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A generalization of [[Vector analysis|vector analysis]], a part of [[Tensor calculus|tensor calculus]] studying differential (and integration) operators on the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t0923601.png" /> of differentiable tensor fields over a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t0923602.png" />. Also the extension of this theory to more general geometric objects than tensor fields, such as tensor densities, vector-valued differential forms, etc. is considered as a part of tensor analysis.
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A generalization of [[Vector analysis|vector analysis]], a part of [[Tensor calculus|tensor calculus]] studying differential (and integration) operators on the algebra $  D( M) $
 +
of differentiable tensor fields over a differentiable manifold $  M $.  
 +
Also the extension of this theory to more general geometric objects than tensor fields, such as tensor densities, vector-valued differential forms, etc. is considered as a part of tensor analysis.
  
 
====References====
 
====References====
Line 7: Line 21:
 
The most important operators of tensor analysis map tensor fields into tensor fields, possibly changing the type of the field.
 
The most important operators of tensor analysis map tensor fields into tensor fields, possibly changing the type of the field.
  
1) The [[Covariant derivative|covariant derivative]] in the direction of a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t0923603.png" /> is a linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t0923604.png" /> of the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t0923605.png" /> into itself satisfying the following conditions:
+
1) The [[Covariant derivative|covariant derivative]] in the direction of a vector field $  X \in D ^ {( 0,1) } ( M) $
 +
is a linear mapping $  \nabla _ {X} $
 +
of the vector space  $  D ^ {( 0,1) } ( M) $
 +
into itself satisfying the following conditions:
 +
 
 +
$$
 +
\nabla _ {fX+ gY }  Z  = f \nabla _ {X} Z + g \nabla _ {Y} Z,\ \
 +
\nabla _ {X} ( fZ)  = f \nabla _ {X} Z + X( f  ) Z,
 +
$$
 +
 
 +
where  $  X, Y, Z \in D ^ {( 0,1) } ( M) $
 +
and  $  f $
 +
and  $  g $
 +
are smooth functions on  $  M $.  
 +
In this interpretation the vector field  $  X $
 +
is considered as a derivation on functions, i.e. in local coordinates  $  u  ^ {1} \dots u  ^ {n} $
 +
with  $  X = \sum _ {i= 1 }  ^ {n} \xi  ^ {i} ( \partial  / \partial  u  ^ {i} ) $,
 +
for the derivative of the function  $  f $
 +
one has  $  X( f  ) = \sum _ {i= 1 }  ^ {n} \xi  ^ {i} ( \partial  f / \partial  u  ^ {i} ) $.  
 +
The prescription of such an operator  $  \nabla :  D ^ {( 0,1) } ( M) \times D ^ {( 0,1) } ( M) \rightarrow D ^ {( 0,1) } ( M) $
 +
defines a [[Linear connection|linear connection]] and a [[Parallel displacement(2)|parallel displacement]] along curves on  $  M $.
 +
 
 +
The extension of the covariant derivative to arbitrary tensor fields on  $  M $
 +
can be characterized by the properties that it is a derivation preserving the type of the tensor field and commuting with contraction. Regarding a tensor field of type  $  ( r, s) $
 +
as a field of multilinear mappings which are defined on  $  r $
 +
copies of the corresponding tangent space and  $  s $
 +
copies of the cotangent space, this extension is defined for  $  T \in D ^ {( r,s) } ( M) $
 +
as follows:
 +
 
 +
$$
 +
( \nabla _ {X} T)( X _ {1} \dots X _ {r} ,\
 +
\omega  ^ {1} \dots \omega  ^ {s} ) =
 +
$$
  
<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/t/t092/t092360/t0923606.png" /></td> </tr></table>
+
$$
 +
= \
 +
\nabla _ {X} ( T( X _ {1} \dots X _ {r} , \omega  ^ {1} \dots \omega  ^ {s} )) -
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t0923607.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t0923608.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t0923609.png" /> are smooth functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236010.png" />. In this interpretation the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236011.png" /> is considered as a derivation on functions, i.e. in local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236012.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236013.png" />, for the derivative of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236014.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236015.png" />. The prescription of such an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236016.png" /> defines a [[Linear connection|linear connection]] and a [[Parallel displacement(2)|parallel displacement]] along curves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236017.png" />.
+
$$
 +
-
 +
\sum _ {i= 1 } ^ { r }  T( X _ {1} \dots \nabla _ {X} X _ {i} \dots X _ {r} , \omega  ^ {1} \dots \omega  ^ {s} ) -
 +
$$
  
