Citizendium - a community developing a quality comprehensive compendium of knowledge, online and free.
Click here to join and contribute—free
CZ thanks our previous donors. Donate here. Treasurer's Financial Report -- Thanks to our content contributors. --

User:John R. Brews/Draft

From Citizendium
Jump to: navigation, search

Coordinate system

The coordinates of a point r in an n-dimensional real numerical space ℝn or a complex n-space ℂn are simply an ordered set of n real or complex numbers:[1][2][3]

Coordinate surfaces, coordinate lines, and basis vectors are components of a coordinate system.[4]

Manifolds

A coordinate system in mathematics is a facet of geometry or of algebra, in particular, a property of manifolds (for example, in physics, configuration spaces or phase spaces).[5][2] The coordinates of a point r in an n-dimensional space are simply an ordered set of n numbers:[1]

In a general Banach space, these numbers could be (for example) coefficients in a functional expansion like a Fourier series. In a physical problem, they could be spacetime coordinates or normal mode amplitudes. In a robot design, they could be angles of relative rotations, linear displacements, or deformations of joints.[6] Here we will suppose these coordinates can be related to a Cartesian coordinate system by a set of functions:

   

where x, y, z, etc. are the n Cartesian coordinates of the point. Given these functions, coordinate surfaces are defined by the relations:

   

The intersection of these surfaces define coordinate lines. At any selected point, tangents to the intersecting coordinate lines at that point define a set of basis vectors {e1, e2, …, en} at that point. That is:

which can be normalized to be of unit length. For more detail see curvilinear coordinates.

Coordinate surfaces, coordinate lines, and basis vectors are components of a coordinate system.[4] If the basis vectors are orthogonal at every point, the coordinate system is an orthogonal coordinate system.

An important aspect of a coordinate system is its metric gik, which determines the arc length ds in the coordinate system in terms of its coordinates:[7]

where repeated indices are summed over.

As is apparent from these remarks, a coordinate system is a mathematical construct, part of an axiomatic system. There is no necessary connection between coordinate systems and physical motion (or any other aspect of reality). However, coordinate systems can be used to describe motion by interpreting one coordinate as time. Thus, Lorentz transformations and Galilean transformations may be viewed as coordinate transformations.


Notes

  1. 1.0 1.1 Granino Arthur Korn, Theresa M. Korn (2000). Mathematical handbook for scientists and engineers : definitions, theorems, and formulas for reference and review. Courier Dover Publications, p. 169. ISBN 0486411478. 
  2. 2.0 2.1 Morita
  3. Fritzche
  4. 4.0 4.1 Wilford Zdunkowski & Andreas Bott (2003). Dynamics of the Atmosphere. Cambridge University Press. ISBN 052100666X. 
  5. According to Hawking and Ellis: "A manifold is a space locally similar to Euclidean space in that it can be covered by coordinate patches. This structure allows differentiation to be defined, but does not distinguish between different coordinate systems. Thus, the only concepts defined by the manifold structure are those that are independent of the choice of a coordinate system." Stephen W. Hawking & George Francis Rayner Ellis (1973). The Large Scale Structure of Space-Time. Cambridge University Press, p. 11. ISBN 0521099064.  A mathematical definition is: A connected Hausdorff space M is called an n-dimensional manifold if each point of M is contained in an open set that is homeomorphic to an open set in Euclidean n-dimensional space.
  6. Katsu Yamane (2004). Simulating and Generating Motions of Human Figures. Springer, 12–13. ISBN 3540203176. 
  7. A. I. Borisenko, I. E. Tarapov, Richard A. Silverman (1979). “§2.8.4 Arc length. Metric coefficients”, Vector and Tensor Analysis with Applications, Reprint of Prentice-Hall 1968 ed. Courier Dover Publications, pp. 86 ff. ISBN 0486638332. 

Choquet-Bruhat Bishop O'Neill Warner