Given that $F(x,y)=(x^2+y, x−y)$ and $G(u,v)=(u^2, u−2v, v^2)$, compute the Jacobian derivative matrix of function $G ◦ F$ at the point $(x, y) = (1, 1)$ using the Chain Rule
I was wondering if my solution is correct because my peer had a different solution where they found the Jacobian derivative matrix of $F(x,y)$ and $G(u,v)$ separately and then multiplied the results.
Thank you.
Well, you did not use the chain rule, but reached the correct answer.
It sounds like your friend tried to use the chain rule but made an error somewhere.
Recall that the chain rule is:
$$\mathbf J_{G\circ F}(x,y) = (\mathbf J_G\circ F(x,y))~\mathbf J_F(x,y)$$
And since $F(1,1)=(2,0)$
$$\mathbf J_{G\circ F}(1,1) = \mathbf J_G(2,0)~~\mathbf J_F(1,1)$$
Check your friend's Jacobian of $G$. Did they make the correct variable assignment?