为了更好地借鉴国外数学教育与研究的成功经验，促使我国数学教育与研究事业的发展，提高高等学校数学教育教学质量，“天元基金影印数学丛书”由数学天元基金赞助，经国内知名数学专家选择、推荐国外反映近代数学发展的优秀数学类教材，在国内影印出版。 本书是“天元基金影印数学丛书”之一，是作者总结自己与工程界及科学界相互配合的研究工作中遇到的问题及解决问题的方法而写成的，偏重于宏观范围的物理世界的问题。本书分成三部分： 一、建模：讲述了建模的一些原则(包括物理定律、本构关系及守恒定律),量纲分析(包括Buckingham的Pi-定理)等，并介绍了一些有趣的例子。 二、分析技巧：讲述了偏微分方程和广义函数的基础知识，然后又介绍了一些生动有趣的例子。 三、渐近分析：该部分大致占全书的一半，讲述了渐近展开的基本概念,正则摄动展开,边界层和多重度规方法等，然后又介绍了不少有趣的例子。 本书可作为高等学校本科高年级和研究生课程教材，对于具有微积分、向量分析、质点动力学初步、无粘流体力学和偏微分方程初步知识的读者,也是一本很好的建模理论的教材。

Preface
PartI Modelling techniques
  1 The basics of modelling
   1.1 Introduction
   1.2 What do we mean by a model?
   1.3 Principles of modelling: physical laws and constitutive relations
   1.4 Conservation laws
   1.5 General remarks
   1.6 Exercises
  2 Units, dimensions and dimensional analysis
   2.1 Introduction
   2.2 Units and dimensions
   2.3 Electric fields and electrostatics
   2.4 Sources and further reading
   2.5 Exercises
  3 Nondimensionalisation
   3.1 Nondimensionalisation and dimensionless parameters
   3.2 The Navier-Stokes equations and Reynolds numbers
   3.3 Buckingham's Pi-theorem
   3.4 Sources and further reading
   3.5 Exercises
  4 Case studies: hair modelling and cable laying
   4.1 The Euler-Bernoulli model for a beam
   4.2 Hair modelling
   4.3 Undersea cable laying
   4.4 Modelling and analysis
   4.5 Sources and further reading
   4.6 Exercises
  5 Case study: the thermistor （1）
   5.1 Heat and current flow in thermistors
   5.2 Nondimensionalisation
   5.3 A thermistor in a circuit
   5.4 Sources and further reading
   5.5 Exercises
  6 Case study: electrostatic painting
   6.1 Electrostatic painting
   6.2 Field equations
   6.3 Boundary conditions
   6.4 Nondimensionalisation
   6.5 Sources and further reading
   6.6 Exercises
Part Ⅱ Analytical techniques
  7 Partial differential equations
   7.1 First-order quasilinear partial differential equations: theory
   7.2 Example: Poisson processes
   7.3 Shocks
   7.4 Fully nonlinear equations: Charpit's method
   7.5 Second-order linear equations in two variables
   7.6 Further reading
   7.7 Exercises
  8 Case study: traffic modelling
   8.1 Simple models for traffic flow
   8.2 Traffic jams and other discontinuous solutions
   8.3 More sophisticated models
   8.4 Sources and further reading
   8.5 Exercises
  9 The delta function and other distributions
   9.1 Introduction
   9.2 A point force on a stretched string; impulses
   9.3 Informal definition of the delta and Heaviside functions
   9.4 Examples
   9.5 Balancing singularities
   9.6 Green's functions
   9.7 Sources and further reading
   9.8 Exercises
  10 Theory of distributions
   10.1 Test functions
   10.2 The action of a test function
   10.3 Definition of a distribution
   10.4 Further properties of distributions
   10.5 The derivative of a distribution
   10.6 Extensions of the theory of distributions
   10.7 Sources and further reading
   10.8 Exercises
  11 Case study: the pantograph
   11.1 What is a pantograph?
   11.2 The model
   11.3 Impulsive attachment for an undamped pantograph
   11.4 Solution near a support
   11.5 Solution for a whole span
   11.6 Sources and further reading
   11.7 Exercises
Part Ⅲ Asymptotic techniques
  12 Asymptotic expansions
   12.1 Introduction
   12.2 Order notation
   12.3 Convergence and divergence
   12.4 Further reading
   12.5 Exercises
  13 Regular perturbation expansions
   13.1 Introduction
   13.2 Example: stability of a spacecraft in orbit
   13.3 Linear stability
   13.4 Example: the pendulum
   13.5 Small perturbations of a boundary
   13.6 Caveat expandator
   13.7 Exercises
  14 Case study: electrostatic painting （2）
   14.1 Small parameters in the electropaint model
   14.2 Exercises
  15 Case study: piano tuning
   15.1 The notes of a piano: the tonal system of Western music
   15.2 Tuning an ideal piano
   15.3 A real piano
   15.4 Sources and further reading
   15.5 Exercises
  16 Boundary layers
   16.1 Introduction
   16.2 Functions with boundary layers; matching
   16.3 Examples from ordinary differential equations
   16.4 Case study: cable laying
   16.5 Examples for partial differential equations
   16.6 Exercises
  17 Case study: the thermistor（2）
   17.1 Strongly temperature-dependent conductivity
   17.2 Exercises
  18 ‘Lubrication theory’analysis in long thin domains
   18.1 ‘Lubrication theory’approximations: slender geometries
   18.2 Heat flow in a bar of variable cross-section
   18.3 Heat flow in a long thin domain with cooling
   18.4 Advection-diffusion in a long thin domain
   18.5 Exercises
  19 Case study: continuous casting of steel
   19.1 Continuous casting of steel
   19.2 Exercises
  20 Lubrication theory for fluids
   20.1 Thin fluid layers: classical lubrication theory
   20.2 Thin viscous fluid sheets on solid substrates
   20.3 Thin fluid sheets and fibres
   20.4 Further reading
   20.5 Exercises
  21 Case study: turning of eggs during incubation
   21.1 Incubating eggs
   21.2 Modelling
   21.3 Exercises
  22 Multiple scales and other methods for nonlinear oscillators
   22.1 The Poincare-Linstedt method
   22.2 The method of multiple scales
   22.3 Relaxation oscillations
   22.4 Exercises
  23 Ray theory and the WKB method
   23.1 Introduction
   23.2 Classical WKB theory
   23.3 Geometric optics and ray theory: why do we say light travels in straight lines?
   23.4 Kelvin's ship waves
   23.5 Exercises
References
Index