WAVELET 469
14.2.8 MULTIRESOLUTION REPRESENTATION 471
14.3 FAST WAVELET TRANSFORMATION 476
14.3.1 GENERALIZED TWO-SCALE RELATIONS 476
14.3.2 DECOMPOSITION ALGORITHM 478
14.3.3 RECONSTRUCTION ALGORITHM 479
PART V DIFFERENTIAL EQUATIONS
15 ORDINARY DIFFERENTIAL EQUATIONS 483
15.1 CONCEPTS OF SOLUTIONS 483
15.1.1 DEFINITION OF ORDINARY DIFFERENTIAL EQUATIONS 483 15.1.2 EXPLICIT
SOLUTION 484
15.1.3 IMPLICIT SOLUTION 485
15.1.4 GENERAL AND PARTICULAR SOLUTIONS 486
15.1.5 SINGULAR SOLUTION 488
15.1.6 INTEGRAL CURVE AND DIRECTION FIELD 489
15.2 EXISTENCE THEOREM FOR THE FIRST-ORDER ODE 491
15.2.1 PICARD METHOD 491
15.2.2 PROPERTIES OF SUCCESSIVE APPROXIMATIONS 493 15.2.3 EXISTENCE
THEOREM AND LIPSCHITZ CONDITION 495 15.2A UNIQUENESS THEOREM 497
15.2.5 REMARKS ON THE TWO THEOREMS 498
15.3 STURM-LIOUVILLE PROBLEMS 500
15.3.1 STURM-LIOUVILLE EQUATION 500
15.3.2 CONVERSION INTO A STURM-LIOUVILLE EQUATION 501 15.3.3
SELF-ADJOINT OPERATORS 502
15.3.4 REQUIRED BOUNDARY CONDITION 503
15.3.5 REALITY OF EIGENVALUES 504
IMAGE 10
XVIII CONTENTS
16 SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS 509
16.1 SYSTEMS OF ODES 509
16.1.1 SYSTEMS OF THE FIRST-ORDER ODES 509
16.1.2 COLUMN-VECTOR NOTATION 510
16.1.3 REDUCING THE ORDER OF ODES 510
16.1.4 LIPSCHITZ CONDITION IN VECTOR SPACES 512
16.2 LINEAR SYSTEM OF ODES 513
16.2.1 BASIC TERMINOLOGY 513
16.2.2 VECTOR SPACE OF SOLUTIONS 514
16.2.3 FUNDAMENTAL SYSTEMS OF SOLUTIONS 516
16.2.4 WRONSKIAN FOR A SYSTEM OF ODES 517
16.2.5 LIOUVILLE FORMULA FOR A WRONSKIAN 518
16.2.6 WRONSKIAN FOR AN NTH-ORDER LINEAR ODE 519
16.2.7 PARTICULAR SOLUTION OF AN INHOMOGENEOUS SYSTEM 522 16.3
AUTONOMOUS SYSTEMS OF ODES 525
16.3.1 AUTONOMOUS SYSTEM 525
16.3.2 TRAJECTORY 526
16.3.3 CRITICAL POINT 527
16.3.4 STABILITY OF A CRITICAL POINT 527
16.3.5 LINEAR AUTONOMOUS SYSTEM 528
16.4 CLASSIFICATION OF CRITICAL POINTS 530
16.4.1 IMPROPER NODE 530
16.4.2 SADDLE POINT 531
16.4.3 PROPER NODE 532
16.4.4 SPIRAL POINT 533
16.4.5 CENTER 533
16.4.6 LIMIT CYCLE 534
16.5 APPLICATIONS IN PHYSICS AND ENGINEERING 536
16.5.1 VAN DER POL GENERATOR 536
17 PARTIAL DIFFERENTIAL EQUATIONS 539
17.1 BASIC PROPERTIES 539
17.1.1 DEFINITIONS 539
17.1.2 SUBSIDIARY CONDITIONS 540
17.1.3 LINEAR AND HOMOGENEOUS PDES 540
17.1.4 CHARACTERISTIC EQUATION 541
17.1.5 SECOND-ORDER PDES 543
17.1.6 CLASSIFICATION OF SECOND-ORDER PDES 544
17.2 THE LAPLACIAN OPERATOR 546
17.2.1 MAXIMUM AND MINIMUM THEOREM 546
17.2.2 UNIQUENESS THEOREM 548
17.2.3 SYMMETRIC PROPERTIES OF THE LAPLACIAN 548
17.3 THE DIFFUSION OPERATOR 550
17.3.1 THE DIFFUSION EQUATIONS IN BOUNDED DOMAINS 550 17.3.2 MAXIMUM AND
MINIMUM THEOREM 551
17.3.3 UNIQUENESS THEOREM 551
IMAGE 11
CONTENTS XIX
17.4 THE WAVE OPERATOR 552
17.4.1 THE CAUCHY PROBLEM 552
17.4.2 HOMOGENEOUS WAVE EQUATIONS 553
17.4.3 INHOMOGENEOUS WAVE EQUATIONS 555
17.4.4 WAVE EQUATIONS IN FINITE DOMAINS 556
17.5 APPLICATIONS IN PHYSICS AND ENGINEERING 559
17.5.1 WAVE EQUATIONS FOR VIBRATING STRINGS 559
17.5.2 DIFFUSION EQUATIONS FOR HEAT CONDUCTION 561
PART VI TENSOR ANALYSES
18 CARTESIAN TENSORS 565
18.1 ROTATION OF COORDINATE AXES 565
18.1.1 TENSORS AND COORDINATE TRANSFORMATIONS 565 18.1.2 SUMMATION
CONVENTION 566
18.1.3 CARTESIAN COORDINATE SYSTEM 567
