Mathematical methods in engineering:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
Cambridge Univ. Press
2015
|
| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XV, 621 S. graph. Darst. |
| ISBN: | 9781107037045 1107037042 |
Staff View
MARC
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| 100 | 1 | |a Powers, Joseph |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Mathematical methods in engineering |c Joseph Powers ; Mihir Sen |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a New York |b Cambridge Univ. Press |c 2015 | |
| 300 | |a XV, 621 S. |b graph. Darst. | ||
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Record in the Search Index
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| adam_text | Contents
Preface page xiii
1 Multivariable Calculus..................................................1
1.1 Implicit Functions 1
1.1.1 One Independent Variable 1
1.1.2 Many Independent Variables 5
1.1.3 Many Dependent Variables 6
1.2 Inverse Function Theorem 11
1.3 Functional Dependence 13
1.4 Leibniz Rule 18
1.5 Optimization 20
1.5.1 Unconstrained Optimization 21
1.5.2 Calculus of Variations 22
1.5.3 Constrained Optimization: Lagrange Multipliers 28
1.6 Non-Cartesian Coordinate Transformations 31
1.6.1 Jacobian Matrices and Metric Tensors 34
1.6.2 Covariance and Contravariance 43
1.6.3 Differentiation and Christoffel Symbols 49
1.6.4 Summary of Identities 53
1.6.5 Nonorthogonal Coordinates: Alternate Approach 54
1.6.6 Orthogonal Curvilinear Coordinates 57
Exercises 59
2 Vectors and Tensors in Cartesian Coordinates..........................64
2.1 Preliminaries 64
2.1.1 Cartesian Index Notation 64
2.1.2 Direction Cosines 67
2.1.3 Scalars 72
2.1.4 Vectors 72
2.1.5 Tensors 73
2.2 Algebra of Vectors 81
2.2.1 Definitions and Properties 81
2.2.2 Scalar Product 82
2.2.3 Cross Product 82
vii
VIII
Contents
2.2.4 Scalar Triple Product 83
2.2.5 Identities 83
2.3 Calculus of Vectors 84
2.3.1 Vector Functions 84
2.3.2 Differential Geometry of Curves 84
2.4 Line Integrals 92
2.5 Surface Integrals 94
2.6 Differential Operators 94
2.6.1 Gradient 95
2.6.2 Divergence 98
2.6.3 Curl 98
2.6.4 Laplacian 99
2.6.5 Identities 99
2.7 Curvature Re visitě d 100
2.7.1 Trajectory 100
2.7.2 Principal 103
2.7.3 Gaussian 103
2.7.4 Mean 104
2.8 Special Theorems 104
2.8.1 Green s Theorem 104
2.8.2 Divergence Theorem 105
2.8.3 Green s Identities 108
2.8.4 Stokes’ Theorem 108
2,8.5 Extended Leibniz Rule 110
Exercises 110
First-Order Ordinary Differential Equations 115
3.1 Paradigm Problem 115
3.2 Separation of Variables 117
3.3 Homogeneous Equations 118
3.4 Exact Equations 120
3.5 Integrating Factors 122
3.6 General Linear Solution 123
3.7 В ernoulli Equation 125
3.8 Riccati Equation 126
3.9 Reduction of Order 129
3.9.1 Dependent Variable у Absent 129
3.9.2 Independent Variable x Absent 129
3.10 Factorable Equations 131
3.11 Uniqueness and Singular Solutions 132
3.12 Clairaut Equation 134
3.13 Picard Iteration 136
3.14 Solution by Taylor Series 139
3.15 Delay Differential Equations 140
Exercises 141
Linear Ordinary Differential Equations 146
4.1 Linearity and Linear Independence 146
4.2 Complementary Functions 149
Contents
IX
4.2.1 Constant Coefficients 149
4.2.2 Variable Coefficients 154
4.3 Particular Solutions 156
4.3.1 Undetermined Coefficients 156
4.3.2 Variation of Parameters 158
4.3.3 Green’s Functions 160
4.3.4 Operator D 166
4.4 Sturm-Liouville Analysis 169
4.4.1 General Formulation 170
4.4.2 Adjoint of Differential Operators 171
