Trying to understand the following calculation：
The key confusion for me is the fact that $\partial_{\bar j} g^{p \bar q} = g^{p \bar s} g^{r \bar q} \partial_{\bar j} g_{r \bar s} $(where $g^{i\bar j}$ is the (i,j)th entry of the inverse of the metric) . According to the notes, they get this formula due to the fact that $\delta A^{-1} = - A^{-1} \delta A A^{-1}$.
My understanding is that componentwisely $(\delta A^{-1})_{ij} = - (A^{-1})_{im} (\delta A)_{mn} (A^{-1})_{nj}$，so we should really get $\partial_{\bar j} g^{p \bar q} = g^{p \bar r} g^{s \bar q} \partial_{\bar j} g_{r \bar s} $ according to this formula.
In our case: $\partial_{\bar j} g^{p \bar q}$ is the pq-entry of the derivative of the inverse of metric. So according to the formula above it should be the product of:
- pr-entry($g^{p \bar r}$) $\times$ rs-entry($\partial_j g_{r \bar s}$) $\times$ sq-entry ($g^{s\bar q}$).(with summation over r and s).
So the formula in the notes really got me confused, and I have been thinking about this for a very long time since I am new to Riemannian geometry. Any help would be very much appreciated.
Note that you dropped a negative sign in your discussion. You have to be careful with barred and unbarred indices, so as to maintain the summation convention. Thus, you should rewrite your formula with barred and unbarred indices more carefully. They also switched dummy summation variables at the end to add some confusion. From $\partial_{\bar j}g^{p\bar q} = -g^{p\bar s}g^{r\bar q}\partial_j g_{r\bar s}$, we get \begin{align*} -g_{p\bar\ell}\partial_{\bar j}g^{p\bar q} &= g_{p\bar\ell}g^{p\bar s}g^{r\bar q}\partial_j g_{r\bar s} = \delta^{\bar s}_{\bar\ell}g^{r\bar q}\partial_j g_{r\bar s} \\ &= g^{r\bar q}\partial_j g_{r\bar\ell} = g^{p\bar q}\partial_j g_{p\bar\ell}, \end{align*} as desired.
EDIT: Just to clarify, let's quickly derive the formula for the derivative of the inverse. From $$0 = D(\delta_i^j) = D(h_{i\bar j}h^{k\bar j})$$ and the product rule, we get $$0 = (Dh_{i\bar j})h^{k\bar j} + h_{i\bar j}(Dh^{k\bar j}),$$ so, multiplying by $h^{i\bar\ell}$ (and summing), we get $$-h^{i\bar\ell}(Dh_{i\bar j})h^{k\bar j} = h^{i\bar\ell}h_{i\bar j}(Dh^{k\bar j}) = \delta^{\bar\ell}_{\bar j}(Dh^{k\bar j}) = Dh^{k\bar\ell},$$ as needed. Note that your "inconsistencies in matrix multiplication" occur throughout.