18.1.4 ROTATION OF COORDINATE AXES 568
18.1.5 ORTHOGONAL RELATIONS 569
18.1.6 MATRIX REPRESENTATIONS 570
18.1.7 DETERMINANT OF A MATRIX 571
18.2 CARTESIAN TENSORS 576
18.2.1 CARTESIAN VECTORS 576
18.2.2 A VECTOR AND A GEOMETRIC ARROW 577
18.2.3 CARTESIAN TENSORS 578
18.2.4 SCALARS 579
18.3 PSEUDOTENSORS 580
18.3.1 IMPROPER ROTATIONS 580
18.3.2 PSEUDOVECTORS 582
18.3.3 PSEUDOTENSORS 584
18.3.4 LEVI-CIVITA SYMBOLS 584
18.4 TENSOR ALGEBRA 586
18.4.1 ADDITION AND SUBTRACTION 586
18.4.2 CONTRACTION 587
18.4.3 OUTER AND INNER PRODUCTS 587
18.4.4 SYMMETRIC AND ANTISYMMETRIC TENSORS 589 18.4.5 EQUIVALENCE OF AN
ANTISYMMETRIC SECOND-ORDER TENSOR TO A PSEUDOVECTOR 590
18.4.6 QUOTIENT THEOREM 592
18.4.7 QUOTIENT THEOREM FOR TWO-SUBSCRIPTED QUANTITIES 593 18.5
APPLICATIONS IN PHYSICS AND ENGINEERING 596
18.5.1 INERTIA TENSOR 596
18.5.2 TENSORS IN ELECTROMAGNETISM IN SOLIDS 598
18.5.3 ELECTROMAGNETIC FIELD TENSOR 598
18.5.4 ELASTIC TENSOR 600
IMAGE 12
XX CONTENTS
19 NON-CARTESIAN TENSORS 601
19.1 CURVILINEAR COORDINATE SYSTEMS 601
19.1.1 LOCAL BASIS VECTORS 601
19.1.2 RECIPROCITY RELATIONS 603
19.1.3 TRANSFORMATION LAW OF COVARIANT BASIS VECTORS 604 19.1.4
TRANSFORMATION LAW OF CONTRAVARIANT BASIS VECTORS . . . 606 19.1.5
COMPONENTS OF A VECTOR 606
19.1.6 COMPONENTS OF A TENSOR 608
19.1.7 MIXED COMPONENTS OF A TENSOR 609
19.1.8 KRONECKER DELTA 610
19.2 METRIC TENSOR 611
19.2.1 DEFINITION 611
19.2.2 GEOMETRIC ROLE OF METRIC TENSORS 612
19.2.3 RIEMANN SPACE AND METRIC TENSOR 613
19.2.4 ELEMENTS OF ARC, AREA, AND VOLUME 614
19.2.5 SCALE FACTORS 616
19.2.6 REPRESENTATION OF BASIS VECTORS IN DERIVATIVES 617 19.2.7 INDEX
LOWERING AND RAISING 617
19.3 CHRISTOFFEL SYMBOLS 621
19.3.1 DERIVATIVES OF BASIS VECTORS 621
19.3.2 NONTENSOR CHARACTER 622
19.3.3 PROPERTIES OF CHRISTOFFEL SYMBOLS 623
19.3.4 ALTERNATIVE EXPRESSION 623
19.4 COVARIANT DERIVATIVES 627
19.4.1 COVARIANT DERIVATIVES OF VECTORS 627
19.4.2 REMARKS ON COVARIANT DERIVATIVES 628
19.4.3 COVARIANT DERIVATIVES OF TENSORS 629
19.4.4 VECTOR OPERATORS IN TENSOR FORM 630
19.5 APPLICATIONS IN PHYSICS AND ENGINEERING 634
19.5.1 GENERAL RELATIVITY THEORY 634
19.5.2 RIEMANN TENSOR 635
19.5.3 ENERGY-MOMENTUM TENSOR 636
19.5.4 EINSTEIN FIELD EQUATION 637
20 TENSOR AS MAPPING 639
20.1 VECTOR AS A LINEAR FUNCTION 639
20.1.1 OVERVIEW 639
20.1.2 VECTOR SPACES REVISITED 640
20.1.3 VECTOR SPACES OF LINEAR FUNCTIONS 640
20.1.4 DUAL SPACES 641
20.1.5 EQUIVALENCE BETWEEN VECTORS AND LINEAR FUNCTIONS . . . 642 20.2
TENSOR AS MULTILINEAR FUNCTION 643
20.2.1 DIRECT PRODUCT OF VECTOR SPACES 643
20.2.2 MULTILINEAR FUNCTIONS 644
20.2.3 TENSOR PRODUCT 644
IMAGE 13
CONTENTS XXI
20.2.4 GENERAL DEFINITION OF TENSORS 645
20.3 COMPONENTS OF TENSORS 646
20.3.1 BASIS OF A TENSOR SPACE 646
20.3.2 TRANSFORMATION LAWS OF TENSORS 648
20.3.3 NATURAL ISOMORPHISM 648
20.3.4 INNER PRODUCT IN TENSOR LANGUAGE 651
20.3.5 INDEX LOWERING AND RAISING IN TENSOR LANGUAGE 652
PART VII APPENDIXES
A PROOF OF THE BOLZANO-WEIERSTRASS THEOREM 657
A.I LIMIT POINTS 657
A.2 CANTOR THEOREM 658
A.3 BOLZANO-WEIERSTRASS THEOREM 659
B DIRAC S FUNCTION 661
B.I BASIC PROPERTIES 661
B.2 REPRESENTATION AS A LIMIT OF FUNCTION 662
B.3 REMARKS ON REPRESENTATION 4 663
