Suppose you have $f(x_1,.....x_n)\colon R^n \to R$ and $g_1,...g_n\colon R^m \to R$.
Then $f(g_1,....g_n)$=$f\circ(g_1,...g_n)$ :$R^m \to R$, right?
Any clear definition on composition of multivariable functions? Also I had problem while studying Spivak's Calculus on Manifolds. The Chain Rule theorem states
$$D(f\circ g)(a)=Df(g(a)) \circ Dg(a).$$ Now suppose $f,g\colon R^n\to R$ and $H(x,y)\colon R^2 \to R$. Then $$H\circ (f,g)=H(f,g)\tag{1}$$
So $$D(H(f,g))=DH \circ (f,g)=DH \circ D(f,g)$$ Now does the last one equal $D(f,g)=(Df,Dg)$? So $(1)$ becomes $DH \circ (Df,Dg)$?? Is that true and why? I'm trying to understand the proofs of the derivatives of multiplication and addition on multivariable functions.