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
4 - Spherical and cylindrical coordinates
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
Many problems in mathematical physics exhibit a spherical or cylindrical symmetry. For example, the gravity field of the Earth is to first order spherically symmetric. Waves excited by a stone thrown into water are usually cylindrically symmetric. Although there is no reason why problems with such a symmetry cannot be analyzed using Cartesian coordinates (i.e. (x, y, z)-coordinates), it is usually not very convenient to use such a coordinate system. The reason for this is that the theory is usually much simpler when one selects a coordinate system with symmetry properties that are the same as the symmetry properties of the physical system that one wants to study. It is for this reason that spherical coordinates and cylindrical coordinates are introduced in this section. It takes a certain effort to become acquainted with these coordinate systems, but this effort is well spent because it makes solving a large class of problems much easier.
Introducing spherical coordinates
In Figure 4.1 a Cartesian coordinate system with its x-, y-, and z-axes is shown as well as the location of a point r. This point can be described either by its x-, y-, and z-components or by the radius r and the angles θ and ϕ shown in Figure 4.1. In the latter case one uses spherical coordinates. Comparing the angles θ and ϕ with the geographical coordinates that define a point on the globe one sees that ϕ can be compared with longitude and θ can be compared with colatitude, which is defined as (latitude – 90 degrees).
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- A Guided Tour of Mathematical MethodsFor the Physical Sciences, pp. 31 - 45Publisher: Cambridge University PressPrint publication year: 2004