4.4.3 Linear Oscillator 175
4.4.4 Legendre Differential Equation 179
4.4.5 Chebyshev Equation 182
4.4.6 Hermite Equation 185
4.4.7 Laguerre Equation 188
4.4.8 Bessel Differential Equation 189
4.5 Fourier Series Representation 193
4.6 Fredholm Alternative 200
4.7 Discrete and Continuous Spectra 201
4.8 Resonance 202
4.9 Linear Difference Equations 207
Exercises 211
5 Approximation Methods 219
5.1 Function Approximation 220
5.1.1 Taylor Series 220
5.1.2 Pade Approximants 222
5.2 Power Series 224
5.2.1 Functional Equations 224
5.2.2 First-Order Differential Equations 226
5.2.3 Second-Order Differential Equations 230
5.2.4 Higher-Order Differential Equations 237
5.3 Taylor Series Solution 238
5.4 Perturbation Methods 240
5.4.1 Polynomial and Transcendental Equations 240
5.4.2 Regular Perturbations 244
5.4.3 Strained Coordinates 247
5.4.4 Multiple Scales 253
5.4.5 Boundary Layers 256
5.4.6 Interior Layers 261
5.4.7 WKBJ Method 263
5.4.8 Solutions of the Type es^ 266
5.4.9 Repeated Substitution 267
5.5 Asymptotic Methods for Integrals 268
Exercises 271
6 Linear Analysis 279
6.1 Sets 279
6.2 Integration 280
X
Contents
6.3 Vector Spaces 283
6.3.1 Normed 288
6.3.2 Inner Product 297
6.4 Gram-Schmidt Procedure 304
6.5 Projection of Vectors onto New Bases 307
6.5.1 Nonorthogonal 307
6.5.2 Orthogonal 313
6.5.3 Orthonormal 314
6.5.4 Reciprocal 324
6.6 Parseval’s Equation, Convergence, and Completeness 330
6.7 Operators 330
6.7.1 Linear 332
6.7.2 Adjoint 334
6.7.3 Inverse 337
6.8 Eigenvalues and Eigenvectors 339
6.9 Rayleigh Quotient 350
6.10 Linear Equations 354
6.11 Method of Weighted Residuals 359
6.12 Ritz and Rayleigh-Ritz Methods 371
6.13 Uncertainty Quantification Via Polynomial Chaos 373
Exercises 379
7 Linear Algebra......................................................390
7.1 Paradigm Problem 390
7.2 Matrix Fundamentals and Operations 391
7.2.1 Determinant and Rank 391
7.2.2 Matrix Addition 392
7.2.3 Column, Row, and Left and Right Null Spaces 392
7.2.4 Matrix Multiplication 394
7.2.5 Definitions and Properties 396
7.3 Systems of Equations 399
7.3.1 Overconstrained 400
7.3.2 Underconstrained 403
7.3.3 Simultaneously Over- and Underconstrained 405
7.3.4 Square 406
7.3.5 Fredholm Alternative 408
7.4 Eigenvalues and Eigenvectors 410
7.4.1 Ordinary 410
7.4.2 Generalized in the Second Sense 414
7.5 Matrices as Linear Mappings 415
7.6 Complex Matrices 416
7.7 Orthogonal and Unitary Matrices 419
7.7.1 Givens Rotation 422
7.7.2 Householder Reflection 423
7.8 Discrete Fourier Transforms 426
7.9 Matrix Decompositions 432
7.9.1 L D U 432
7.9.2 Cholesky 434
7.9.3 Row Echelon Form 435
Contents
xi
7.9.4 Q U 439
7.9.5 Diagonalization 441
7.9.6 Jordan Canonical Form 447
7.9.7 Schur 449
7.9.8 Singular Value 450
7.9.9 Polar 453
7.9.10 Hessenberg 456
7.10 Projection Matrix 456
7.11 Least Squares 458
7.11.1 Unweighted 459
7.11.2 Weighted 460
7.12 Neumann Series 461
7.13 Matrix Exponential 462
7.14 Quadratic Form 464
7.15 Moore-Penrose Pseudoinverse 467
Exercises 470
8 Linear Integral Equations 480
8.1 Definitions 480
8.2 Homogeneous Fredholm Equations 481
8.2.1 First Kind 481
8.2.2 Second Kind 482
8.3 Inhomogeneous Fredholm Equations 487
8.3.1 First Kind 487
8.3.2 Second Kind 489
8.4 Fredholm Alternative 490
8.5 Fourier Series Projection 490
Exercises 495
9 Dynamical Systems 497
9.1 Iterated Maps 497
9.2 Fractals 501
9.2.1 Cantor Set 501
9.2.2 Koch Curve 502
9.2.3 Menger Sponge 502
9.2.4 Weierstrass Function 503
9.2.5 Mandelbrot and Julia Sets 503
