Jacobian and chain rule

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Problem

Let $F(x,y) = (xy, x+y^2)$ and $g(u,v) = u+v^2$ . Compute $J_F$ at $(1,2)$ and the gradient of $g(F(x,y))$ at $(1,2)$ .

Reveal answer or reference solution

J_F(1,2) = \begin{bmatrix} 2 & 1 \\ 1 & 4 \end{bmatrix}.

Since $F(1,2)=(2,5)$ , we have $\nabla g=(1,10)$ in $(u,v)$ coordinates. Therefore,

J_F(1,2)^{\mathsf T}\nabla g(F(1,2)) = (12,41).

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Appears in

Mathematics Diagnostic
Calculus Practice