The extension of the covariant derivative to arbitrary tensor fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236018.png" /> can be characterized by the properties that it is a derivation preserving the type of the tensor field and commuting with contraction. Regarding a tensor field of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236019.png" /> as a field of multilinear mappings which are defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236020.png" /> copies of the corresponding tangent space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236021.png" /> copies of the cotangent space, this extension is defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236022.png" /> as follows:
+
$$
 +
-
 +
\sum _ {j= 1 } ^ { s }  T( X _ {1} \dots X _ {r} , \omega
 +
^ {1} \dots \nabla _ {X} \omega  ^ {j} \dots \omega  ^ {s} ),
 +
$$
  
<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/t/t092/t092360/t09236023.png" /></td> </tr></table>
+
where  $  ( \nabla _ {X} \omega )( Y) = \nabla _ {X} ( \omega ( Y))- \omega ( \nabla _ {X} Y) $,
 +
$  X, Y, X _ {1} \dots X _ {r} \in D ^ {( 0,1) } ( M) $
 +
and  $  \omega , \omega _ {1} \dots \omega _ {s} \in D ^ {( 1,0) } ( M) $.  
 +
This implies in local coordinates, using the notations introduced above and the connection coefficients  $  \Gamma _ {ij }  ^ {k} $
 +
given by  $  \nabla _ {\partial  / \partial  u  ^ {i}  } \partial  / \partial  u  ^ {j} = \sum _ {k= 1 }  ^ {n} \Gamma _ {ij }  ^ {k} \partial  / \partial  u  ^ {k} $,
 +
the following formula in terms of the components  $  T _ {i _ {1}  \dots i _ {r} } ^ {j _ {1} \dots j _ {s} } $
 +
of  $  T $:
  
<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/t/t092/t092360/t09236024.png" /></td> </tr></table>
+
$$
 +
( \nabla _ {X} T) _ {i _ {1}  \dots i _ {r} } ^ {j _ {1} \dots j _ {s} }  = \
 +
\sum _ {k= 1 } ^ { n }  \xi  ^ {k} \left [
 +
\frac \partial {\partial  u  ^ {k} }
  
<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/t/t092/t092360/t09236025.png" /></td> </tr></table>
+
( T _ {i _ {1}  \dots i _ {r} } ^ {j _ {1} \dots j _ {s} } ) \right . -
 +
$$
  
<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/t/t092/t092360/t09236026.png" /></td> </tr></table>
+
$$
 +
-
 +
\left . \sum _ {\nu = 1 } ^ { r }  \sum _ {l= 1 } ^ { n }  \Gamma _ {ki _  \nu  }  ^ {l} T _ {i _ {1}  \dots i _ {\nu - 1 }  l i _ {\nu + 1 }  \dots i _ {r} } ^ {j _ {1} \dots
 +
j _ {s} } + \sum _ {\mu = 1 } ^ { s }  \sum _ {l= 1 } ^ { n }  \Gamma _ {kl }  ^ {j _  \mu  } T _ {i _ {1}  \dots i _ {r} } ^ {j _ {1} \dots j _ {\mu - 1 }  l j _ {\mu + 1 }  \dots j _ {s} } \right ] .
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236029.png" />. This implies in local coordinates, using the notations introduced above and the connection coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236030.png" /> given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236031.png" />, the following formula in terms of the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236033.png" />:
+
From this definition one easily obtains the [[Covariant differential|covariant differential]]  $  \nabla T \in D ^ {( r+ 1,s) } ( M) $
 +
of the tensor field  $  T \in D ^ {( r,s) } ( M) $
 +
by
  
<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/t/t092/t092360/t09236034.png" /></td> </tr></table>
+
$$
 +
( \nabla T)( X _ {0} , X _ {1} \dots X _ {r} ,\
 +
\omega _ {1} \dots \omega _ {s} ) =
 +
$$
  
<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/t/t092/t092360/t09236035.png" /></td> </tr></table>
+
$$
 +
= \
 +
( \nabla _ {X _ {0}  } T)( X _ {1} \dots X _ {r} , \omega _ {1} \dots \omega _ {s} ),
 +
$$
  
From this definition one easily obtains the [[Covariant differential|covariant differential]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236036.png" /> of the tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236037.png" /> by
+
where the tensorial character of the first entry can be seen from the rules for  $  \nabla $
 +
presented above. In the case of the Levi-Civita connection of a Riemannian metric on  $  M $,
 +
the trace of the covariant differential of  $  T $
 +
with respect to the first and some other covariant entry leads to an extension of the [[Divergence|divergence]] to tensor fields.
  
