User:John R. Brews/Draft

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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:

$\mathbf {r} =[x^{1},\ x^{2},\ \dots \ ,x^{n}]\ .$

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

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). The coordinates of a point r in an n-dimensional space are simply an ordered set of n numbers:

$\mathbf {r} =[x^{1},\ x^{2},\ \dots \ ,x^{n}]\ .$

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. Here we will suppose these coordinates can be related to a Cartesian coordinate system by a set of functions:

$x^{j}=x^{j}(x,\ y,\ z,\ \dots )\ ,$    $j=1,\ \dots \ ,\ n\$

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

$x^{j}(x,y,z,\dots )=\mathrm {constant} \ ,$    $j=1,\ \dots \ ,\ n\ .$

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, …, e_n} at that point. That is:

$\mathbf {e} _{i}(\mathbf {r} )=\lim _{\epsilon \rightarrow 0}{\frac {\mathbf {r} \left(x^{1},\ \dots ,\ x^{i}+\epsilon ,\ \dots ,\ x^{n}\right)-\mathbf {r} \left(x^{1},\ \dots ,\ x^{i},\ \dots ,\ x^{n}\right)}{\epsilon }}\ ,$

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. 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 g_ik, which determines the arc length ds in the coordinate system in terms of its coordinates:

$(ds)^{2}=g_{ik}\ dx^{i}\ dx^{k}\ ,$

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

Choquet-Bruhat Bishop O'Neill Warner