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The Princeton Companion to Applied Mathematics

Book Description

This is the most authoritative and accessible single-volume reference book on applied mathematics. Featuring numerous entries by leading experts and organized thematically, it introduces readers to applied mathematics and its uses; explains key concepts; describes important equations, laws, and functions; looks at exciting areas of research; covers modeling and simulation; explores areas of application; and more.

Modeled on the popular Princeton Companion to Mathematics, this volume is an indispensable resource for undergraduate and graduate students, researchers, and practitioners in other disciplines seeking a user-friendly reference book on applied mathematics.

  • Features nearly 200 entries organized thematically and written by an international team of distinguished contributors
  • Presents the major ideas and branches of applied mathematics in a clear and accessible way
  • Explains important mathematical concepts, methods, equations, and applications
  • Introduces the language of applied mathematics and the goals of applied mathematical research
  • Gives a wide range of examples of mathematical modeling
  • Covers continuum mechanics, dynamical systems, numerical analysis, discrete and combinatorial mathematics, mathematical physics, and much more
  • Explores the connections between applied mathematics and other disciplines
  • Includes suggestions for further reading, cross-references, and a comprehensive index

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Preface
  6. Contributors (1/2)
  7. Contributors (2/2)
  8. Part I Introduction to Applied Mathematics
    1. I.1 What Is Applied Mathematics?
    2. I.2 The Language of Applied Mathematics (1/4)
    3. I.2 The Language of Applied Mathematics (2/4)
    4. I.2 The Language of Applied Mathematics (3/4)
    5. I.2 The Language of Applied Mathematics (4/4)
    6. I.3 Methods of Solution (1/3)
    7. I.3 Methods of Solution (2/3)
    8. I.3 Methods of Solution (3/3)
    9. I.4 Algorithms (1/2)
    10. I.4 Algorithms (2/2)
    11. I.5 Goals of Applied Mathematical Research (1/2)
    12. I.5 Goals of Applied Mathematical Research (2/2)
    13. I.6 The History of Applied Mathematics (1/6)
    14. I.6 The History of Applied Mathematics (2/6)
    15. I.6 The History of Applied Mathematics (3/6)
    16. I.6 The History of Applied Mathematics (4/6)
    17. I.6 The History of Applied Mathematics (5/6)
    18. I.6 The History of Applied Mathematics (6/6)
  9. Part II Concepts
    1. II.1 Asymptotics
    2. II.2 Boundary Layer
    3. II.3 Chaos and Ergodicity
    4. II.4 Complex Systems
    5. II.5 Conformal Mapping
    6. II.6 Conservation Laws
    7. II.7 Control
    8. II.8 Convexity
    9. II.9 Dimensional Analysis and Scaling
    10. II.10 The Fast Fourier Transform
    11. II.11 Finite Differences
    12. II.12 The Finite-Element Method
    13. II.13 Floating-Point Arithmetic
    14. II.14 Functions of Matrices
    15. II.15 Function Spaces
    16. II.16 Graph Theory
    17. II.17 Homogenization
    18. II.18 Hybrid Systems
    19. II.19 Integral Transforms and Convolution
    20. II.20 Interval Analysis
    21. II.21 Invariants and Conservation Laws (1/2)
    22. II.21 Invariants and Conservation Laws (2/2)
    23. II.22 The Jordan Canonical Form
    24. II.23 Krylov Subspaces
    25. II.24 The Level Set Method
    26. II.25 Markov Chains
    27. II.26 Model Reduction
    28. II.27 Multiscale Modeling
    29. II.28 Nonlinear Equations and Newton’s Method