<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/t/t092/t092360/t09236038.png" /></td> </tr></table>
+
2) The [[Lie derivative|Lie derivative]] along a vector field  $  X $
 +
is the mapping  $  L _ {X} : D ^ {( 0,1) } ( M) \rightarrow D ^ {( 0,1) } ( M) $
 +
defined by  $  L _ {X} Y = [ X, Y] $,
 +
where  $  [ X, Y]( f  ) = X( Y( f  )) - Y( X( f  )) $
 +
for any smooth function  $  f $
 +
on  $  M $.
 +
The extension of the Lie derivative to arbitrary tensor fields  $  T \in D ^ {( r,s) } ( M) $
 +
can be defined in the same way as the covariant derivative:
  
<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/t/t092/t092360/t09236039.png" /></td> </tr></table>
+
$$
 +
( L _ {X} T)( X _ {1} \dots X _ {r} , \omega  ^ {1} \dots \omega  ^ {s} ) =
 +
$$
  
where the tensorial character of the first entry can be seen from the rules for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236040.png" /> presented above. In the case of the Levi-Civita connection of a Riemannian metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236041.png" />, the trace of the covariant differential of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236042.png" /> with respect to the first and some other covariant entry leads to an extension of the [[Divergence|divergence]] to tensor fields.
+
$$
 +
= \
 +
\nabla _ {X} ( T( X _ {1} \dots X _ {r} , \omega  ^ {1} \dots \omega  ^ {s} )) -
 +
$$
  
2) The [[Lie derivative|Lie derivative]] along a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236043.png" /> is the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236044.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236045.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236046.png" /> for any smooth function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236047.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236048.png" />. The extension of the Lie derivative to arbitrary tensor fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236049.png" /> can be defined in the same way as the covariant derivative:
+
$$
 +
-
 +
\sum _ {i= 1 } ^ { r }  T( X _ {1} \dots L _ {X} X _ {i} \dots X _ {r} , \omega  ^ {1} \dots \omega  ^ {s} ) -
 +
$$
  
<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/t/t092/t092360/t09236050.png" /></td> </tr></table>
+
$$
 +
-  
 +
\sum _ {j= 1 } ^ { s }  T( X _ {1} \dots X _ {r} , \omega  ^ {1} \dots L _ {X} \omega  ^ {j} \dots \omega  ^ {s} ),
 +
$$
  
<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/t/t092/t092360/t09236051.png" /></td> </tr></table>
+
where  $  ( L _ {X} \omega )( Y) = \nabla _ {X} ( \omega ( Y))- \omega ( L _ {X} Y) $,
 +
$  X, Y, X _ {1} \dots X _ {r} \in D ^ {( 0,1) } ( M) $
 +
and  $  \omega , \omega _ {1} \dots \omega _ {s} \in D ^ {( 1,0) } ( M) $.  
 +
This implies in local coordinates,
  
<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/t/t092/t092360/t09236052.png" /></td> </tr></table>
+
$$
 +
( L _ {X} T) _ {i _ {1}  \dots i _ {r} } ^ {j _ {1} \dots j _ {s} }  = \
 +
\sum _ {k= 1 } ^ { n }  \xi  ^ {k}
 +
\frac \partial {\partial  u  ^ {k} }
  
<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/t/t092/t092360/t09236053.png" /></td> </tr></table>
+
( T _ {i _ {1}  \dots i _ {r} } ^ {j _ {1} \dots j _ {s} } ) +
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236056.png" />. This implies in local coordinates,
+
$$
 +
+
 +
\sum _ {k= 1 } ^ { n }  \sum _ {\nu = 1 } ^ { r } 
 +
\frac{
 +
\partial  \xi  ^ {k} }{\partial  u ^ {i _  \nu  } }
 +
T _ {i _ {1}  \dots i _ {
 +
\nu - 1 }  k i _ {\nu + 1 }  \dots i _ {r} } ^ {j _ {1} \dots j _ {s} }
 +
- \sum _ {k= 1 } ^ { n }  \sum _ {\mu = 1 } ^ { s } 
 +
\frac{
 +
\partial  \xi ^ {j _  \mu  } }{\partial  u  ^ {k} }
 +
T _ {i _ {1}  \dots i _ {r} } ^ {j _ {1} \dots j _ {\mu - 1 }  k j _ {\mu + 1 }  \dots j _ {s} } .
 +
$$
  