C PROOF OF WEIERSTRASS APPROXIMATION THEOREM 667
D TABULATED LIST OF ORTHONORMAL POLYNOMIAL FUNCTIONS 671
INDEX 677 |
| any_adam_object | 1 |
| author | Shima, Hiroyuki Nakayama, Tsuneyoshi 1945- |
| author_GND | (DE-588)141263954 (DE-588)124884199 |
| author_facet | Shima, Hiroyuki Nakayama, Tsuneyoshi 1945- |
| author_role | aut aut |
| author_sort | Shima, Hiroyuki |
| author_variant | h s hs t n tn |
| building | Verbundindex |
| bvnumber | BV036568536 |
| classification_rvk | SK 400 SK 950 |
| ctrlnum | (OCoLC)640128803 (DE-599)DNB993291163 |
| dewey-full | 515.02462 515.02453 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.02462 515.02453 |
| dewey-search | 515.02462 515.02453 |
| dewey-sort | 3515.02462 |
| dewey-tens | 510 - Mathematics |
| discipline | Maschinenbau / Maschinenwesen Physik Mathematik |
| format | Book |
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| genre_facet | Lehrbuch |
| id | DE-604.BV036568536 |
| illustrated | Illustrated |
| indexdate | 2024-07-20T10:42:03Z |
| institution | BVB |
| isbn | 9783540878636 9783642425912 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020489705 |
| oclc_num | 640128803 |
| open_access_boolean | |
| owner | DE-634 DE-83 DE-703 DE-573 DE-706 |
| owner_facet | DE-634 DE-83 DE-703 DE-573 DE-706 |
| physical | XXI, 688 S. graph. Darst. 24 cm |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Springer |
| record_format | marc |
| spelling | Shima, Hiroyuki Verfasser (DE-588)141263954 aut Higher mathematics for physics and engineering Hiroyuki Shima ; Tsuneyoshi Nakayama Berlin [u.a.] Springer 2010 XXI, 688 S. graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Analysis (DE-588)4001865-9 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Analysis (DE-588)4001865-9 s DE-604 Nakayama, Tsuneyoshi 1945- Verfasser (DE-588)124884199 aut Erscheint auch als Online-Ausgabe 978-3-540-87864-3 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3261705&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020489705&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Shima, Hiroyuki Nakayama, Tsuneyoshi 1945- Higher mathematics for physics and engineering Analysis (DE-588)4001865-9 gnd |
| subject_GND | (DE-588)4001865-9 (DE-588)4123623-3 |
| title | Higher mathematics for physics and engineering |
| title_auth | Higher mathematics for physics and engineering |
| title_exact_search | Higher mathematics for physics and engineering |
| title_full | Higher mathematics for physics and engineering Hiroyuki Shima ; Tsuneyoshi Nakayama |
| title_fullStr | Higher mathematics for physics and engineering Hiroyuki Shima ; Tsuneyoshi Nakayama |
| title_full_unstemmed | Higher mathematics for physics and engineering Hiroyuki Shima ; Tsuneyoshi Nakayama |
| title_short | Higher mathematics for physics and engineering |
| title_sort | higher mathematics for physics and engineering |
| topic | Analysis (DE-588)4001865-9 gnd |
| topic_facet | Analysis Lehrbuch |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3261705&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020489705&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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