    30. II.29 Orthogonal Polynomials
    31. II.30 Shocks
    32. II.31 Singularities
    33. II.32 The Singular Value Decomposition
    34. II.33 Tensors and Manifolds
    35. II.34 Uncertainty Quantification
    36. II.35 Variational Principle
    37. II.36 Wave Phenomena
  10. Part III Equations, Laws, and Functions of Applied Mathematics
    1. III.1 Benford’s Law
    2. III.2 Bessel Functions
    3. III.3 The Black–Scholes Equation
    4. III.4 The Burgers Equation
    5. III.5 The Cahn–Hilliard Equation
    6. III.6 The Cauchy–Riemann Equations
    7. III.7 The Delta Function and Generalized Functions
    8. III.8 The Diffusion Equation
    9. III.9 The Dirac Equation
    10. III.10 Einstein’s Field Equations
    11. III.11 The Euler Equations
    12. III.12 The Euler–Lagrange Equations
    13. III.13 The Gamma Function
    14. III.14 The Ginzburg–Landau Equation
    15. III.15 Hooke’s Law
    16. III.16 The Korteweg–de Vries Equation
    17. III.17 The Lambert W Function
    18. III.18 Laplace’s Equation
    19. III.19 The Logistic Equation
    20. III.20 The Lorenz Equations
    21. III.21 Mathieu Functions
    22. III.22 Maxwell’s Equations
    23. III.23 The Navier–Stokes Equations
    24. III.24 The Painlevé Equations
    25. III.25 The Riccati Equation
    26. III.26 Schrödinger’s Equation
    27. III.27 The Shallow-Water Equations
    28. III.28 The Sylvester and Lyapunov Equations
    29. III.29 The Thin-Film Equation
    30. III.30 The Tricomi Equation
    31. III.31 The Wave Equation
  11. Part IV Areas of Applied Mathematics
    1. IV.1 Complex Analysis
    2. IV.2 Ordinary Differential Equations (1/2)
    3. IV.2 Ordinary Differential Equations (2/2)
    4. IV.3 Partial Differential Equations (1/2)
    5. IV.3 Partial Differential Equations (2/2)
    6. IV.4 Integral Equations (1/2)
    7. IV.4 Integral Equations (2/2)
    8. IV.5 Perturbation Theory and Asymptotics (1/2)
    9. IV.5 Perturbation Theory and Asymptotics (2/2)
    10. IV.6 Calculus of Variations (1/2)
    11. IV.6 Calculus of Variations (2/2)
    12. IV.7 Special Functions (1/2)
    13. IV.7 Special Functions (2/2)
    14. IV.8 Spectral Theory (1/3)
    15. IV.8 Spectral Theory (2/3)
    16. IV.8 Spectral Theory (3/3)
    17. IV.9 Approximation Theory (1/3)
    18. IV.9 Approximation Theory (2/3)
    19. IV.9 Approximation Theory (3/3)
    20. IV.10 Numerical Linear Algebra and Matrix Analysis (1/4)
    21. IV.10 Numerical Linear Algebra and Matrix Analysis (2/4)
    22. IV.10 Numerical Linear Algebra and Matrix Analysis (3/4)
    23. IV.10 Numerical Linear Algebra and Matrix Analysis (4/4)
    24. IV.11 Continuous Optimization (Nonlinear and Linear Programming) (1/3)
    25. IV.11 Continuous Optimization (Nonlinear and Linear Programming) (2/3)
    26. IV.11 Continuous Optimization (Nonlinear and Linear Programming) (3/3)
    27. IV.12 Numerical Solution of Ordinary Differential Equations (1/3)
    28. IV.12 Numerical Solution of Ordinary Differential Equations (2/3)
    29. IV.12 Numerical Solution of Ordinary Differential Equations (3/3)
    30. IV.13 Numerical Solution of Partial Differential Equations (1/3)
    31. IV.13 Numerical Solution of Partial Differential Equations (2/3)
    32. IV.13 Numerical Solution of Partial Differential Equations (3/3)
    33. IV.14 Applications of Stochastic Analysis (1/2)
    34. IV.14 Applications of Stochastic Analysis (2/2)
    35. IV.15 Inverse Problems (1/2)
    36. IV.15 Inverse Problems (2/2)
    37. IV.16 Computational Science (1/3)
    38. IV.16 Computational Science (2/3)
    39. IV.16 Computational Science (3/3)
