I'm having some trouble understanding the first fundamental form of a manifold. Here are the definitions I'm working with (from Milnor's Morse Theory):
Let $M$ be a $k$-manifold differentiably embedded in $\mathbb R^n$ for some $k<n$. Suppose a region of $M$ has coordinates $(u^1,\dots,u^k)$. The inclusion map $M\hookrightarrow\mathbb R^n$ defines functions $x_1,\dots,x_n$. Let $\vec x=(x_1,\dots,x_n)$.
Then define the first fundamental form to be $$(g_{ij})=\left(\frac{\partial\vec x}{\partial u_i}\cdot\frac{\partial\vec x}{\partial u_j}\right).$$
(I assume Milnor's $u_i$ and $u_j$ are supposed to be $u^i$ and $u^j$ here?)
From what I could tell from Wikipedia, this form is supposed to act as the inner product, namely by taking $(v,w)$ to $v^T(g_{ij})w$. But this doesn't make a lot of sense, because Milnor says later that it will help to assume that the coordinates have been chosen so that $g_{ij}$, evaluated at $\vec q$, is the identity matrix, which means that $(g_{ij})$ really does just act on a single vector. Could someone explain this? I feel like I'm missing something obvious in the definitions here.