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''pseudo-tensor''
 
''pseudo-tensor''
  
A geometric object described in a coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092390/t0923901.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092390/t0923902.png" /> components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092390/t0923903.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092390/t0923904.png" />, transforming under a change of coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092390/t0923905.png" /> according to the formula
+
A geometric object described in a coordinate system $  x = ( x  ^ {1} \dots x  ^ {n} ) $
 +
by $  n  ^ {p+} q $
 +
components $  a _ {j _ {1}  \dots j _ {q} } ^ {i _ {1} \dots i _ {p} } $,
 +
$  1 \leq  i _  \nu  , j _  \mu  \leq  n $,  
 +
transforming under a change of coordinates $  x \mapsto y = ( y  ^ {1} \dots y  ^ {n} ) $
 +
according to the formula
 +
 
 +
$$
 +
a _ {j _ {1}  \dots j _ {q} } ^ {i _ {1} \dots i _ {p} }  = \
 +
\Delta ^ {- \kappa }
 +
a _ {\beta _ {1}  \dots \beta _ {q} } ^ {\alpha _ {1} \dots \alpha _ {p} }
  
<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/t092390/t0923906.png" /></td> </tr></table>
+
\frac{\partial  y ^ {i _ {1} } }{\partial  x ^ {\alpha _ {1} } }
 +
\dots
  
<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/t092390/t0923907.png" /></td> </tr></table>
+
\frac{\partial  y ^ {i _ {p} } }{\partial  x ^ {\alpha _ {p} } }
 +
\cdot
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092390/t0923908.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092390/t0923909.png" /> is called the weight of the tensor density. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092390/t09239010.png" />, the tensor density is a tensor (cf. [[Tensor on a vector space|Tensor on a vector space]]). Concepts such as type, valency, covariance, contravariance, etc. are introduced similar to the corresponding tensor concepts. Tensor densities of types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092390/t09239011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092390/t09239012.png" /> are called vector densities. Tensor densities of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092390/t09239013.png" /> are called scalar densities.
+
$$
 +
\cdot
  
 +
\frac{\partial  x ^ {\beta _ {1} } }{\partial  y ^
 +
{j _ {1} } }
 +
\dots
 +
\frac{\partial  x ^ {\beta _ {q} } }{\partial  y ^ {j _ {q} } }
 +
,
 +
$$
  
 +
where  $  \Delta =  \mathop{\rm det} ( \partial  y  ^ {i} / \partial  x _ {k} ) $.
 +
The number  $  \kappa $
 +
is called the weight of the tensor density. When  $  \kappa = 0 $,
 +
the tensor density is a tensor (cf. [[Tensor on a vector space|Tensor on a vector space]]). Concepts such as type, valency, covariance, contravariance, etc. are introduced similar to the corresponding tensor concepts. Tensor densities of types  $  ( 1, 0) $
 +
and  $  ( 0, 1) $
 +
are called vector densities. Tensor densities of type  $  ( 0, 0) $
 +
are called scalar densities.
  
 
====Comments====
 
====Comments====
A tensor density as defined above is also called a relative tensor. One distinguishes between odd relative tensors of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092390/t09239014.png" />, which transform as above, and even relative tensors, which transform according to the same formula except that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092390/t09239015.png" /> is replaced by its absolute value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092390/t09239016.png" />. In [[#References|[a2]]] an even tensor density is simply called a  "tensor density"  and an odd one is called a tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092390/t09239018.png" />-density.
+
A tensor density as defined above is also called a relative tensor. One distinguishes between odd relative tensors of weight $  k $,  
 +
which transform as above, and even relative tensors, which transform according to the same formula except that $  \Delta $
 +
is replaced by its absolute value $  | \Delta | $.  
 +
In [[#References|[a2]]] an even tensor density is simply called a  "tensor density"  and an odd one is called a tensor $  \Delta $-
 +
density.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''I''' , Publish or Perish  (1970)  pp. 437ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.A. Schouten,  "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer  (1954)  pp. 12  (Translated from German)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''I''' , Publish or Perish  (1970)  pp. 437ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.A. Schouten,  "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer  (1954)  pp. 12  (Translated from German)</TD></TR></table>

Revision as of 08:25, 6 June 2020


pseudo-tensor

A geometric object described in a coordinate system $ x = ( x ^ {1} \dots x ^ {n} ) $ by $ n ^ {p+} q $ components $ a _ {j _ {1} \dots j _ {q} } ^ {i _ {1} \dots i _ {p} } $, $ 1 \leq i _ \nu , j _ \mu \leq n $, transforming under a change of coordinates $ x \mapsto y = ( y ^ {1} \dots y ^ {n} ) $ according to the formula

$$ a _ {j _ {1} \dots j _ {q} } ^ {i _ {1} \dots i _ {p} } = \ \Delta ^ {- \kappa } a _ {\beta _ {1} \dots \beta _ {q} } ^ {\alpha _ {1} \dots \alpha _ {p} } \frac{\partial y ^ {i _ {1} } }{\partial x ^ {\alpha _ {1} } } \dots \frac{\partial y ^ {i _ {p} } }{\partial x ^ {\alpha _ {p} } } \cdot $$

$$ \cdot \frac{\partial x ^ {\beta _ {1} } }{\partial y ^ {j _ {1} } } \dots \frac{\partial x ^ {\beta _ {q} } }{\partial y ^ {j _ {q} } } , $$

where $ \Delta = \mathop{\rm det} ( \partial y ^ {i} / \partial x _ {k} ) $. The number $ \kappa $ is called the weight of the tensor density. When $ \kappa = 0 $, the tensor density is a tensor (cf. Tensor on a vector space). Concepts such as type, valency, covariance, contravariance, etc. are introduced similar to the corresponding tensor concepts. Tensor densities of types $ ( 1, 0) $ and $ ( 0, 1) $ are called vector densities. Tensor densities of type $ ( 0, 0) $ are called scalar densities.

Comments

A tensor density as defined above is also called a relative tensor. One distinguishes between odd relative tensors of weight $ k $, which transform as above, and even relative tensors, which transform according to the same formula except that $ \Delta $ is replaced by its absolute value $ | \Delta | $. In [a2] an even tensor density is simply called a "tensor density" and an odd one is called a tensor $ \Delta $- density.

References

[a1] M. Spivak, "A comprehensive introduction to differential geometry" , I , Publish or Perish (1970) pp. 437ff
[a2] J.A. Schouten, "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer (1954) pp. 12 (Translated from German)
How to Cite This Entry:
Tensor density. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tensor_density&oldid=48956
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article