9.3 Introduction to Differential Systems 503
9.3.1 Autonomous Example 504
9.3.2 Nonautonomous Example 508
9.3.3 General Approach 510
9.4 High-Order Scalar Differential Equations 512
9.5 Linear Systems 514
9.5.1 Inhomogeneous with Variable Coefficients 514
9.5.2 Homogeneous with Constant Coefficients 515
9.5.3 Inhomogeneous with Constant Coefficients 525
9.6 Nonlinear Systems 528
9.6.1 Definitions 529
9.6.2 Linear Stability 532
Contents
9.6.3 Heteroclinic and Homoclinic Trajectories 533
9.6.4 Nonlinear Forced Mass-Spring-Damper 539
9.6.5 Lyapunov Functions 541
9.6.6 Hamiltonian Systems 543
9.7 Differential-Algebraic Systems 545
9.7.1 Linear Homogeneous 545
9.7.2 Nonlinear 548
9.8 Fixed Points at Infinity 549
9.8.1 Poincare Sphere 549
9.8.2 Projective Space 553
9.9 Bifurcations 554
9.9.1 Pitchfork 555
9.9.2 Transcritical 556
9.9.3 Saddle-Node 557
9.9.4 Hopf 558
9.10 Projection of Partial Differential Equations 559
9.11 Lorenz Equations 562
9.11.1 Linear Stability 563
9.11.2 Nonlinear Stability: Center Manifold Projection 565
9.11.3 Transition to Chaos 569
Exercises 573
Appendix A............................................................585
A.l Roots of Polynomial Equations 585
A. 1.1 First-Order 585
A. 1.2 Quadratic 585
A.1.3 Cubic 586
A. 1.4 Quartic 587
A.1.5 Quintic and Higher 589
A.2 Cramer’s Rule 589
A.3 Gaussian Elimination 590
A.4 Trapezoidal Rule 591
A.5 Trigonometric Relations 591
A.6 Hyperbolic Functions 593
A.7 Special Functions 593
A.7.1 Gamma 593
A.7.2 Error 594
A.7.3 Sine, Cosine, and Exponential Integral 594
A.7.4 Hypergeometric 595
A.7.5 Airy 596
A.7.6 Dirac 6 and Heaviside 596
A.8 Complex Numbers 598
A.S.l Euler’s Formula 598
A.8.2 Polar and Cartesian Representations 599
Exercises 600
Bibliography
Index
603
609
|
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| indexdate | 2025-02-20T06:38:34Z |
| institution | BVB |
| isbn | 9781107037045 1107037042 |
| language | English |
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| physical | XV, 621 S. graph. Darst. |
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| publisher | Cambridge Univ. Press |
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| spellingShingle | Powers, Joseph Sen, Mihir Mathematical methods in engineering Ingenieurwissenschaften (DE-588)4137304-2 gnd Technische Mathematik (DE-588)4827059-3 gnd Mathematik (DE-588)4037944-9 gnd |
| subject_GND | (DE-588)4137304-2 (DE-588)4827059-3 (DE-588)4037944-9 (DE-588)4123623-3 |
| title | Mathematical methods in engineering |
| title_auth | Mathematical methods in engineering |
| title_exact_search | Mathematical methods in engineering |
| title_full | Mathematical methods in engineering Joseph Powers ; Mihir Sen |
| title_fullStr | Mathematical methods in engineering Joseph Powers ; Mihir Sen |
| title_full_unstemmed | Mathematical methods in engineering Joseph Powers ; Mihir Sen |
| title_short | Mathematical methods in engineering |
| title_sort | mathematical methods in engineering |
| topic | Ingenieurwissenschaften (DE-588)4137304-2 gnd Technische Mathematik (DE-588)4827059-3 gnd Mathematik (DE-588)4037944-9 gnd |
| topic_facet | Ingenieurwissenschaften Technische Mathematik Mathematik Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027766361&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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Indexes
Sonderstandort Fakultät
| Call Number: |
2000 SK 950 P888 |
|---|---|
| Copy 1 | nicht ausleihbar Checked out – Due: 31.12.2099 Recall This |