Confusion about the first fundamental form? I am reading about the first fundamental form. The definition on Wolfram is:

Let $M$ be a regular surface with $v_p, w_p$ points in the tangent space $M_p$ of $M$. Then, the first fundamental form is the inner product of tangent vectors: $I(v_p, w_p) = v_p \cdot w_p$.

So, if I understand correctly, the first fundamental form is actually a map $I: M_p \times M_p \to \mathbb{R}$. Is that correct?
Now, Wikipedia claims something similar.

The first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of $\mathbb{R}^3: I(x,y) = \langle x, y \rangle$.

However, it then proceeds to say that the first fundamental form may be represented as a symmetric matrix: $$ I(x,y) = x^T \begin{bmatrix}  E & F \\ F & G   \end{bmatrix} y,$$ where $E, F, G$ are the coefficients of the first fundamental form.
This is super weird to me, because this second representation strikes me as different from the first one. However, I suspect that $x$ and $y$ in the second interpretation are not the same $x$ and $y$ as in the definition. However, I can't make much sense of what exactly they are and in what sense they are different.
I would appreciate it if someone clarified these things to me as  the confusion is driving me crazy.
 A: $\newcommand{\brak}[1]{\left\langle#1\right\rangle}\newcommand{\dd}{\partial}$The first definition meshes with the second if we let the point $p$ of tangency range over $M$, and for each $p$ we let $v_{p}$ and $w_{p}$ range over all ordered pairs of tangent vectors to $M$ at $p$.
The functions in the wikipedia definition are not intrinsic to the geometry of $M$, but instead depend on a coordinate system $(s, t)$ for $M$. If $(\dd_{s}, \dd_{t})$ denotes the associated coordinate frame, the resulting components of the first fundamental form are defined by
\begin{align*}
  E(s, t) &= \brak{\dd_{s}, \dd_{s}}, \\
  F(s, t) &= \brak{\dd_{s}, \dd_{t}}, \\
  G(s, t) &= \brak{\dd_{t}, \dd_{t}}.
\end{align*}
We may then let $x$ and $y$ denote ordinary vectors in Euclidean three-space that are tangent to $M$ at some point $p$, write each as a linear combination of the coordinate frame $(\dd_{s}, \dd_{t})$, and use the resulting ordered pairs to form Wikipedia's matrix product.
There are technical definitions, for which search terms include tangent bundle, Riemannian metric, and tensor components. Those are all well-explained elsewhere on-site.
