I am trying to understand how to apply the chain rule. Suppose that I have a sufficiently smooth real valued function of two real variables $w(x,y)$.
Let us say that we also have a change of variables $(x,y) = F(\hat{x},\hat{y})$, and we define $\hat{w} = w \circ F$. In this case, I know that the chain rule gives us $$ \frac{\partial w}{\partial x} = \frac{\partial \hat{w}}{\partial \hat{x} }\frac{\partial \hat{x}}{\partial x} + \frac{\partial \hat{w}}{\partial \hat{y}}\frac{\partial \hat{y}}{\partial x}. $$
Now, I want to assume that $\hat{x}$ and $\hat{y}$ are both functions of a variable $s$. Say, $(\hat{x},\hat{y})=(\gamma_1(s),\gamma_2(s))$. In this situation, how can I compute $ \frac{\partial [w\circ F \circ (\gamma_1(s),\gamma_2(s))]}{\partial x}$?
EDIT: The comment below says I should examine the Jacobain matrices. So, for the first case we would have $$ \bigg[\frac{\partial w}{\partial x},\frac{\partial w}{\partial y}\bigg] = \bigg[\frac{\partial \hat{w}}{\partial \hat{x}},\frac{\partial \hat{w}}{\partial \hat{y}}\bigg] \begin{bmatrix} \frac{\partial \hat{x}}{\partial x}&\frac{\partial \hat{x}}{\partial y} \\ \frac{\partial \hat{y}}{\partial x}&\frac{\partial \hat{y}}{\partial y} \end{bmatrix} . $$ It is still not clear to me what the Jacobian for the second mapping would be. Is it $[\frac{\partial s}{\partial \hat{x}},\frac{\partial s}{\partial \hat{y}}]^T$?