    40. IV.17 Data Mining and Analysis (1/2)
    41. IV.17 Data Mining and Analysis (2/2)
    42. IV.18 Network Analysis (1/6)
    43. IV.18 Network Analysis (2/6)
    44. IV.18 Network Analysis (3/6)
    45. IV.18 Network Analysis (4/6)
    46. IV.18 Network Analysis (5/6)
    47. IV.18 Network Analysis (6/6)
    48. IV.19 Classical Mechanics (1/2)
    49. IV.19 Classical Mechanics (2/2)
    50. IV.20 Dynamical Systems (1/2)
    51. IV.20 Dynamical Systems (2/2)
    52. IV.21 Bifurcation Theory (1/2)
    53. IV.21 Bifurcation Theory (2/2)
    54. IV.22 Symmetry in Applied Mathematics (1/2)
    55. IV.22 Symmetry in Applied Mathematics (2/2)
    56. IV.23 Quantum Mechanics (1/2)
    57. IV.23 Quantum Mechanics (2/2)
    58. IV.24 Random-Matrix Theory (1/2)
    59. IV.24 Random-Matrix Theory (2/2)
    60. IV.25 Kinetic Theory (1/4)
    61. IV.25 Kinetic Theory (2/4)
    62. IV.25 Kinetic Theory (3/4)
    63. IV.25 Kinetic Theory (4/4)
    64. IV.26 Continuum Mechanics (1/3)
    65. IV.26 Continuum Mechanics (2/3)
    66. IV.26 Continuum Mechanics (3/3)
    67. IV.27 Pattern Formation (1/2)
    68. IV.27 Pattern Formation (2/2)
    69. IV.28 Fluid Dynamics (1/2)
    70. IV.28 Fluid Dynamics (2/2)
    71. IV.29 Magnetohydrodynamics (1/2)
    72. IV.29 Magnetohydrodynamics (2/2)
    73. IV.30 Earth System Dynamics (1/3)
    74. IV.30 Earth System Dynamics (2/3)
    75. IV.30 Earth System Dynamics (3/3)
    76. IV.31 Effective Medium Theories
    77. IV.32 Mechanics of Solids (1/3)
    78. IV.32 Mechanics of Solids (2/3)
    79. IV.32 Mechanics of Solids (3/3)
    80. IV.33 Soft Matter (1/2)
    81. IV.33 Soft Matter (2/2)
    82. IV.34 Control Theory (1/2)
    83. IV.34 Control Theory (2/2)
    84. IV.35 Signal Processing (1/3)
    85. IV.35 Signal Processing (2/3)
    86. IV.35 Signal Processing (3/3)
    87. IV.36 Information Theory (1/2)
    88. IV.36 Information Theory (2/2)
    89. IV.37 Applied Combinatorics and Graph Theory (1/3)
    90. IV.37 Applied Combinatorics and Graph Theory (2/3)
    91. IV.37 Applied Combinatorics and Graph Theory (3/3)
    92. IV.38 Combinatorial Optimization (1/2)
    93. IV.38 Combinatorial Optimization (2/2)
    94. IV.39 Algebraic Geometry (1/2)
    95. IV.39 Algebraic Geometry (2/2)
    96. IV.40 General Relativity and Cosmology (1/3)
    97. IV.40 General Relativity and Cosmology (2/3)
    98. IV.40 General Relativity and Cosmology (3/3)
  12. Part V Modeling
    1. V.1 The Mathematics of Adaptation (Or the Ten Avatars of Vishnu)
    2. V.2 Sport (1/2)
    3. V.2 Sport (2/2)
    4. V.3 Inerters
    5. V.4 Mathematical Biomechanics (1/2)
    6. V.4 Mathematical Biomechanics (2/2)
    7. V.5 Mathematical Physiology (1/2)
    8. V.5 Mathematical Physiology (2/2)
    9. V.6 Cardiac Modeling
    10. V.7 Chemical Reactions (1/2)
    11. V.7 Chemical Reactions (2/2)
    12. V.8 Divergent Series: Taming the Tails (1/2)
    13. V.8 Divergent Series: Taming the Tails (2/2)
    14. V.9 Financial Mathematics (1/2)
    15. V.9 Financial Mathematics (2/2)
    16. V.10 Portfolio Theory (1/2)
    17. V.10 Portfolio Theory (2/2)
    18. V.11 Bayesian Inference in Applied Mathematics
    19. V.12 A Symmetric Framework with Many Applications
    20. V.13 Granular Flows (1/2)
    21. V.13 Granular Flows (2/2)
    22. V.14 Modern Optics (1/2)
    23. V.14 Modern Optics (2/2)
    24. V.15 Numerical Relativity (1/2)
    25. V.15 Numerical Relativity (2/2)
    26. V.16 The Spread of Infectious Diseases (1/2)
    27. V.16 The Spread of Infectious Diseases (2/2)
    28. V.17 The Mathematics of Sea Ice (1/3)
    29. V.17 The Mathematics of Sea Ice (2/3)
