Calculus problem (Shame on me)

[economicus]

math reviewing process before mphil/phd...

Can someone please help me at one math problem that is driving me crazy? ...especially because I think that is such a damn simple thing...

We have: G(x,y) = (x^2+1), y^2) and F(u,v) = (u+v, v^2). Compute the Jacobian derivative matrix F(G(x,y)).

How do ve get u and v each in terms of x and y? These double functions have impaired my vision...

[ForTheWin!_08]

To get the composition F(G(x,y)):

Take your expression for F in terms of u and v. These are just placeholders. Set (u,v) = G(x,y) = (x^2 + 1, y^2). Now everywhere where you see a "u" in the expression for F, substitute "x^2 + 1" and likewise, everywhere where you see a "v" in F(u,v), substitute "y^2".

Therefore F(G(x,y)) = (x^2 + y^2 + 1, (y^2)^2 )
= (x^2 + y^2 + 1, y^4 ). Now you can calculate the Jacobian of that in terms of x and y.

[economicus]

Thanks!