Pseudotensor

The term Pseudotensordichte denotes a tuple of numbers whose values ​​depend on the chosen basis of a vector space. Here, this dependence is sufficient for a change of basis similar transformation formulas as they apply to the components of a tensor. The difference from a tensor consists only in that each is multiplied at a Pseudotensordichte for transformation with a power of the magnitude of the Jacobian as well as with their signs.

Definition and Example

The sizes like for a basic transformation of a parent base to another parent basis for any ordered bases B of an n- dimensional vector space V always the formula

. meet It denotes the transformation matrix for the base transition from C to C ', ie, And denote the determinant of the transformation matrix.

Then we call the set of an m -fold covariant Pseudotensordichte by weight.

Correspondingly, one can define tensors and contravariant and mixed Pseudotensordichten analogy.

For one speaks of a pseudotensor. A simple co- or contravariant pseudotensor called a pseudo vector.

An example of a covariant Pseudotensordichte the weight -1 ( m = n) is the Levi- Civita symbol. For him, the sizes remain unchanged at a change of basis.