In the following picture, $U \subset \Bbb{R}^m$ is an open set, $f: U \to \Bbb{R}^{m+n}$ is a parameterization, $\gamma: I \to U$ is a smooth curve and $\tilde{\gamma} = f \circ \gamma$.
The function $$g_{ij}(x) = \delta_{mn}\frac{\partial f^m}{\partial x_i}(\gamma(t))\frac{\partial f^j}{\partial x_j}(\gamma(t))$$ is the First Fundamental Form.
I think the index $m$ and $n$ are just a bad notation and, certainly $1/2$ is a misprint. My question is about the summation of the inner product and the chain rule. By definition of inner product $$\langle \frac{d}{dt}\tilde{\gamma}(t), \frac{d}{dt}\tilde{\gamma}(t) \rangle = \sum_{i,j = 1}^{m+n}\delta_{ij}\frac{d}{dt}\tilde{\gamma}_i(t)\frac{d}{dt}\tilde{\gamma}_j(t)$$ and $$\frac{d}{dt}\tilde{\gamma}_i(t) = \frac{d}{dt}(f_i \circ \gamma)(t) = \sum_{r=1}^{m} \frac{\partial}{\partial x_r}f_i(\gamma(t))\frac{d}{dt}\gamma_r(t).$$ So, I don't know why the equality on the picture is true.