View source for Pseudo-tensor

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A tensor considered up to multiplication by an arbitrary function (cf. [[Tensor on a vector space|Tensor on a vector space]]).

====Comments====
More precisely, a pseudo-tensor (also called relative tensor) is a quantity  $  p _ {j _ {1}  \dots j _ {n} } ^ {i _ {1} \dots i _ {m} } $
which under a coordinate change transforms as

$$ 
\overline{p}\; {} _ {j _ {1}  \dots j _ {n} } ^ {i _ {1} \dots i _ {m} }  = \ 
\tau ( \overline{x}\; )
p _ {k _ {1}  \dots k _ {n} } ^ {l _ {1} \dots l _ {m} }

\frac{\partial  \overline{x}\; {} ^ {i _ {1} } }{\partial  x ^ {l _ {1} } }
 \dots 
\frac{\partial 
\overline{x}\; {} ^ {i _ {m} } }{\partial  x ^ {l _ {m} } }
 \cdot

\frac{\partial  x ^ {k _ {1} } }{\partial  \overline{x}\; {} ^ {j _ {1} } }
 \dots 
\frac{\partial 
x ^ {k _ {n} } }{\partial  \overline{x}\, {} ^ {j _ {n} } }
 ,
$$

where  $  \tau $
is a scalar-valued function. Most frequently (in applications), the function  $  \tau $
depends in a simple manner on the Jacobian determinant  $  \Delta =  \mathop{\rm det}  ( {\partial  \overline{x}\; {}  ^ {i} } / {\partial  x  ^ {j} } ) $
of the coordinate transformation. In [[#References|[a1]]] the following cases are distinguished:

i)  $  \tau = \Delta  ^ {-w} \overline \Delta \, {} ^ {- w  ^  \prime  } $, 
a tensor  $  \Delta $-density of weight  $  w $
and anti-weight  $  w  ^  \prime  $;

ii)  $  \tau = | \Delta |  ^ {w} $, 
a tensor density of weight  $  w $;

iii)  $  \tau = \Delta / | \Delta | $, 
a  $  W $-
tensor.

Here  $  \overline \Delta $
is the complex conjugate of  $  \Delta $. 
A tensor density of weight zero is an ordinary tensor (cf. [[Tensor on a vector space|Tensor on a vector space]]).

In [[#References|[a2]]] a tensor  $  \Delta $-density of weight 1 and anti-weight 0 is called a tensor density and a tensor  $  \Delta $-density of weight  $  - 1 $
and anti-weight 0 a tensor capacity.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.A. Schouten,   "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer  (1954)  pp. 11ff  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Sauer (ed.)  I. Szabó (ed.) , ''Mathematische Hilfsmittel des Ingenieurs'' , '''III''' , Springer  (1968)  pp. Sect. G.II.6</TD></TR></table>