What is the first fundamental form? The first fundamental form of a surface $S$ at a point $p$ is "the quadratic form on the tangent plane $S_p$ inherited from the inner product structure of $\mathbb R^3$".
At the same time, the first fundamental form is apparently supposed to describe the surface at $p$ in some way, for instance you can compute Gaussian curvature from it. In particular, the first fundamental form should be different for different surfaces $S$.
So what is the first fundamental form?


*

*It can't be the actual quadratic form $\langle x, x\rangle$ on $S_p$, that is, it can't simply be a function from $\mathbb R^3$ into $\mathbb R$, because that's the same for any surface $S$ that has the same tangent plane at $p$.

*It can't be the triplet of coefficients $(E, F, G)$ either, since they depend on the parameterization used, and in fact I think by using the right parameterization we can get any triplet $(E, F, G)$ we want.
 A: $\newcommand{\Vec}[1]{\mathbf{#1}}\newcommand{\Brak}[1]{\left\langle #1\right\rangle}$There's a specific, formally simple relationship between a coordinate system on a regular surface and the components of the first fundamental form: In customary notation, if $\Vec{x}$ is a regular parametrization, then
$$
E(u, v) = \Brak{\Vec{x}_{u}, \Vec{x}_{u}},\qquad
F(u, v) = \Brak{\Vec{x}_{u}, \Vec{x}_{v}},\qquad
G(u, v) = \Brak{\Vec{x}_{v}, \Vec{x}_{v}}.
$$
The subtle points are:


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*Although two surfaces sharing a tangent plane at a point $p$ have "the same first fundamental form at $p$", they do not generally have equivalent first fundamental forms in any open neighborhood of $p$;

*Perhaps surprisingly, it's not true that by using the right parameterization we can get any triplet $(E, F, G)$ we want. If it were, there would be no local invariants of a surface, because one could always pick, say, $E = G = 1$ and $F = 0$.
Globally, the first fundamental form of a surface is an inner-product-valued function on a surface.
Locally, the first fundamental form of a surface is represented by an association to each coordinate system of a $2 \times 2$ matrix-valued function whose value at each point is positive-definite, and which "transforms like an inner product" under change of coordinate. The first fundamental form itself may be viewed as the resulting equivalence class of all such pairs of "coordinate system and matrix-valued function".
