Editing Frame of reference (section)

== Coordinate systems ==
{{main|Coordinate systems}}
{{see also|Generalized coordinates|Axes conventions}}
[[File:Reference frame and observer.svg|thumb|250px|right| An observer O, situated at the origin of a local set of coordinates – a frame of reference '''F'''. The observer in this frame uses the coordinates (''x, y, z, t'') to describe a spacetime event, shown as a star.]]
Although the term "coordinate system" is often used (particularly by physicists) in a nontechnical sense, the term "coordinate system" does have a precise meaning in mathematics, and sometimes that is what the physicist means as well.

A coordinate system in mathematics is a facet of [[geometry]] or of [[algebra]],<ref name=Barker>{{cite book |title=Continuous symmetry: from Euclid to Klein |author1=William Barker |author2=Roger Howe |page=18 ff |url=https://books.google.com/books?id=NIxExnr2EjYC&q=geometry++axiom+%22coordinate+system%22&pg=PA17
|isbn=978-0-8218-3900-3 |year=2008 |publisher=American Mathematical Society }}</ref><ref name=Ramsay>{{cite book |title=Introduction to Hyperbolic Geometry |author1=Arlan Ramsay |author2=Robert D. Richtmyer |page=[https://archive.org/details/introductiontohy0000rams/page/11 11] |url=https://archive.org/details/introductiontohy0000rams
|url-access=registration |quote=geometry  axiom coordinate system. |isbn=0-387-94339-0 |publisher=Springer |year=1995}}</ref> in particular, a property of [[manifold]]s (for example, in physics, [[Configuration space (physics)|configuration space]]s or [[phase space]]s).<ref name=Hawking>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." {{cite book |title=The Large Scale Structure of Space-Time |author1=Stephen W. Hawking |author2=George Francis Rayner Ellis |isbn=0-521-09906-4 |year=1973 |publisher=Cambridge University Press |page=11 |url=https://books.google.com/books?id=QagG_KI7Ll8C&q=manifold+%22The+Large+Scale+Structure+of+Space-Time%22&pg=PA59
}} 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.''</ref><ref name=Morita>{{cite book |title=Geometry of Differential Forms |author1=Shigeyuki Morita |author2=Teruko Nagase |author3=Katsumi Nomizu |page=[https://archive.org/details/geometryofdiffer00mori/page/12 12] |url=https://archive.org/details/geometryofdiffer00mori
|url-access=registration |quote=geometry  axiom coordinate system. |isbn=0-8218-1045-6 |year=2001 |publisher=American Mathematical Society Bookstore}}</ref> The [[Cartesian coordinate system|coordinates]] of a point '''r''' in an ''n''-dimensional space are simply an ordered set of ''n'' numbers:<ref name=Korn>{{cite book |title=Mathematical handbook for scientists and engineers : definitions, theorems, and formulas for reference and review |author1=Granino Arthur Korn |author2=Theresa M. Korn |author2-link= Theresa M. Korn |page=169 |url=https://books.google.com/books?id=xHNd5zCXt-EC&q=curvilinear+%22coordinate+system%22&pg=PA169
|isbn=0-486-41147-8 |year=2000 |publisher=Courier Dover Publications}}</ref><ref name=encarta>See [http://encarta.msn.com/encyclopedia_761579532/Coordinate_System_(mathematics).html Encarta definition]. [https://web.archive.org/web/20091030054251/http://encarta.msn.com/encyclopedia_761579532/Coordinate_System_(mathematics).html Archived] 2009-10-31.</ref> 

: <math>\mathbf{r} = [x^1,\ x^2,\ \dots,\ x^n].</math>

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 [[Robotics|robot design]], they could be angles of relative rotations, linear displacements, or deformations of [[linkage (mechanical)|joints]].<ref name=Yamane>{{cite book |author=Katsu Yamane |title=Simulating and Generating Motions of Human Figures |isbn=3-540-20317-6 |year=2004 |publisher=Springer |pages=12–13 |url=https://books.google.com/books?id=tNrMiIx3fToC&q=generalized+coordinates+%22kinematic+chain%22&pg=PA12}}</ref> Here we will suppose these coordinates can be related to a [[Cartesian coordinate]] system by a set of functions:

: <math>x^j = x^j (x,\ y,\ z,\ \dots),\quad j = 1,\ \dots,\ n,</math>

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

: <math> x^j (x, y, z, \dots) = \mathrm{constant},\quad j = 1,\ \dots,\ n.</math>

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''' {'''e'''<sub>1</sub>, '''e'''<sub>2</sub>, ..., '''e'''<sub>n</sub>} at that point. That is:<ref name=Papapetrou>{{cite book |title=Lectures on General Relativity |author=Achilleus Papapetrou |page=5 |url=https://books.google.com/books?id=SWeOggyp1ZsC&q=relativistic++%22general+coordinates%22&pg=PA3
|isbn=90-277-0540-2 |year=1974 |publisher=Springer}}</ref>

: <math>\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},\quad i = 1,\ \dots,\ n,</math>

which can be normalized to be of unit length. For more detail see [[Curvilinear coordinates#Covariant basis|curvilinear coordinates]].

Coordinate surfaces, coordinate lines, and [[Basis (linear algebra)|basis vectors]] are components of a '''coordinate system'''.<ref name=Zdunkowski>{{cite book |title=Dynamics of the Atmosphere |page=84 |isbn=0-521-00666-X |year=2003 |author1=Wilford Zdunkowski |author2=Andreas Bott |publisher=Cambridge University Press |url=https://books.google.com/books?id=GuYvC21v3g8C&q=%22curvilinear+coordinate+system%22&pg=RA1-PA84}}</ref> If the basis vectors are orthogonal at every point, the coordinate system is an [[Orthogonal coordinates|orthogonal coordinate system]].

An important aspect of a coordinate system is its [[metric tensor]] ''g<sub>ik</sub>'', which determines the [[arc length]] ''ds'' in the coordinate system in terms of its coordinates:<ref name=Borisenko>{{cite book |title=Vector and Tensor Analysis with Applications |author1=A. I. Borisenko |author2=I. E. Tarapov |author3=Richard A. Silverman |page=86 |url=https://books.google.com/books?id=CRIjIx2ac6AC&q=coordinate+metric&pg=PA86|isbn=0-486-63833-2 |publisher=Courier Dover Publications |year=1979}}</ref>

: <math>(ds)^2 = g_{ik}\ dx^i\ dx^k,</math>

where repeated indices are summed over.

As is apparent from these remarks, a coordinate system is a [[Model theory|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 include time as a coordinate, and can be used to describe motion. Thus, [[Lorentz transformation]]s and [[Galilean transformation]]s may be viewed as [[Coordinate system#Transformations|coordinate transformation]]s.