<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/t/t092/t092360/t09236057.png" /></td> </tr></table>
+
3) The exterior differential or exterior derivative is a linear operator  $  d $
 +
assigning to an (exterior) differential form  $  \omega \in F  ^ {p} ( M) $
 +
of degree  $  p $
 +
a differential form  $  d \omega $
 +
of degree  $  p+ 1 $
 +
such that the following compatibility with the wedge product (exterior product) of differential forms is satisfied:
  
<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/t/t092/t092360/t09236058.png" /></td> </tr></table>
+
$$
 +
d( \omega _ {1} \wedge \omega _ {2} )  = \
 +
d \omega _ {1} \wedge \omega _ {2} + (- 1) ^ {p }
 +
\omega _ {1} \wedge d \omega _ {2} ,
 +
$$
  
3) The exterior differential or exterior derivative is a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236059.png" /> assigning to an (exterior) differential form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236060.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236061.png" /> a differential form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236062.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236063.png" /> such that the following compatibility with the wedge product (exterior product) of differential forms is satisfied:
+
where  $  \omega _ {i} \in F ^ {p } ( M) $.
 +
Furthermore, one assumes that for a smooth function  $  f $(
 +
a differential form of degree 0 $),
 +
$  df $
 +
is given by the usual differential of  $  f $
 +
and that  $  ddf= 0 $.  
 +
This implies  $  dd \omega = 0 $
 +
in general. For  $  \omega \in F  ^ {p} ( M) $,
 +
the exterior differential can be described by:
  
<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/t/t092/t092360/t09236064.png" /></td> </tr></table>
+
$$
 +
( d \omega )( X _ {1} \dots X _ {p+ 1 }  ) =
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236065.png" />. Furthermore, one assumes that for a smooth function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236066.png" /> (a differential form of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236067.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236068.png" /> is given by the usual differential of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236069.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236070.png" />. This implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236071.png" /> in general. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236072.png" />, the exterior differential can be described by:
+
$$
 +
= \
 +
\sum _ {j= 1 } ^ { p }  (- 1) ^ {j+ 1 } X _ {j} \omega ( X _ {1} \dots X _ {j- 1 }  , X _ {j+ 1 }  \dots X _ {p+ 1 }  ) +
 +
$$
  
<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/t/t092/t092360/t09236073.png" /></td> </tr></table>
+
$$
 +
+
 +
\sum _ {i< j } (- 1) ^ {i+ j } \omega ([ X _ {i} , X _ {j} ], X _ {1} \dots X _ {i- 1 }  , \
 +
$$
  
<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/t/t092/t092360/t09236074.png" /></td> </tr></table>
+
$$
 +
 +
{} X _ {i+ 1 }  \dots X _ {j- 1 }  , X _ {j+ 1 }  \dots X _ {p+ 1 }  ) .
 +
$$
  
<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/t/t092/t092360/t09236075.png" /></td> </tr></table>
+
If in local coordinates  $  \omega = \sum _ {i _ {1}  \dots i _ {p} }  ^ {n} \omega _ {i _ {1}  \dots i _ {p} } du ^ {i _ {1} } \wedge \dots \wedge du ^ {i _ {p} } $,
 +
then
  
<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/t/t092/t092360/t09236076.png" /></td> </tr></table>
+
$$
 +
d \omega  = \sum _ {i _ {1} \dots i _ {p} } ^ { n }  \sum _ {k= 1 } ^ { n }
  
If in local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236077.png" />, then
+
\frac{\partial  \omega _ {i _ {1}  \dots i _ {p} } }{\partial  u  ^ {k} }
  
<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/t/t092/t092360/t09236078.png" /></td> </tr></table>
+
du  ^ {k}  \wedge  du ^ {i _ {1} } \wedge \dots \wedge  du ^ {i _ {p} } .
 +
$$
  
