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.

Table of Contents

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