The derivatives of a diffeomorphism I read this:

Let $F_t^s:\Omega_s \to \Omega_t$ be a diffeomorphism between hypersurfaces. Let $\text{det}D_{\Omega_s}F_t^s(\cdot)$ be the Jacobian of the matrix representation of $D_{\Omega_s}F_t^s(\cdot)$ with respect to the orthogonal basis of the respective tangent space.

My question is, what exactly is $D_{\Omega_s}F_t^s(\cdot)$? Why should it not be just the usual matrix of partial derivatives of the components of $F_t^s$? And what is meant by "with respect to the orthogonal basis...". I understand there is a tangent space associated to each point in the surface $\Omega_s$, which is a vector space, so it has a basis. But that's about it.

$D_{\Omega_s}F_t^s(\cdot)$ means the differential of $F_t^s$, which is the induced linear map between tangent spaces. We can always choose basis from the two tangent spaces, and represent this linear map, as a matrix.
