목차
A: An Overview of the Interaction of Mathematics and Natural Science. Chapter 1: What is Applied Mathematics
On the nature of applied mathematics
Introduction to the analysis of galactic structure
Aggregation of slime mold amebae
Chapter 2: Deterministic Systems and Ordinary Differential Equations
Planetary orbits
Elements of perturbation theory, including Poincare's method for periodic orbits
A system of ordinary differential equations
Chapter 3: Random Processes and ial Differential Equations
Random walk in one dimension
Langevin's equation
Asymptotic series, Laplace's method, gamma function, Stirling's formula
A difference equation and its limit
Further considerations pertinent to the relationship between probability and ial differential equations
Chapter 4: Superposition, Heat Flow, and Fourier Analysis
Conduction of heat
Fourier's theorem
On the nature of Fourier series
Chapter 5: Further Developments in Fourier Analysis
Other aspects of heat conduction
Sturn Liouville systems
Brief introduction to Fourier transform
Generalized harmonic analysis
B: Some Fundamental Procedures Illustrated on Ordinary Differential Equations. Chapter 6: Simplification, Dimensional Analysis, and Scaling
The basic simplification procedure
Dimensional analysis
Scaling
Chapter 7: Regular Perturbation Theory
The series method applied to the simple pendulum
Projectile problem solved by perturbation theory
Chapter 8: Illustration of Techniques on a Physiological Flow Problem
Physical formulation and dimensional analysis of a model for """"standing gradient"" osmotically driven flow
A mathematical model and its dimensional analysis
Obtaining the final scaled dimensionless form of the mathematical model
Solution and interpretation
Chapter 9: Introduction to Singular Perturbation Theory
Roots of polynomial equations
Boundary value problems for ordinary differential equations
Chapter 10: Singular Perturbation Theory Applied to a Problem in Biochemical Kinetics
Formulation of an initial value problem for a one enzyme one substrate chemical reaction
Approximate solution by singular perturbation methods
Chapter 11: Three Techniques Applied to the Simple Pendulum
Stability of normal and inverted equilibrium of the pendulum
A multiple scale expansion
The phase plane
C: Introduction to Theories of Continuous Fields. Chapter 12: Longitudinal Motion of a Bar
Derivation of the governing equations
One dimensional elastic wave propagation
Discontinuous solutions
Work, energy, and vibrations
Chapter 13: The Continuous Medium
The continuum model
Kinematics of deformable media
the material derivative
The Jacobian and its material derivative
Chapter 14: Field Equations of Continuum Mechanics
Conservation of mass
Balance of linear momentum
Balance of angular momentum
Energy and entropy
On constitutive equations, covariance
and the continuum model
Chapter 15: Inviscid Fluid Flow
Stress in motionless and inviscid fluids
Stability of a stratified fluid
Compression waves in gases
Uniform flow past a circular cylinder
Chapter 16: Potential Theory
Equations of Laplace and Poisson
Green's functions
Diffraction of acoustic waves by a hole.