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curvilinear coordinate system, parametrization of a set

A one-to-one mapping

of a set onto an open subset of the real vector space . The integer is called the dimension of the chart, and the components of the vector are called the coordinates of with respect to the chart .

An example of a chart is the Cartesian coordinate system in the plane and in space, introduced by P. Fermat and R. Descartes, and taken by them as the basis for analytic geometry. e root','../p/p074630.htm','Series','../s/s084670.htm','Stability of an elastic system','../s/s087010.htm','Siegel disc','../s/s110120.htm','Theta-function','../t/t092600.htm','Trigonometric series','../t/t094240.htm','Two-term congruence','../t/t094620.htm','Umbral calculus','../u/u095050.htm','Variation of constants','../v/v096160.htm','Variational calculus','../v/v096190.htm','Variational calculus, numerical methods of','../v/v096210.htm','Venn diagram','../v/v096550.htm','Zeta-function','../z/z099260.htm')" style="background-color:yellow;">L. Euler was the first to employ charts (curvilinear coordinates) on surfaces in geometric research. B. Riemann took up the notion of a chart as the basis for a new infinitesimal approach to the foundations of geometry (see [1]). In Riemann's view, the basic object of study in geometry is a manifold — a set endowed with a chart. The modern concept of a manifold is a natural generalization of Riemann's definition.

A chart of some subset of is called a local chart of with domain of definition . If is endowed with the structure of a topological space, then it is further required that be an open subset of and that the mapping be a homeomorphism. A chart can similarly be defined with values in , where is any normed field, and more generally, a chart can take values in a topological vector space. Two local charts , with domains of definition in are said to be compatible of class if 1) their common domain of definition is mapped by both charts onto an open set (that is, the sets and are open in ); and 2) the coordinates of a point of with respect to one of these charts are times continuously-differentiable functions of the coordinates of the same point with respect to the other chart, that is, the vector function

is times continuously differentiable. A family of pairwise-compatible local charts of that cover (that is, ) is called an atlas of . The specification of an atlas defines on the structure of a differentiable manifold, and local charts that are compatible with all the charts of this atlas are said to be admissible (or -smooth).

The infinitesimal analogue of the notion of a chart is the concept of an infinitesimal chart of order (or a -jet (of a chart) or a co-frame of order ). Two compatible local charts , of a set are said to be tangent to each other up to order at a point if and if all the partial derivatives up to order , inclusive, of the vector function vanish at . The class of local charts tangent (up to order ) at a point of an admissible local chart of a differentiable manifold is called the infinitesimal chart of order at , or -jet at .

The choice of a chart on a manifold allows one to consider various field quantities on as numerical functions and to apply to them the methods of analysis. In general, the value of a field quantity at a point depends on the choice of the chart. (Quantities which are independent of the choice of the chart are called scalars and are described by functions on .) However, for a wide and most important class of quantities (see Geometric objects, theory of), their value at a point depends only on the structure of the chart in the -th order infinitesimal neighbourhood of this point. Such quantities (examples of which are the tensor fields) are described by functions on the set of all co-frames of order on . Along with these one studies the properties of quantities which do not depend on the choice of a chart. In this connection, the invariant coordinate-free approach to problems of differential geometry proves to be highly effective.


[1] B. Riemann, "Ueber die Hypothesen, welche der Geometrie zuGrunde liegen" , Das Kontinuum und andere Monographien , Chelsea, reprint (1973)
[2] P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
[3] R. Sulanke, P. Wintgen, "Differentialgeometrie und Faserbündel" , Birkhäuser (1972)
[4] A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1976) (Translated from French)
[5] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963)


For Riemann's view see, in particular, [1].


[a1] O. Veblen, J.H.C. Whitehead, "The foundations of differential geometry" , Cambridge Univ. Press (1967)
[a2] R.L. Bishop, R.J. Crittenden, "Geometry of manifolds" , Acad. Press (1964)
How to Cite This Entry:
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This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article