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==Coordinate system== | |||
The 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 |author=Granino Arthur Korn, Theresa M. Korn |pages=p. 169 |url=http://books.google.com/books?id=xHNd5zCXt-EC&pg=PA169&dq=curvilinear+%22coordinate+system%22&lr=&as_brr=0&sig=ACfU3U3psSqwpBtA3U40e46VPPaMNMEw4g#PPA169,M1 | |||
|isbn=0486411478 |year=2000 |publisher=Courier Dover Publications}} | |||
</ref> | |||
:<math>\mathbf{r} =[x^1,\ x^2,\ \dots\ , x^n] \ .</math> | |||
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=052100666X |year=2003 |author=Wilford Zdunkowski & Andreas Bott |publisher=Cambridge University Press |url=http://books.google.com/books?id=GuYvC21v3g8C&pg=RA1-PA84&dq=%22curvilinear+coordinate+system%22&lr=&as_brr=0&sig=ACfU3U2g2k7kY5u-CVcJ1pH5ZxsbEb9Rig }}</ref> | |||
==Manifolds== | |||
A coordinate system in mathematics is a facet of [[geometry]] or of [[algebra]], in particular, a property of [[Manifold (geometry)|manifold]]s (for example, in 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 |author=Stephen W. Hawking & George Francis Rayner Ellis |isbn=0521099064 |year=1973 |publisher=Cambridge University Press |pages=p. 11 |url=http://books.google.com/books?id=QagG_KI7Ll8C&pg=PA59&dq=manifold+%22The+Large+Scale+Structure+of+Space-Time%22&lr=&as_brr=0&sig=ACfU3U1q-iaRTBDo6J8HMEsyPeFi8cJNWg#PPA11,M1 | |||
}} 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 |author=Shigeyuki Morita, Teruko Nagase, Katsumi Nomizu |pages=p. 12 |url=http://books.google.com/books?id=5N33Of2RzjsC&pg=PA12&dq=geometry++axiom+%22coordinate+system%22&lr=&as_brr=0&sig=ACfU3U3Vi7xsLiYiWCK0erF6X2gczHOkJA#PPA12,M1 | |||
|isbn=0821810456 |year=2001 |publisher=American Mathematical Society Bookstore }} | |||
</ref> The 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 |author=Granino Arthur Korn, Theresa M. Korn |pages=p. 169 |url=http://books.google.com/books?id=xHNd5zCXt-EC&pg=PA169&dq=curvilinear+%22coordinate+system%22&lr=&as_brr=0&sig=ACfU3U3psSqwpBtA3U40e46VPPaMNMEw4g#PPA169,M1 | |||
|isbn=0486411478 |year=2000 |publisher=Courier Dover Publications}} | |||
</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=3540203176 |year=2004 |publisher=Springer |pages=12–13 |url=http://books.google.com/books?id=tNrMiIx3fToC&pg=PA12&dq=generalized+coordinates+%22kinematic+chain%22&lr=&as_brr=0&sig=ACfU3U3LRGJJTAHs21CHdOvuu08vw0cAuw#PPA13,M1 }} | |||
</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)\ , </math>    <math> 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}\ , </math>    <math> 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: | |||
:<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 }\ ,</math> | |||
which can be normalized to be of unit length. For more detail see [[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=052100666X |year=2003 |author=Wilford Zdunkowski & Andreas Bott |publisher=Cambridge University Press |url=http://books.google.com/books?id=GuYvC21v3g8C&pg=RA1-PA84&dq=%22curvilinear+coordinate+system%22&lr=&as_brr=0&sig=ACfU3U2g2k7kY5u-CVcJ1pH5ZxsbEb9Rig }}</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 (mathematics)|metric]] ''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 |author= A. I. Borisenko, I. E. Tarapov, Richard A. Silverman |page=86 |url=http://books.google.com/books?id=CRIjIx2ac6AC&pg=PA86&dq=coordinate+metric&lr=&as_brr=0&sig=ACfU3U1osXaT2hg7Md57cJ9katl3ttL43Q | |||
|isbn=0486638332 |publisher=Courier Dover Publications |year=1979 |pages=pp. 86 ''ff'' |chapter=§2.8.4 Arc length. Metric coefficients |edition=Reprint of Prentice-Hall 1968 ed }}</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 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 transformation]]s and [[Galilean transformation]]s may be viewed as [[coordinate transformation]]s. | |||
==Notes== | ==Notes== | ||
<references/> | <references/> | ||
[http://books.google.com/books?id=hUWEXphqLo8C&pg=PA111&dq=manifold+%22coordinate+system%22&hl=en&ei=I5GGTbWsPIz2tgOmoIzoAQ&sa=X&oi=book_result&ct=result&resnum=5&ved=0CEQQ6AEwBA#v=onepage&q=manifold%20%22coordinate%20system%22&f=false Choquet-Bruhat] | |||
[http://books.google.com/books?id=sRaSuentwngC&pg=PA2&dq=manifold+%22coordinate+system%22&hl=en&ei=I5GGTbWsPIz2tgOmoIzoAQ&sa=X&oi=book_result&ct=result&resnum=2&ved=0CDIQ6AEwAQ#v=onepage&q=manifold%20%22coordinate%20system%22&f=false Bishop] | |||
Revision as of 17:48, 20 March 2011
Coordinate system
The coordinates of a point r in an n-dimensional space are simply an ordered set of n numbers:[1]
![{\displaystyle \mathbf {r} =[x^{1},\ x^{2},\ \dots \ ,x^{n}]\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d77ca41fbd7ae5a9deac0187b55635d1afae0092)
Coordinate surfaces, coordinate lines, and basis vectors are components of a coordinate system.[2]
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).[3][4] 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.[5] 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.[2] 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:[6]
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
- ↑ 1.0 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.
- ↑ 2.0 2.1 Wilford Zdunkowski & Andreas Bott (2003). Dynamics of the Atmosphere. Cambridge University Press. ISBN 052100666X.
- ↑ 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.
- ↑ Shigeyuki Morita, Teruko Nagase, Katsumi Nomizu (2001). Geometry of Differential Forms. American Mathematical Society Bookstore, p. 12. ISBN 0821810456.
- ↑ Katsu Yamane (2004). Simulating and Generating Motions of Human Figures. Springer, 12–13. ISBN 3540203176.
- ↑ 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.