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Difference between revisions of "Pseudo-tensor"

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m (tex encoded by computer)
m (fix tex)
 
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  \dots  
 
  \dots  
 
\frac{\partial  
 
\frac{\partial  
x ^ {k _ {n} } }{\partial  \overline{x}\; {} ^ {j _ {n} } }
+
x ^ {k _ {n} } }{\partial  \overline{x}\, {} ^ {j _ {n} } }
 
  ,
 
  ,
 
$$
 
$$
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of the coordinate transformation. In [[#References|[a1]]] the following cases are distinguished:
 
of the coordinate transformation. In [[#References|[a1]]] the following cases are distinguished:
  
i)  $  \tau = \Delta  ^ {-} w \overline \Delta \; {} ^ {- w  ^  \prime  } $,  
+
i)  $  \tau = \Delta  ^ {-w} \overline \Delta \, {} ^ {- w  ^  \prime  } $,  
a tensor  $  \Delta $-
+
a tensor  $  \Delta $-density of weight  $  w $
density of weight  $  w $
 
 
and anti-weight  $  w  ^  \prime  $;
 
and anti-weight  $  w  ^  \prime  $;
  
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tensor.
 
tensor.
  
Here  $  \overline \Delta \; $
+
Here  $  \overline \Delta $
 
is the complex conjugate of  $  \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]]).
 
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 $-
+
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 $
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.
 
and anti-weight 0 a tensor capacity.
  
 
====References====
 
====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>
 
<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>

Latest revision as of 17:42, 3 January 2021


A tensor considered up to multiplication by an arbitrary function (cf. 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 [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).

In [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

[a1] J.A. Schouten, "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer (1954) pp. 11ff (Translated from German)
[a2] R. Sauer (ed.) I. Szabó (ed.) , Mathematische Hilfsmittel des Ingenieurs , III , Springer (1968) pp. Sect. G.II.6
How to Cite This Entry:
Pseudo-tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-tensor&oldid=48354