The fundamental operators of classical vector analysis may be described in terms of forms and exterior differentiation. For instance, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236079.png" />, acting on differential forms of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236080.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236081.png" />, corresponds to the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236082.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236083.png" />). Differential forms are the suitable objects for the theory of [[Integration on manifolds|integration on manifolds]]. Using the exterior derivative, the general form of the [[Stokes theorem|Stokes theorem]] can be given.
+
The fundamental operators of classical vector analysis may be described in terms of forms and exterior differentiation. For instance, the operator $  d $,  
 +
acting on differential forms of degree $  2 $
 +
in $  \mathbf R  ^ {3} $,  
 +
corresponds to the operator $  \mathop{\rm rot} $(
 +
$  \mathop{\rm curl} $).  
 +
Differential forms are the suitable objects for the theory of [[Integration on manifolds|integration on manifolds]]. Using the exterior derivative, the general form of the [[Stokes theorem|Stokes theorem]] can be given.
  
4) As an example of the application of tensor analysis in differential geometry, the [[Curvature tensor|curvature tensor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236084.png" /> of a linear connection should be mentioned. This tensor is of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236085.png" />. In terms of the corresponding covariant derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236086.png" /> and vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236088.png" /> is given as follows:
+
4) As an example of the application of tensor analysis in differential geometry, the [[Curvature tensor|curvature tensor]] $  R $
 +
of a linear connection should be mentioned. This tensor is of type $  ( 3, 1) $.  
 +
In terms of the corresponding covariant derivative $  \nabla $
 +
and vector fields $  X, Y, Z \in D ^ {( 0,1) } ( M) $,
 +
$  R $
 +
is given as follows:
  
<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/t/t092/t092360/t09236089.png" /></td> </tr></table>
+
$$
 +
R( X, Y) Z  = \nabla _ {X} \nabla _ {Y} Z - \nabla _ {Y} \nabla _ {X} Z - \nabla _ {[ X,Y] }  Z,
 +
$$
  
where, as usual, the contravariant entry is identified with a vector-valued multilinear mapping. In case of the Levi-Civita connection of a Riemannian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236090.png" /> one obtains for the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236091.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092360/t09236092.png" />:
+
where, as usual, the contravariant entry is identified with a vector-valued multilinear mapping. In case of the Levi-Civita connection of a Riemannian metric $  g $
 +
one obtains for the components $  R _ {ikj }  ^ {l} $
 +
of $  R $:
  
<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/t/t092/t092360/t09236093.png" /></td> </tr></table>
+
$$
 +
R _ {ikj }  ^ {l}  = \
 +
 
 +
\frac{\partial  \Gamma _ {ik }  ^ {l} }{\partial  u  ^ {j} }
 +
-
 +
 
 +
\frac{\partial  \Gamma _ {ij }  ^ {l} }{\partial  u  ^ {k} }
 +
+
 +
\sum _ {\nu = 1 } ^ { n }  \Gamma _ {ik }  ^  \nu  \Gamma _ {\nu j }  ^ {l} -
 +
\sum _ {\nu = 1 } ^ { n }  \Gamma _ {ij }  ^  \nu  \Gamma _ {\nu k }  ^ {l} ,
 +
$$
  
 
where the connection coefficients are obtained from the metric as follows:
 
where the connection coefficients are obtained from the metric as follows:
  
<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/t/t092/t092360/t09236094.png" /></td> </tr></table>
+
$$
 +
\Gamma _ {jk }  ^ {i}  = {
 +
\frac{1}{2}
 +
} \sum _ {\nu = 1 } ^ { n }  g ^ {i \nu }
 +
\left (
 +
\frac{\partial  g _ {j \nu }  }{\partial  u  ^ {k} }
 +
-
 +
 
 +
\frac{\partial  g _ {jk }  }{\partial  u  ^  \nu  }
 +
+
 +
 
 +
\frac{\partial  g _ {\nu k }  }{\partial  u  ^ {j} }
 +
\right ) .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Klingenberg,  "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.J. Hicks,  "Notes on differential geometry" , v. Nostrand  (1965)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1–2''' , Interscience  (1963)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Klingenberg,  "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.J. Hicks,  "Notes on differential geometry" , v. Nostrand  (1965)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1–2''' , Interscience  (1963)</TD></TR></table>

Latest revision as of 08:25, 6 June 2020


A generalization of vector analysis, a part of tensor calculus studying differential (and integration) operators on the algebra $ D( M) $ of differentiable tensor fields over a differentiable manifold $ M $. Also the extension of this theory to more general geometric objects than tensor fields, such as tensor densities, vector-valued differential forms, etc. is considered as a part of tensor analysis.

