In physics, a tensor is defined as a multidimensional array with a special transformation law.
Therefore, a tensor of type $(r, s)$ is an geometric object
$T_{i_{1}\dots i_{s}}^{j_{1}\dots j_{r}}[\,\underline{e}\,]$
to each basis $\underline{e} = (e_1, ..., e_n)$ of an n-dimensional vector space such that the multidimensional array obeys the transformation law
$T_{i^{\prime}_{1}\dots i^{\prime}_{s}}^{j^{\prime}_{1}\dots j^{\prime}_{r}}[M\cdot \underline{e}\,]=(M^{-1})_{j_{1}}^{j^{\prime}_{1}}\cdot\dots\cdot (M^{-1})_{j_{r}}^{j^{\prime}_{r}}\cdot T_{i_{1}\dots i_{s}}^{j_{1}\dots j_{r}}\cdot M_{i_{1}^{\prime}}^{i_{1}}\cdot\dots\cdot M_{i_{s}^{\prime}}^{i_{s}}$
where $M$ is the transformation matrix.
In calculus this objects can be defined more abstract: There a tensor of type (r,s) is an element of the abstract tensor product
$T\in \underbrace{V\otimes \dots \otimes V}_{r-\text{times}}\otimes \underbrace{V^{\ast}\otimes\dots\otimes V^{\ast}}_{s-\text{times}}$
or, because of can. isomorphy, a tensor can be viewed as a multilinear function
$T:\underbrace{V^{\ast}\times\dots\times V^{\ast}}_{r-\text{times}}\times \underbrace{V\times \dots \times V}_{s-\text{times}}\to \mathbb{R}$.
Now to my question: In physics, there is also the definition of a ''pseudo-tensor'', which is an geometric object
$T_{i_{1}\dots i_{s}}^{j_{1}\dots j_{r}}[\,\underline{e}\,]$
with the the transformation law
$T_{i^{\prime}_{1}\dots i^{\prime}_{s}}^{j^{\prime}_{1}\dots j^{\prime}_{r}}[M\cdot \underline{e}\,]=\mathrm{sign}(\mathrm{det}(M))\cdot(M^{-1})_{j_{1}}^{j^{\prime}_{1}}\cdot\dots\cdot (M^{-1})_{j_{r}}^{j^{\prime}_{r}}\cdot T_{i_{1}\dots i_{s}}^{j_{1}\dots j_{r}}\cdot M_{i_{1}^{\prime}}^{i_{1}}\cdot\dots\cdot M_{i_{s}^{\prime}}^{i_{s}}$
Is there also an abstract definition for pseudo-tensors?
Thank you!
