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''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|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 [[#References|[a2]]] an even tensor density is simply called a  "tensor density"  and an odd one is called a tensor  $  \Delta $-
density.

====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>