References

[1] P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
[2] J.A. Schouten, "Tensor analysis for physicists" , Cambridge Univ. Press (1951)
[3] A.J. MacConnel, "Applications of tensor analysis" , Dover, reprint (1957)
[4] I.S. Sokolnikoff, "Tensor analysis" , Wiley (1964)

Comments

The most important operators of tensor analysis map tensor fields into tensor fields, possibly changing the type of the field.

1) The covariant derivative in the direction of a vector field $ X \in D ^ {( 0,1) } ( M) $ is a linear mapping $ \nabla _ {X} $ of the vector space $ D ^ {( 0,1) } ( M) $ into itself satisfying the following conditions:

$$ \nabla _ {fX+ gY } Z = f \nabla _ {X} Z + g \nabla _ {Y} Z,\ \ \nabla _ {X} ( fZ) = f \nabla _ {X} Z + X( f ) Z, $$

where $ X, Y, Z \in D ^ {( 0,1) } ( M) $ and $ f $ and $ g $ are smooth functions on $ M $. In this interpretation the vector field $ X $ is considered as a derivation on functions, i.e. in local coordinates $ u ^ {1} \dots u ^ {n} $ with $ X = \sum _ {i= 1 } ^ {n} \xi ^ {i} ( \partial / \partial u ^ {i} ) $, for the derivative of the function $ f $ one has $ X( f ) = \sum _ {i= 1 } ^ {n} \xi ^ {i} ( \partial f / \partial u ^ {i} ) $. The prescription of such an operator $ \nabla : D ^ {( 0,1) } ( M) \times D ^ {( 0,1) } ( M) \rightarrow D ^ {( 0,1) } ( M) $ defines a linear connection and a parallel displacement along curves on $ M $.

The extension of the covariant derivative to arbitrary tensor fields on $ M $ can be characterized by the properties that it is a derivation preserving the type of the tensor field and commuting with contraction. Regarding a tensor field of type $ ( r, s) $ as a field of multilinear mappings which are defined on $ r $ copies of the corresponding tangent space and $ s $ copies of the cotangent space, this extension is defined for $ T \in D ^ {( r,s) } ( M) $ as follows:

$$ ( \nabla _ {X} T)( X _ {1} \dots X _ {r} ,\ \omega ^ {1} \dots \omega ^ {s} ) = $$

$$ = \ \nabla _ {X} ( T( X _ {1} \dots X _ {r} , \omega ^ {1} \dots \omega ^ {s} )) - $$

$$ - \sum _ {i= 1 } ^ { r } T( X _ {1} \dots \nabla _ {X} X _ {i} \dots X _ {r} , \omega ^ {1} \dots \omega ^ {s} ) - $$

$$ - \sum _ {j= 1 } ^ { s } T( X _ {1} \dots X _ {r} , \omega ^ {1} \dots \nabla _ {X} \omega ^ {j} \dots \omega ^ {s} ), $$

where $ ( \nabla _ {X} \omega )( Y) = \nabla _ {X} ( \omega ( Y))- \omega ( \nabla _ {X} Y) $, $ X, Y, X _ {1} \dots X _ {r} \in D ^ {( 0,1) } ( M) $ and $ \omega , \omega _ {1} \dots \omega _ {s} \in D ^ {( 1,0) } ( M) $. This implies in local coordinates, using the notations introduced above and the connection coefficients $ \Gamma _ {ij } ^ {k} $ given by $ \nabla _ {\partial / \partial u ^ {i} } \partial / \partial u ^ {j} = \sum _ {k= 1 } ^ {n} \Gamma _ {ij } ^ {k} \partial / \partial u ^ {k} $, the following formula in terms of the components $ T _ {i _ {1} \dots i _ {r} } ^ {j _ {1} \dots j _ {s} } $ of $ T $:

$$ ( \nabla _ {X} T) _ {i _ {1} \dots i _ {r} } ^ {j _ {1} \dots j _ {s} } = \ \sum _ {k= 1 } ^ { n } \xi ^ {k} \left [ \frac \partial {\partial u ^ {k} } ( T _ {i _ {1} \dots i _ {r} } ^ {j _ {1} \dots j _ {s} } ) \right . - $$

$$ - \left . \sum _ {\nu = 1 } ^ { r } \sum _ {l= 1 } ^ { n } \Gamma _ {ki _ \nu } ^ {l} T _ {i _ {1} \dots i _ {\nu - 1 } l i _ {\nu + 1 } \dots i _ {r} } ^ {j _ {1} \dots j _ {s} } + \sum _ {\mu = 1 } ^ { s } \sum _ {l= 1 } ^ { n } \Gamma _ {kl } ^ {j _ \mu } T _ {i _ {1} \dots i _ {r} } ^ {j _ {1} \dots j _ {\mu - 1 } l j _ {\mu + 1 } \dots j _ {s} } \right ] . $$