    30. V.17 The Mathematics of Sea Ice (3/3)
    31. V.18 Numerical Weather Prediction (1/2)
    32. V.18 Numerical Weather Prediction (2/2)
    33. V.19 Tsunami Modeling (1/2)
    34. V.19 Tsunami Modeling (2/2)
    35. V.20 Shock Waves
    36. V.21 Turbulence (1/2)
    37. V.21 Turbulence (2/2)
  13. Part VI Example Problems
    1. VI.1 Cloaking
    2. VI.2 Bubbles
    3. VI.3 Foams
    4. VI.4 Inverted Pendulums
    5. VI.5 Insect Flight
    6. VI.6 The Flight of a Golf Ball
    7. VI.7 Automatic Differentiation
    8. VI.8 Knotting and Linking of Macromolecules
    9. VI.9 Ranking Web Pages
    10. VI.10 Searching a Graph
    11. VI.11 Evaluating Elementary Functions
    12. VI.12 Random Number Generation
    13. VI.13 Optimal Sensor Location in the Control of Energy-Efficient Buildings
    14. VI.14 Robotics
    15. VI.15 Slipping, Sliding, Rattling, and Impact: Nonsmooth Dynamics and Its Applications
    16. VI.16 From the N-Body Problem to Astronomy and Dark Matter
    17. VI.17 The N-Body Problem and the Fast Multipole Method
    18. VI.18 The Traveling Salesman Problem
  14. Part VII Application Areas
    1. VII.1 Aircraft Noise
    2. VII.2 A Hybrid Symbolic–Numeric Approach to Geometry Processing and Modeling
    3. VII.3 Computer-Aided Proofs via Interval Analysis
    4. VII.4 Applications of Max-Plus Algebra
    5. VII.5 Evolving Social Networks, Attitudes, and Beliefs—and Counterterrorism
    6. VII.6 Chip Design
    7. VII.7 Color Spaces and Digital Imaging
    8. VII.8 Mathematical Image Processing
    9. VII.9 Medical Imaging (1/2)
    10. VII.9 Medical Imaging (2/2)
    11. VII.10 Compressed Sensing
    12. VII.11 Programming Languages: An Applied Mathematics View (1/3)
    13. VII.11 Programming Languages: An Applied Mathematics View (2/3)
    14. VII.11 Programming Languages: An Applied Mathematics View (3/3)
    15. VII.12 High-Performance Computing
    16. VII.13 Visualization
    17. VII.14 Electronic Structure Calculations (Solid State Physics)
    18. VII.15 Flame Propagation
    19. VII.16 Imaging the Earth Using Green’s Theorem
    20. VII.17 Radar Imaging
    21. VII.18 Modeling a Pregnancy Testing Kit
    22. VII.19 Airport Baggage Screening with X-Ray Tomography
    23. VII.20 Mathematical Economics
    24. VII.21 Mathematical Neuroscience (1/2)
    25. VII.21 Mathematical Neuroscience (2/2)
    26. VII.22 Systems Biology
    27. VII.23 Communication Networks
    28. VII.24 Text Mining
    29. VII.25 Voting Systems (1/2)
    30. VII.25 Voting Systems (2/2)
  15. Part VIII Final Perspectives
    1. VIII.1 Mathematical Writing
    2. VIII.2 How to Read and Understand a Paper
    3. VIII.3 How to Write a General Interest Mathematics Book (1/2)
    4. VIII.3 How to Write a General Interest Mathematics Book (2/2)
    5. VIII.4 Workflow
    6. VIII.5 Reproducible Research in the Mathematical Sciences (1/2)
    7. VIII.5 Reproducible Research in the Mathematical Sciences (2/2)
    8. VIII.6 Experimental Applied Mathematics (1/2)
    9. VIII.6 Experimental Applied Mathematics (2/2)
    10. VIII.7 Teaching Applied Mathematics (1/2)
    11. VIII.7 Teaching Applied Mathematics (2/2)
    12. VIII.8 Mediated Mathematics: Representations of Mathematics in Popular Culture and Why These Matter (1/2)
    13. VIII.8 Mediated Mathematics: Representations of Mathematics in Popular Culture and Why These Matter (2/2)
    14. VIII.9 Mathematics and Policy (1/2)
    15. VIII.9 Mathematics and Policy (2/2)
  16. Index (1/7)
  17. Index (2/7)
  18. Index (3/7)
  19. Index (4/7)
  20. Index (5/7)
  21. Index (6/7)
  22. Index (7/7)