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
22 - Cartesian tensors
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 physics and mathematics, coordinate transformations play an important role because many problems are much simpler when a suitable coordinate system is used. Furthermore, the requirement that physical laws do not change under certain transformations imposes constraints on the physical laws. An example of this is presented in Section 22.11 where it is shown that the fact that the pressure in a fluid is isotropic follows from the requirement that some physical laws may not change under a rotation of the coordinate system. In this chapter it is shown how the change of vectors and matrices under coordinate transformations is derived. The derived transformation properties can be generalized to other mathematical objects which are called tensors. In this chapter, only transformations of rectangular coordinate systems are considered. Since these coordinate systems are called Cartesian coordinate systems, the associated tensors are called Cartesian tensors. The transformation properties of tensors in Cartesian and curvilinear coordinate systems are described in detail by Butkov [24] and Riley et al. [87].
Coordinate transforms
In this section we consider the transformation of a coordinate system in two dimensions. In Figure 22.1 an old coordinate system with coordinates xold and yold is shown. The unit vectors along the old coordinate axis are denoted by êx,old and êy,old. In a coordinate transformation, these old unit vectors are transformed to new unit vectors êx,new and êy,new, respectively.
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- A Guided Tour of Mathematical MethodsFor the Physical Sciences, pp. 379 - 411Publisher: Cambridge University PressPrint publication year: 2004