I am trying to figure out how to compute $\frac{\partial |Z|}{Z_{ij}}$ where $|Z|$ is the volume element and $Z_{ij}$ is the covariant metric tensor entries. Apparently the answer is given as:
$$ \frac{\partial Z}{\partial Z_{ij} }= ZZ^{ij}$$
My question is how did the determinant of $Z$ come again in RHS? I can't figure out how to prove the above. Source: Pavel Grinfeld's Tensor Calculus book
Here is what I know:
$$ \frac{\partial |A|}{\partial a_l^u} = A_u^l$$
Where $A_u^l$ is the cofactor's determinant, $|A|$ is det of matrix $A$ and $a_l^u$. This is also why I find the derivative expression weird. Here there is no determinant of the original matrix on RHS.
