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''curvilinear coordinate system, parametrization of a set  $  M $''

A one-to-one mapping

$$ 
x:  M  \rightarrow  D,\ \ 
p  \rightarrow  x ( p) =
( x  ^ {1} ( p) \dots x  ^ {n} ( p)) ,
$$

of a set  $  M $
onto an open subset  $  D $
of the real vector space  $  \mathbf R  ^ {n} $. 
The integer  $  n $
is called the dimension of the chart, and the components  $  x  ^ {i} ( p) $
of the vector  $  x ( p) \in \mathbf R  ^ {n} $
are called the coordinates of  $  p \in M $
with respect to the chart  $  x $.

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 [[#References|[1]]]). In Riemann's view, the basic object of study in geometry is a manifold — a set  $  M $
endowed with a chart. The modern concept of a manifold is a natural generalization of Riemann's definition.

A chart  $  x:  U \rightarrow D $
of some subset  $  U $
of  $  M $
is called a local chart of  $  M $
with domain of definition  $  U $. 
If  $  M $
is endowed with the structure of a topological space, then it is further required that  $  U $
be an open subset of  $  M $
and that the mapping  $  x $
be a homeomorphism. A chart can similarly be defined with values in  $  F ^ { n } $, 
where  $  F $
is any normed field, and more generally, a chart can take values in a topological vector space. Two local charts  $  ( x, U) $, 
$  ( y, V) $
with domains of definition  $  U, V $
in  $  M $
are said to be compatible of class  $  C ^ { l } $
if 1) their common domain of definition  $  W = U \cap V $
is mapped by both charts onto an open set (that is, the sets  $  x ( W) $
and  $  y ( W) $
are open in  $  \mathbf R  ^ {n} $); 
and 2) the coordinates of a point of  $  W $
with respect to one of these charts are  $  l $
times continuously-differentiable functions of the coordinates of the same point with respect to the other chart, that is, the vector function

$$ 
y \circ x  ^ {-} 1 : \ 
x ( W)  \rightarrow  y ( W)
$$

is  $  l $
times continuously differentiable. A family  $  A = \{ ( x _  \alpha  , U _  \alpha  ) \} $
of pairwise-compatible local charts  $  ( x _  \alpha  , U _  \alpha  ) $
of  $  M $
that cover  $  M $(
that is,  $  \cup _  \alpha  U _  \alpha  = M $) 
is called an atlas of  $  M $. 
The specification of an atlas defines on  $  M $
the structure of a [[Differentiable manifold|differentiable manifold]], and local charts that are compatible with all the charts of this atlas are said to be admissible (or  $  C ^ { l } $-
smooth).

The infinitesimal analogue of the notion of a chart is the concept of an infinitesimal chart of order  $  k $(
or a  $  k $-
jet (of a chart) or a co-frame of order  $  k $). 
Two compatible local charts  $  ( x, U) $, 
$  ( y, V) $
of a set  $  M $
are said to be tangent to each other up to order  $  k $
at a point  $  p \in U \cap V $
if  $  x ( p) = y ( p) $
and if all the partial derivatives up to order  $  k $, 
inclusive, of the vector function  $  y \circ x  ^ {-} 1 :  x \rightarrow y ( x) $
vanish at  $  x ( p) $. 
The class  $  j _ {p}  ^ {k} ( x) $
of local charts tangent (up to order  $  k $) 
at a point  $  p \in U $
of an admissible local chart  $  ( x, U) $
of a differentiable manifold  $  M $
is called the infinitesimal chart of order  $  k $
at  $  p $, 
or  $  k $-
jet at  $  p $.

The choice of a chart on a manifold  $  M $
allows one to consider various field quantities on  $  M $
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  $  M $.) 
However, for a wide and most important class of quantities (see [[Geometric objects, theory of|Geometric objects, theory of]]), their value at a point depends only on the structure of the chart in the  $  k $-
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  $  k $
on  $  M $. 
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.

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

====Comments====
For Riemann's view see, in particular, [[#References|[1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  O. Veblen,   J.H.C. Whitehead,   "The foundations of differential geometry" , Cambridge Univ. Press  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.L. Bishop,   R.J. Crittenden,   "Geometry of manifolds" , Acad. Press  (1964)</TD></TR></table>