I'm self-study Riemannian Geometry in order to be able to understand this lecture notes about Mean Curvature Flow. I'm reading the first chapter, which is a review of Riemannian Geometry, and I'm stuck in the following part:
For local choice of unit normal vector field $\nu$, the second fundamental form of $M$, a $(0,2)-$tensor is given by
$$A(V,W) = - \langle S_V W, \nu \rangle_e = \langle W, \overline{\nabla}_V \nu \rangle_e$$
or in coordinates $A = (h_{ij})$ by
$$h_{ij} = - \left\langle \frac{\partial^2 F}{\partial x_jx_i},\nu \right\rangle_e = \left\langle \frac{\partial F}{\partial x_i},\frac{\partial \nu}{\partial x_j} \right\rangle_e.$$
A relevant information on the lecture notes is
We restrict ourselves to manifolds of codimension $1$ in an Euclidean ambient space, i.e. we consider a $n$-dimensional smooth manifold $M$, without boundary, either closed or complete and non-compact and an immersion (or embedding)
$$F: M \longrightarrow \mathbb{R}^{n+1}$$
We call the image $F(M)$ a hypersurface. We will often identify points on M with their image under the immersion, if there is no risk of confusion. Let $x = (x_1, \cdots , x_n)$ be a local coordinate system on M[...] We denote by
$$g_{ij} = \left\langle \frac{\partial F}{\partial x_i}, \frac{\partial F}{\partial x_j} \right\rangle$$
My doubt is why the expression for $h_{ij}$ is that from above?
$\textbf{My attempt in order to understand the coordinates $h_{ij}$:}$
Firstly, I think that $h_{ij} = A \left( \frac{\partial F}{\partial x_i}, \frac{\partial F}{\partial x_j} \right)$, then
$h_{ij} = - \left\langle S_{\frac{\partial F}{\partial x_i}} \frac{\partial F}{\partial x_j}, \nu \right\rangle_e = - \left\langle \left( \overline{\nabla}_{\frac{\partial F}{\partial x_i}} \frac{\partial F}{\partial x_j} \right)^{\perp}, \nu \right\rangle_e$
and
$h_{ij} = \left\langle \frac{\partial F}{\partial x_j}, \overline{\nabla}_{\frac{\partial F}{\partial x_i}} \nu \right\rangle_e$
Secondly, I think that if $U_{\alpha} \subset \mathbb{R}^n$ are open sets and $\phi_{\alpha}: U_{\alpha} \longrightarrow M$ are charts of the manifold $M$, then $F \circ \phi_{\alpha}: U_{\alpha} \longrightarrow F(M) \subset \mathbb{R}^{n+1}$ are charts of the submanifold $F(M)$, therefore the $\frac{\partial}{\partial x_i} = d (F \circ \phi_{\alpha}) (e_i)$ for $i = 1, \cdots, n$, where $e_i$ is an element of canonic basis of $\mathbb{R}^n$, but I don't sure about this because in the literature of Riemmanian Geometry $\frac{\partial}{\partial x_i}$ denote an element of coordinate basis of $T_pM$.
Thanks in advance!