Book contents
- Frontmatter
- Contents
- Preface to Second Edition
- Acknowledgements
- 1 Introduction
- 2 Dimensional analysis
- 3 Power series
- 4 Spherical and cylindrical coordinates
- 5 Gradient
- 6 Divergence of a vector field
- 7 Curl of a vector field
- 8 Theorem of Gauss
- 9 Theorem of Stokes
- 10 Laplacian
- 11 Conservation laws
- 12 Scale analysis
- 13 Linear algebra
- 14 Dirac delta function
- 15 Fourier analysis
- 16 Analytic functions
- 17 Complex integration
- 18 Green's functions: principles
- 19 Green's functions: examples
- 20 Normal modes
- 21 Potential theory
- 22 Cartesian tensors
- 23 Perturbation theory
- 24 Asymptotic evaluation of integrals
- 25 Variational calculus
- 26 Epilogue, on power and knowledge
- References
- Index
12 - Scale analysis
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface to Second Edition
- Acknowledgements
- 1 Introduction
- 2 Dimensional analysis
- 3 Power series
- 4 Spherical and cylindrical coordinates
- 5 Gradient
- 6 Divergence of a vector field
- 7 Curl of a vector field
- 8 Theorem of Gauss
- 9 Theorem of Stokes
- 10 Laplacian
- 11 Conservation laws
- 12 Scale analysis
- 13 Linear algebra
- 14 Dirac delta function
- 15 Fourier analysis
- 16 Analytic functions
- 17 Complex integration
- 18 Green's functions: principles
- 19 Green's functions: examples
- 20 Normal modes
- 21 Potential theory
- 22 Cartesian tensors
- 23 Perturbation theory
- 24 Asymptotic evaluation of integrals
- 25 Variational calculus
- 26 Epilogue, on power and knowledge
- References
- Index
Summary
In most situations, the equations that we would like to solve in mathematical physics are too complicated to solve analytically. One of the reasons for this is often that an equation contains many different terms which make the problem simply too complex to be manageable. However, many of these terms may in practice be very small. Ignoring these small terms can simplify the problem to such an extent that it can be solved in closed form. Moreover, by deleting terms that are small one is able to focus on the terms that are significant and that contain the relevant physics. In this sense, ignoring small terms can actually give a better physical insight into the processes that really do matter.
Scale analysis is a technique in which one estimates the different terms in an equation by considering the scale over which the relevant parameters vary. This is an extremely powerful tool for simplifying problems. A comprehensive overview of this technique with many applications is given by Kline and in Chapter 6 of Lin et al. Interesting examples of the application of scaling arguments to biology are given by Vogel.
With the application of scale analysis one caveat must be made. One of the major surprises of classical physics of the twentieth century was the discovery of chaos in dynamical systems. In a chaotic system small changes in the initial conditions lead to a change in the time evolution of the system that grows exponentially with time.
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- A Guided Tour of Mathematical MethodsFor the Physical Sciences, pp. 153 - 172Publisher: Cambridge University PressPrint publication year: 2004