I am taking a course on tensor calculus and differential geometry, and I am encountering some trouble understanding the following procedure on tensor transformation laws presented in the lecture notes. Let me give a short summary here of what is discussed.
Given:
- $n$ - dimensional vector space $V$ over $\mathbb{R}$ with corresponding dual space $V^*$
- Fixed vector basis $\left\{\bf{e_i} \right\}$
- A unit volume form $\bf{\mu}$ relative to the fixed basis such that $\bf{\mu}(\bf{e_1,..., e_n}) = 1$
If we denote the basis of the dual space as $\left\{\bf{\hat{e}^i}\right\}$, and introduce a change of basis such that $\bf{\hat{e}^i} = A^i_ j\bf{\hat{f}^j}$ with $\left\{\bf{\hat{e}^j}\right\}$ the new basis covectors and $\bf{A}$ the Jacobian of the transformation then we can write the following decompositions. $$\bf{\mu} = \mu_{i_1....i_n}\bf{\hat{e}^{i_1}} \otimes ... \otimes \bf{\hat{e}^{i_n}} = \bar{\mu}_{i_1....i_n}\bf{\hat{f}^{i_1}} \otimes ... \otimes \bf{\hat{f}^{i_n}}.$$
Here $\mu_{i_1....i_n}$ and $\bar{\mu}_{i_1....i_n}$ denote the coefficients to the two bases. When one substitutes the change of basis for the covectors $\left\{\bf{\hat{e}^i}\right\}$ into these expressions it is straightforward to derive that, $$\bar{\mu}_{i_1....i_n} = A^{i_1}_{j_n}...A^{i_1}_{j_n}\mu_{i_1....i_n}.$$
Up to this point I understand. Now however they state that to make the decomposition coefficients invariant under a change of basis, one should actually switch to the relative transformation law, which reads as follows.
$$\bar{\mu}_{i_1....i_n} = \frac{1}{\det{A}} A^{i_1}_{j_n}...A^{i_1}_{j_n}\mu_{i_1....i_n}.$$
I understand that given this definition it holds that $\bar{\mu}_{i_1....i_n} = \bar{\mu}_{i_1....i_n}$, which can be derived by using the fact that for the unit volume form $\mu_{i_1....i_n} = \left[i_1, ..., i_n \right]$, i.e. the fully antisymmetric symbol.
What I don't understand however is why one is suddenly allowed to change the tensor transformation law, when it was derived using the change of basis and thus to me seems as a given. Why is it okay to suddenly define a different transformation law, which to my eye is off by a factor $\frac{1}{\det{A}} $ from the actual transformation law that one can actually derive. Any help would be greatly appreciated!
Edit: After reading into this a little more I think my question can actually be rephrased. Why do relative tensors (or tensor densities) that obey a relative transformation law exist at all? If the tensor transformation law is directly defined from the change of basis, then how can there exist "relative tensors" for which this law does not hold?