Let $f:\mathbb{R}^{3}\to\mathbb{R}^{2}$ be given by $f(r,s,t)=(r^{3}s+t^{2},rst)$. Let $g:\mathbb{R}^{2}\to\mathbb{R}^{11}$ and $h:\mathbb{R}^{11}\to\mathbb{R}^{4}$ be two differentiable functions. Compute $$ J_{h\circ g\circ f}(0,21,0). $$
I have the above question which I solved to be a zero matrix so not asking anyone to solve the question for me, however I do want to ask what the size of the resulting Jacobian matrix would be for this question?
The Jacobian of any differentiable function $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$ has dimensions $m\times n$, so determine just determine the $m,n$ values for the composite function $h\circ g\circ f$.