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''of a tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058560/l0585601.png" /> in the direction of a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058560/l0585602.png" /> on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058560/l0585603.png" />''
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The tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058560/l0585604.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058560/l0585605.png" />, of the same type as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058560/l0585606.png" />, given by the formula
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<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/l/l058/l058560/l0585607.png" /></td> </tr></table>
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''of a tensor field  $  Q $
 +
in the direction of a vector field  $  X $
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on a manifold  $  M $''
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058560/l0585608.png" /> is the local one-parameter group of transformations of the space of tensor fields generated by the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058560/l0585609.png" />. In local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058560/l05856010.png" />, the Lie derivative of a tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058560/l05856011.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058560/l05856012.png" /> in the direction of the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058560/l05856013.png" /> has coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058560/l05856014.png" />:
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The tensor field $  {\mathcal L} _ {X} Q $
 +
on  $  M $,  
 +
of the same type as  $  Q $,
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given by the formula
  
<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/l/l058/l058560/l05856015.png" /></td> </tr></table>
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$$
 +
( {\mathcal L} _ {X} Q ) _ {x}  = \lim\limits _ {t \rightarrow 0 } 
 +
\frac{1}{t}
 +
((
 +
\phi _ {t}  ^ {*} Q ) _ {x} - Q _ {x} ) ,\  x \in M ,
 +
$$
  
<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/l/l058/l058560/l05856016.png" /></td> </tr></table>
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where  $  \phi _ {t}  ^ {*} $
 +
is the local one-parameter group of transformations of the space of tensor fields generated by the vector field  $  X $.
 +
In local coordinates  $  x  ^ {i} $,
 +
the Lie derivative of a tensor field  $  Q = ( Q _ {j _ {1}  \dots j _ {l} } ^ {i _ {1} {} \dots i _ {k} } ) $
 +
of type  $  ( k , l ) $
 +
in the direction of the vector field  $  X = ( X  ^ {i} ) $
 +
has coordinates  $  ( \partial  _ {i} = \partial  / {\partial  x  ^ {i} } ) $:
  
<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/l/l058/l058560/l05856017.png" /></td> </tr></table>
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$$
 +
( {\mathcal L} _ {X} Q ) _ {j _ {1}  \dots j _ {l} } ^ {i _ {1} \dots i _ {k} }  = X  ^ {i} \partial  _ {i} Q _ {j _ {1}  \dots j _ {l} } ^ {i _ {1} \dots i _ {k} } +
 +
$$
 +
 
 +
$$
 +
- \sum _ {\alpha = 1 } ^ { k }  \partial  _ {i} X ^ {i _  \alpha  } Q _ {j _ {1}  \dots j _ {k} } ^ {i _ {1} \dots \widehat{i}  _  \alpha  ii _ {\alpha + 1 }  \dots i _ {k} } +
 +
$$
 +
 
 +
$$
 +
+
 +
\sum _ {\beta = 1 } ^ { l }  \partial  _ {j _  \beta  } X  ^ {j} Q _ {
 +
j _ {1}  \dots \widehat{j}  _  \beta  jj _ {\beta + 1 }  \dots
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j _ {l} } ^ {i _ {1} \dots i _ {k} } .
 +
$$
  
 
See also [[Lie differentiation|Lie differentiation]].
 
See also [[Lie differentiation|Lie differentiation]].

Latest revision as of 22:16, 5 June 2020


of a tensor field $ Q $ in the direction of a vector field $ X $ on a manifold $ M $

The tensor field $ {\mathcal L} _ {X} Q $ on $ M $, of the same type as $ Q $, given by the formula

$$ ( {\mathcal L} _ {X} Q ) _ {x} = \lim\limits _ {t \rightarrow 0 } \frac{1}{t} (( \phi _ {t} ^ {*} Q ) _ {x} - Q _ {x} ) ,\ x \in M , $$

where $ \phi _ {t} ^ {*} $ is the local one-parameter group of transformations of the space of tensor fields generated by the vector field $ X $. In local coordinates $ x ^ {i} $, the Lie derivative of a tensor field $ Q = ( Q _ {j _ {1} \dots j _ {l} } ^ {i _ {1} {} \dots i _ {k} } ) $ of type $ ( k , l ) $ in the direction of the vector field $ X = ( X ^ {i} ) $ has coordinates $ ( \partial _ {i} = \partial / {\partial x ^ {i} } ) $:

$$ ( {\mathcal L} _ {X} Q ) _ {j _ {1} \dots j _ {l} } ^ {i _ {1} \dots i _ {k} } = X ^ {i} \partial _ {i} Q _ {j _ {1} \dots j _ {l} } ^ {i _ {1} \dots i _ {k} } + $$

$$ - \sum _ {\alpha = 1 } ^ { k } \partial _ {i} X ^ {i _ \alpha } Q _ {j _ {1} \dots j _ {k} } ^ {i _ {1} \dots \widehat{i} _ \alpha ii _ {\alpha + 1 } \dots i _ {k} } + $$

$$ + \sum _ {\beta = 1 } ^ { l } \partial _ {j _ \beta } X ^ {j} Q _ { j _ {1} \dots \widehat{j} _ \beta jj _ {\beta + 1 } \dots j _ {l} } ^ {i _ {1} \dots i _ {k} } . $$

See also Lie differentiation.

How to Cite This Entry:
Lie derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_derivative&oldid=47628
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article