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''curvilinear coordinate system, parametrization of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c0218001.png" />''
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''curvilinear coordinate system, parametrization of a set  $  M $''
  
 
A one-to-one mapping
 
A one-to-one mapping
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c0218002.png" /></td> </tr></table>
+
$$
 +
x: M  \rightarrow  D,\ \
 +
p  \rightarrow  x ( p) =
 +
( x  ^ {1} ( p) \dots x  ^ {n} ( p)) ,
 +
$$
  
of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c0218003.png" /> onto an open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c0218004.png" /> of the real vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c0218005.png" />. The integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c0218006.png" /> is called the dimension of the chart, and the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c0218007.png" /> of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c0218008.png" /> are called the coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c0218009.png" /> with respect to the chart <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180010.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180011.png" /> endowed with a chart. The modern concept of a manifold is a natural generalization of Riemann's definition.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180012.png" /> of some subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180014.png" /> is called a local chart of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180015.png" /> with domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180016.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180017.png" /> is endowed with the structure of a topological space, then it is further required that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180018.png" /> be an open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180019.png" /> and that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180020.png" /> be a homeomorphism. A chart can similarly be defined with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180022.png" /> is any normed field, and more generally, a chart can take values in a topological vector space. Two local charts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180024.png" /> with domains of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180025.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180026.png" /> are said to be compatible of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180028.png" /> if 1) their common domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180029.png" /> is mapped by both charts onto an open set (that is, the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180031.png" /> are open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180032.png" />); and 2) the coordinates of a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180033.png" /> with respect to one of these charts are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180034.png" /> times continuously-differentiable functions of the coordinates of the same point with respect to the other chart, that is, the vector function
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180035.png" /></td> </tr></table>
+
$$
 +
y \circ x  ^ {-} 1 : \
 +
x ( W)  \rightarrow  y ( W)
 +
$$
  
is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180036.png" /> times continuously differentiable. A family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180037.png" /> of pairwise-compatible local charts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180038.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180039.png" /> that cover <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180040.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180041.png" />) is called an atlas of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180042.png" />. The specification of an atlas defines on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180043.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180044.png" />-smooth).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180046.png" /> (or a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180048.png" />-jet (of a chart) or a co-frame of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180050.png" />). Two compatible local charts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180052.png" /> of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180053.png" /> are said to be tangent to each other up to order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180054.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180055.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180056.png" /> and if all the partial derivatives up to order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180057.png" />, inclusive, of the vector function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180058.png" /> vanish at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180059.png" />. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180060.png" /> of local charts tangent (up to order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180061.png" />) at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180062.png" /> of an admissible local chart <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180063.png" /> of a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180064.png" /> is called the infinitesimal chart of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180065.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180066.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180067.png" />-jet at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180068.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180069.png" /> allows one to consider various field quantities on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180070.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180071.png" />.) 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180072.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180073.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021800/c02180074.png" />. 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.
+
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====
 
====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>
 
<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====
 
====Comments====

Latest revision as of 16:43, 4 June 2020


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)

Comments

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

References

[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:
Chart. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chart&oldid=46326
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article