What does it mean for a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ of several variables to be differentiable at a point $x$ ? State and prove the chain rule for functions of several variables. For each of the following two functions from $\mathbb{R}^{2}$ to $\mathbb{R}$, give with proof the set of points at which it is differentiable:

$$\begin{aligned} &g_{1}(x, y)= \begin{cases}\left(x^{2}-y^{2}\right) \sin \frac{1}{x^{2}-y^{2}} & \text { if } x \neq \pm y \\ 0 & \text { otherwise; }\end{cases} \\ &g_{2}(x, y)= \begin{cases}\left(x^{2}+y^{2}\right) \sin \frac{1}{x^{2}+y^{2}} & \text { if at least one of } x, y \text { is not } 0 \\ 0 & \text { if } x=y=0\end{cases} \end{aligned}$$