Let $F : \mathbb{R}^2 \rightarrow \mathbb{R}^2, u : \mathbb{R}^2 \rightarrow \mathbb{R},$ both $C^{\infty}$ and $ g (x,y) = u(F(x,y))$. What is $\partial_x g$? And $\partial^2_{x,x} g$?
Let $ F(x,y) = (F_1(x,y), F_2(x,y)), u=u(a,b)$.
$$ DF =
\begin{pmatrix}
\partial_x F_1 & \partial_y F_1\\
\partial_x F_2 & \partial_y F_2
\end{pmatrix}, \;
Du=
\begin{pmatrix}
\partial_a u & \partial_b u
\end{pmatrix}
$$
How should I apply $ ((Du) \circ F) \cdot (DF) $ ?