The Princeton companion to applied mathematics

Introduction to applied mathematics : What is applied mathematics? ; The language of applied mathematics ; Methods of solution ; Algorithms ; Goals of applied mathematical research ; The history of applied mathematics

Concepts : Asymptotics ; Boundary layer ; Chaos and ergodicity ; Complex systems ; Conformal mapping ; Conservation laws ; Control ; Convexity ; Dimensional analysis and scaling ; The fast Fourier transform ; Finite differences ; The finite-element method ; Floating-point arithmetic ; Functions of matrices ; Function spaces ; Graph theory ; Homogenization ; Hybrid systems ; Integral transforms and convolution ; Interval analysis ; Invariants and conservation laws ; The Jordan canonical form ; Krylov subspaces ; The level set method ; Markov chains ; Model reduction ; Multiscale modeling ; Nonlinear equations and Newton’s method ; Orthogonal polynomials ; Shocks ; Singularities ; The singular value decomposition ; Tensors and manifolds ; Uncertainty and quantification ; Variational principle ; Wave phenomena

Equations, laws, and functions of applied mathematics : Benford’s law ; Bessel functions ; The black-scholes equation ; The burgers equation ; The Cahn-Hilliard equation ; The Cauchy-Riemann equations ; The delta function and generalized functions ; The diffusion equation ; The Dirac equation ; Einstein’s field equations ; The Euler equations ; The Euler-Lagrange equations ; The gamma function ; The Ginzburg-Landau equation ; Hooke’s law ; The Korteweg-de Vries equation ; The Lambert W function ; Laplace’s equation ; The logistic equation ; The Lorenz equations ; Mathieu functions ;Maxwell’s equations ; The Navier-Stokes equations ; The Painleve equations ; The Riccati equation ; Schrodinger’s equation ; The shallow-water equations ; The Sylvester and Lyapunov equations ; The thin-film equation ; The tricomi equation ; The wave equation

Areas of applied mathematics : Complex analysis ; Ordinary differential equations ; Integral equations ; Perturbation theory and asymptotics ; Calculus of variations ; Special functions ; Spectral theory ; Approximation theory ; Numerical linear algebra and matrix analysis ; Continuous optimization (nonlinear and linear programming) ; Numerical solution of ordinary differential equations ; Numerical solution of partial differential equations ; Applications of stochastic analysis ; Inverse problems ; Computational science ; Data mining and analysis ; Network analysis ; Classical mechanics ; Dynamical systems ; Bifurcation theory ; Symmetry in applied mathematics ; Quantum mechanics ; Random-matrix theory ; Kinetic theory ; Continuum mechanics ; Pattern formation ; Fluid dynamics ; Magnetohydrodynamics ; Earth system dynamics ; Effective medium theories ; Mechanics of solids ; Soft matter ; Control theory ; Signal processing ; Information theory ; Applied combinatorics and graph theory ; Combinatorial optimization ; Algebraic geometry ; General relativity and cosmology

Modeling : The mathematics of adaptation (or the ten avatars of Vishnu) ; Sport ; Inerters ; Mathematical biomechanics ; Mathematical physiology ; Cardiac modeling ; Chemical reactions ; Divergent series: taming the tails ; Financial mathematics ; Portfolio theory ; Bayesian inference in applied mathematics ; A symmetric framework with many applications ; Granular flows ; Modern optics ; Numerical relativity ; The spread of infectious diseases ; The mathematics of sea ice ; Numerical weather prediction ; Tsunami modeling ; Shock waves ; Turbulence

Example problems : Cloaking ; Bubbles ; Foams ; Inverted pendulums ; Insect flight ; The flight of a golf ball ; Automatic differentiation ; Knotting and linking of macromolecules ; Ranking web pages ; Searching a graph ; Evaluating elementary functions ; Random number generator ; Optimal sensor location in the control of energy-efficient buildings ; Robotics ; Slipping, sliding, rattling, and impact: nonsmooth dynamics and its applications ; From the N-body problem to astronomy and dark matter ; The N-body problem and the fast multipole method ; The traveling salesman problem

Application areas : Aircraft noise ; A hybrid symbolic-numeric approach to geometry processing and modeling ; Computer-aided proofs via interval analysis ; Applications of max-plus algebra ; Evolving social networks, attitudes, and beliefs - and counterterrorism ; Chip design ; Color spaces and digital imaging ; Mathematical image processing ; Medical imaging ; Compressed sensing ; Programming languages: an applied mathematics view ; High-performance computing ; Visualization ; Electronic structure calculations (solid state physics) ; Flame propagation ; Imaging the earth using Green’s theorem ; Radar imaging ; Modeling a pregnancy testing kit ; Airport baggage screening with X-ray tomography ; Mathematical economics ; Mathematical neuroscience ; Systems biology ; Communication networks ; Text mining ; Voting systems

Final perspectives : Mathematical writing ; How to read and understand a paper ; How to write a general interest mathematics book ; Workflow ; Reproducible research in the mathematical sciences ; Experimental applied mathematics ; Teaching applied mathematics ; Mediated mathematics representations of mathematics in popular culture and why these matter ; Mathematics and policy