State and prove the chain rule for differentiable mappings $F: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ and $G: \mathbb{R}^{m} \rightarrow \mathbb{R}^{k}$.

Suppose now $F: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ has image lying on the unit circle in $\mathbb{R}^{2}$. Prove that the determinant $\operatorname{det}\left(\left.D F\right|_{x}\right)$ vanishes for every $x \in \mathbb{R}^{2}$.