TITLE: Implicit function theorem problem from Munkres Analysis on Manifolds text
QUESTION [3 upvotes]: For the problem below, which is from Munkres' Analysis on Manifolds section on the implicit function theorem, I'm getting the following solution $\frac{\partial u}{\partial y} \neq \frac{18}{5}$, but I saw a solution giving the answer as $\frac{\partial u}{\partial y} \neq \frac{2}{5}$
My attempt:
I began by checking the conditions for both $G$, and $H$, where $det\big(\frac{\partial G(2,-1,1)}{\partial y} \big) \neq 0$ and the same for $H$
(1) $\det \big(\frac{\partial H(2,-1,1)}{\partial y} \big)=2\frac{\partial u}{\partial y}+9+3\frac{\partial u}{\partial y}=0$
(2) $\det \big(\frac{\partial G(2,-1,1)}{\partial y} \big)=f'(x,y)\frac{\partial u}{\partial y}+2u\frac{\partial u}{\partial y}=0$
and substituting (1) into (2) gives
$f'(x,y)\frac{\partial u}{\partial y}=\frac{18}{5}$
My thinking is that I was supposed to do something more with $f'(x,y)$ so that I could isolate $\frac{\partial u}{\partial y}$?
Thanks

REPLY [3 votes]: No, you have to look at the determinant of the $2\times 2$ matrix $\dfrac{\partial(G,H)}{\partial(x,u)}$ and see that it is nonzero.
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