# Chart

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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 [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, 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), 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

 [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)