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Difference between revisions of "Tensor density"

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components  $  a _ {j _ {1}  \dots j _ {q} } ^ {i _ {1} \dots i _ {p} } $,  
 
components  $  a _ {j _ {1}  \dots j _ {q} } ^ {i _ {1} \dots i _ {p} } $,  
 
$  1 \leq  i _  \nu  , j _  \mu  \leq  n $,  
 
$  1 \leq  i _  \nu  , j _  \mu  \leq  n $,  
transforming under a change of coordinates  $  x \mapsto y = ( y  ^ {1} \dots y  ^ {n} ) $
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transforming under a change of coordinates  $  x \mapsto y = ( y  ^ {1}, \dots, y  ^ {n} ) $
 
according to the formula
 
according to the formula
  
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\frac{\partial  y ^ {i _ {p} } }{\partial  x ^ {\alpha _ {p} } }
 
\frac{\partial  y ^ {i _ {p} } }{\partial  x ^ {\alpha _ {p} } }
\cdot
 
$$
 
  
$$
 
\cdot
 
  
 
\frac{\partial  x ^ {\beta _ {1} } }{\partial  y ^  
 
\frac{\partial  x ^ {\beta _ {1} } }{\partial  y ^  

Latest revision as of 10:34, 5 March 2022


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

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

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=52196
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article