实用数学（影印版）

书籍目录

PrefacePart Ⅰ  Modelling techniques1  The basics of modelling1.1  Introduction1.2  What do we mean by a model?1.3  Principles of modelling: physical laws and constitutive relations1.4  Conservation laws1.5  General remarks1.6  Exercises2  Units, dimensions and dimensional analysis2.1  Introduction2.2  Units and dimensions2.3  Electric fields and electrostatics2.4  Sources and further reading2.5  Exercises3  Nondimensionalisation3.1  Nondimensionalisation and dimensionless parameters3.2  The Navier-Stokes equations and Reynolds numbers3.3  Buckingham's Pi-theorem3.4  Sources and further reading3.5  Exercises4  Case studies: hair modelling and cable laying4.1  The Euler-Bernoulli model for a beam4.2  Hair modelling4.3  Undersea cable laying4.4  Modelling and analysis4.5  Sources and further reading4.6  Exercise5  Case study: the thermistor (1)5.1  Heat and current flow in thermistors5.2  Nondimensionalisation5.3  A thermistor in a circuit5.4  Sources and further reading5.5  Exercises6  Case study: electrostatic painting6.1  Electrostatic painting6.2  Field equations6.3  Boundary conditions6.4  Nondimensionalisation6.5  Sources and further reading6.6  ExercisesPart Ⅱ  Analytical techniques7  Partial differential equations7.1  First-order quasilinear partial differential equations: theory7.2  Example: Poisson processes7.3  Shocks7.4  Fully nonlinear equations: Charpit's method7.5  Second-order linear equations in two variables7.6  Further reading7.7  Exercises8  Case study: traffic modelling8.1  Simple models for traffic flow8.2  Traffic jams and other discontinuous solutions8.3  More sophisticated models8.4  Sources and further reading8.5  Exercises9  The delta function and other distributions9.1  Introduction9.2  A point force on a stretched string; impulses9.3  Informal definition of the delta and Heaviside functions9.4  Examples9.5  Balancing singularities9.6  Green's functions9.7  Sources and further reading9.8  Exercises10  Theory of distributions10.1 Test functions10.2  The action of a test function10.3  Definition of a distribution10.4  Further properties of distributions10.5  The derivative of a distribution10.6  Extensions of the theory of distributions10.7  Sources and further reading10.8  Exercises11  Case study: the pantograph11.1  What is a pantograph?11.2  The model11.3  Impulsive attachment for an undamped pantograph11.4  Solution near a support11.5  Solution for a whole span11.6  Sources and further reading11.7  ExercisesPart Ⅲ  Asymptotic techniques12  Asymptotic expansions12.1  Introduction12.2  Order notation12.3  Convergence and divergence12.4  Further reading12.5  Exercises13  Regular perturbation expansions13.1  Introduction13.2  Example: stability of a spacecraft in orbit13.3  Linear stability13.4  Example: the pendulum13.5  Small perturbations of a boundary13.6  Caveat expandator13.7  Exercises14  Case study: electrostatic painting (2)14.1  Small parameters in the electropaint model14.2  Exercises15  Case study: piano tuning15.1  The notes of a piano: the tonal system of Western music15.2  Tuning an ideal piano15.3  A real piano15.4  Sources and further reading15.5  Exercises16  Boundary layers16.1  Introduction16.2  Functions with boundary layers; matching16.3  Examples from ordinary differential equations16.4  Case study: cable laying16.5  Examples for partial differential equations16.6  Exercises17  Case study: the thermistor (2)17.1  Strongly temperature-dependent conductivity17.2  Exercises18  Lubrication theory' analysis in long thin domains18.1  Lubrication theory' approximations: slender geometries18.2  Heat flow in a bar of variable cross-section18.3  Heat flow in a long thin domain with cooling18.4  Advection-diffusion in a long thin domain18.5  Exercises19  Case study: continuous casting of steel19.1  Continuous casting of steel19.2  Exercises20  Lubrication theory for fluids20.1  Thin fluid layers: classical lubrication theory20.2  Thin viscous fluid sheets on solid substrates20.3  Thin fluid sheets and fibres20.4  Further reading20.5  Exercises21  Case study: turning of eggs during incubation21.1  Incubating eggs21.2  Modelling21.3  Exercises22  Multiple scales and other methods for nonlinear oscillators22.1  The Poincare-Linstedt method22.2  The method of multiple scales22.3  Relaxation oscillations22.4  Exercises23  Ray theory and the WKB method23.1  Introduction23.2  Classical WKB theory23.3  Geometric optics and ray theory: why do we say light travels in straight lines?23.4  Kelvin's ship waves23.5  ExercisesReferencesIndex

作者简介

本书内容分为三部分: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;分析技巧，讲述了偏微分方程和广义函数的基础知识;渐近分析，讲述了渐近展开的基本概念，正则摄动展开，边界层和多重尺度法等。

目录

Preface

Part Ⅰ 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 Exercise

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

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