 Pseudotensor

Under an orthogonal coordinate transformation $R$, a Cartesian pseudotensor has the usual tensor transformation law multiplied by $\det R$. Thus proper rotation matrices act as on ordinary tensors, while an orientation-reversing change contributes an extra minus sign. Interpreted as a rank-three pseudotensor, the Levi-Civita symbol has the same numerical components in every orthonormal frame, because its ordinary three-index transformation already contributes a factor $\det R$.

A pseudotensor is a mathematical object similar to a tensor, but it behaves differently under transformations, specifically under improper transformations such as reflections or parity transformations. While a regular tensor (like a vector or a second-order tensor) transforms according to certain rules under coordinate changes, a pseudotensor will change its sign under these transformations. To be more specific, pseudotensors come into play in various areas of physics, especially in the context of fields such as general relativity and continuum mechanics.