From this definition one easily obtains the covariant differential $ \nabla T \in D ^ {( r+ 1,s) } ( M) $ of the tensor field $ T \in D ^ {( r,s) } ( M) $ by

$$ ( \nabla T)( X _ {0} , X _ {1} \dots X _ {r} ,\ \omega _ {1} \dots \omega _ {s} ) = $$

$$ = \ ( \nabla _ {X _ {0} } T)( X _ {1} \dots X _ {r} , \omega _ {1} \dots \omega _ {s} ), $$

where the tensorial character of the first entry can be seen from the rules for $ \nabla $ presented above. In the case of the Levi-Civita connection of a Riemannian metric on $ M $, the trace of the covariant differential of $ T $ with respect to the first and some other covariant entry leads to an extension of the divergence to tensor fields.

2) The Lie derivative along a vector field $ X $ is the mapping $ L _ {X} : D ^ {( 0,1) } ( M) \rightarrow D ^ {( 0,1) } ( M) $ defined by $ L _ {X} Y = [ X, Y] $, where $ [ X, Y]( f ) = X( Y( f )) - Y( X( f )) $ for any smooth function $ f $ on $ M $. The extension of the Lie derivative to arbitrary tensor fields $ T \in D ^ {( r,s) } ( M) $ can be defined in the same way as the covariant derivative:

$$ ( L _ {X} T)( X _ {1} \dots X _ {r} , \omega ^ {1} \dots \omega ^ {s} ) = $$

$$ = \ \nabla _ {X} ( T( X _ {1} \dots X _ {r} , \omega ^ {1} \dots \omega ^ {s} )) - $$

$$ - \sum _ {i= 1 } ^ { r } T( X _ {1} \dots L _ {X} X _ {i} \dots X _ {r} , \omega ^ {1} \dots \omega ^ {s} ) - $$

$$ - \sum _ {j= 1 } ^ { s } T( X _ {1} \dots X _ {r} , \omega ^ {1} \dots L _ {X} \omega ^ {j} \dots \omega ^ {s} ), $$

where $ ( L _ {X} \omega )( Y) = \nabla _ {X} ( \omega ( Y))- \omega ( L _ {X} Y) $, $ X, Y, X _ {1} \dots X _ {r} \in D ^ {( 0,1) } ( M) $ and $ \omega , \omega _ {1} \dots \omega _ {s} \in D ^ {( 1,0) } ( M) $. This implies in local coordinates,

$$ ( L _ {X} T) _ {i _ {1} \dots i _ {r} } ^ {j _ {1} \dots j _ {s} } = \ \sum _ {k= 1 } ^ { n } \xi ^ {k} \frac \partial {\partial u ^ {k} } ( T _ {i _ {1} \dots i _ {r} } ^ {j _ {1} \dots j _ {s} } ) + $$

$$ + \sum _ {k= 1 } ^ { n } \sum _ {\nu = 1 } ^ { r } \frac{ \partial \xi ^ {k} }{\partial u ^ {i _ \nu } } T _ {i _ {1} \dots i _ { \nu - 1 } k i _ {\nu + 1 } \dots i _ {r} } ^ {j _ {1} \dots j _ {s} } - \sum _ {k= 1 } ^ { n } \sum _ {\mu = 1 } ^ { s } \frac{ \partial \xi ^ {j _ \mu } }{\partial u ^ {k} } T _ {i _ {1} \dots i _ {r} } ^ {j _ {1} \dots j _ {\mu - 1 } k j _ {\mu + 1 } \dots j _ {s} } . $$

