If I have a function $f(x,y,z)$, $f: R^3 \rightarrow R^3$ and $g(s,t)=(x(s,t),y(s,t),z(s,t))$, $g:R^2 \rightarrow R^3$. Can I say that:
$\frac{\partial f(x(s,t),y(s,t),z(s,t))}{\partial (x(s,t),y(s,t),z(s,t))}\frac{\partial (x(s,t),y(s,t),z(s,t))}{\partial s}=\frac{\partial f(x(s,t),y(s,t),z(s,t))}{\partial s}$, that is:
$Df(g(s,t))\frac{\partial g(s,t)}{\partial s}=\frac{\partial f(g(s,t))}{\partial s}$ ?
If not, is there a way to write $\frac{\partial f(g(s,t))}{\partial s}$ in terms of the Jacobian matrix of $f$?
