Pseudotensor density

The term pseudotensor density denotes a tuple of numbers whose values depend on the chosen basis of a vector space . This dependency is sufficient for a change of basis, similar to transformation formulas that apply to the components of a tensor . The only difference compared to a tensor is that in the case of a pseudotensor density, the transformation is always multiplied by a power of the amount of the Jacobite terminant and by its sign .

Definition and example

For any ordered bases B of an n-dimensional vector space V, the quantities in a basis transformation from one ordered basis to another ordered basis always like the formula ${\displaystyle T_{{j_{1}}\dots {j_{m}}}(B);\ j_{1},\ldots ,j_{m}\in \{1,\ldots ,n\}}$ ${\displaystyle C=(c_{1},\ldots ,c_{n})}$ ${\displaystyle C'=(c_{1}',\ldots ,c_{n}')}$

${\displaystyle T_{{i_{1}}\dots {i_{m}}}(C')=\sum _{j_{1}=1}^{n}\dots \sum _{j_{m}=1}^{n}\operatorname {sgn}(\Delta )|\Delta |^{\beta }\,a_{i_{1}}^{j_{1}}\dots a_{i_{m}}^{j_{m}}\,T_{{j_{1}}\dots {j_{m}}}(C);\ i_{1},\ldots ,i_{m}\in \{1,\ldots ,n\}}$

fulfill. Here denote the transformation matrix for the basic transition from C to C ', i.e. H. , and denote the determinant of this transformation matrix. ${\displaystyle (a_{i}^{j})}$ ${\displaystyle c_{i}'=\sum _{j=1}^{n}a_{i}^{j}c_{j},\ i\in \{1,\ldots ,n\}}$ ${\displaystyle \Delta }$

Then we call the set of an m-fold covariant pseudotensor density of weight . ${\displaystyle T_{{j_{1}}\dots {j_{m}}}(B);\ j_{1},\ldots ,j_{m}\in \{1,\ldots ,n\}}$ ${\displaystyle \beta }$

In analogy to tensors, one can also define contravariant and mixed pseudotensor densities.

For one speaks of a pseudotensor . A simply co- or contravariant pseudotensor is called a pseudo vector . ${\displaystyle \beta =0}$

An example of a covariant pseudotensor density of weight −1 (with m = n) is the Levi-Civita symbol . With him, the sizes remain unchanged when the base is changed . ${\displaystyle T_{j_{1}\dots j_{n}}(B)}$

See also

• Tensor density

Individual evidence

1. ^ G. Grosche, V. Ziegler, D. Ziegler, E. Zeidler (Eds.) Teubner-Taschenbuch der Mathematik, Part II . 8th edition. BG Teubner Verlag, Wiesbaden, November 2003, ISBN 3-519-21008-8 , p. 242.