Let $F:A \to B$ map subsets of $\mathbb{R}^n$ with inverse $F^{-1}$.
Let $d(\cdot) = \text{det} \mathbf{D}F(\cdot)$ with $\mathbf{D}$ denoting the total derivative matrix.
Am I correct that $d \circ F^{-1} = \text{det} \mathbf{D}F(F^{-1}) = \text{det} \mathbf{D}\text{(Id)} = 1$.
I am not entirely sure of the what the definition of $d$ really means so I better ask this question. Thanks.