My question is rather straightforward.
Let $d \geq 3$, then given the following parametrization $$ z_i = a_i/|a|,\ i=1,2,...,d-1 $$ where $|a|^2 =1+ \sum_{i=1}^{d-1} a_i^2$. What is $$ d z_1 \wedge d z_2 \wedge ... \wedge d z_{d-1} =? $$
Computing the differential for $d=3,4$ one would expect that the following is true for general $d$ ( if my calculations are correct): $$ d z_1 \wedge d z_2 \wedge ... \wedge d z_{d-1} = \frac{1}{|a|^{d+1}} da_1 \wedge da_2 \wedge ... \wedge d a_{d-1}, $$ however proving that this holds in generality seems to elude me. So far I've attempted to do induction on $d$ as well as trying to derive some general expression for $$ d z_1 \wedge ... \wedge d z_k,\ 1 \leq k \leq d-1, $$ but I seem to fail to obtain anything useful.
Any help would be greatly appreciated.
I found the answer by applying the Matrix determinant lemma: https://en.wikipedia.org/wiki/Matrix_determinant_lemma, and recognizing that the Jacobian is given by $J_{ij} = \frac{\delta_{ij}}{|a|} - \frac{a_i a_j}{|a|^3}$ where $\delta_{ij}$ is the Kronecker delta.