3) The exterior differential or exterior derivative is a linear operator $ d $ assigning to an (exterior) differential form $ \omega \in F ^ {p} ( M) $ of degree $ p $ a differential form $ d \omega $ of degree $ p+ 1 $ such that the following compatibility with the wedge product (exterior product) of differential forms is satisfied:

$$ d( \omega _ {1} \wedge \omega _ {2} ) = \ d \omega _ {1} \wedge \omega _ {2} + (- 1) ^ {p } \omega _ {1} \wedge d \omega _ {2} , $$

where $ \omega _ {i} \in F ^ {p } ( M) $. Furthermore, one assumes that for a smooth function $ f $( a differential form of degree $ 0 $), $ df $ is given by the usual differential of $ f $ and that $ ddf= 0 $. This implies $ dd \omega = 0 $ in general. For $ \omega \in F ^ {p} ( M) $, the exterior differential can be described by:

$$ ( d \omega )( X _ {1} \dots X _ {p+ 1 } ) = $$

$$ = \ \sum _ {j= 1 } ^ { p } (- 1) ^ {j+ 1 } X _ {j} \omega ( X _ {1} \dots X _ {j- 1 } , X _ {j+ 1 } \dots X _ {p+ 1 } ) + $$

$$ + \sum _ {i< j } (- 1) ^ {i+ j } \omega ([ X _ {i} , X _ {j} ], X _ {1} \dots X _ {i- 1 } , \ $$

$$ {} X _ {i+ 1 } \dots X _ {j- 1 } , X _ {j+ 1 } \dots X _ {p+ 1 } ) . $$

If in local coordinates $ \omega = \sum _ {i _ {1} \dots i _ {p} } ^ {n} \omega _ {i _ {1} \dots i _ {p} } du ^ {i _ {1} } \wedge \dots \wedge du ^ {i _ {p} } $, then

$$ d \omega = \sum _ {i _ {1} \dots i _ {p} } ^ { n } \sum _ {k= 1 } ^ { n } \frac{\partial \omega _ {i _ {1} \dots i _ {p} } }{\partial u ^ {k} } du ^ {k} \wedge du ^ {i _ {1} } \wedge \dots \wedge du ^ {i _ {p} } . $$

The fundamental operators of classical vector analysis may be described in terms of forms and exterior differentiation. For instance, the operator $ d $, acting on differential forms of degree $ 2 $ in $ \mathbf R ^ {3} $, corresponds to the operator $ \mathop{\rm rot} $( $ \mathop{\rm curl} $). Differential forms are the suitable objects for the theory of integration on manifolds. Using the exterior derivative, the general form of the Stokes theorem can be given.

4) As an example of the application of tensor analysis in differential geometry, the curvature tensor $ R $ of a linear connection should be mentioned. This tensor is of type $ ( 3, 1) $. In terms of the corresponding covariant derivative $ \nabla $ and vector fields $ X, Y, Z \in D ^ {( 0,1) } ( M) $, $ R $ is given as follows:

$$ R( X, Y) Z = \nabla _ {X} \nabla _ {Y} Z - \nabla _ {Y} \nabla _ {X} Z - \nabla _ {[ X,Y] } Z, $$

where, as usual, the contravariant entry is identified with a vector-valued multilinear mapping. In case of the Levi-Civita connection of a Riemannian metric $ g $ one obtains for the components $ R _ {ikj } ^ {l} $ of $ R $:

$$ R _ {ikj } ^ {l} = \ \frac{\partial \Gamma _ {ik } ^ {l} }{\partial u ^ {j} } - \frac{\partial \Gamma _ {ij } ^ {l} }{\partial u ^ {k} } + \sum _ {\nu = 1 } ^ { n } \Gamma _ {ik } ^ \nu \Gamma _ {\nu j } ^ {l} - \sum _ {\nu = 1 } ^ { n } \Gamma _ {ij } ^ \nu \Gamma _ {\nu k } ^ {l} , $$

where the connection coefficients are obtained from the metric as follows:

$$ \Gamma _ {jk } ^ {i} = { \frac{1}{2} } \sum _ {\nu = 1 } ^ { n } g ^ {i \nu } \left ( \frac{\partial g _ {j \nu } }{\partial u ^ {k} } - \frac{\partial g _ {jk } }{\partial u ^ \nu } + \frac{\partial g _ {\nu k } }{\partial u ^ {j} } \right ) . $$

References

[a1] W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)
[a2] N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)
[a3] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1963)
How to Cite This Entry:
Tensor analysis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tensor_analysis&oldid=48954
This article was adapted from an original article by B. Wegner (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article