So i was reading the following note on the implicit function theorem:
https://rutherglen.science.mq.edu.au/wchen/lnmvafolder/mva03.pdf
I understand everything for the most part with regards to the theoretical part, but on page 6 of the pdf halfway down on the right after we have established that the partial derivative is $-8z_0w_0$ the author says that we can solve for $z$ and $w$ and proceeds to show that
$$z = g_1(x,y) = \sqrt{1-x^2}$$ and $$w = g_2(x,y) = \sqrt{1-y^2}$$
Now i understand by the fact that the partial doesn't equal $0$ at certain points we can solve for the variables from $F(x,y,z,w) = 0$, but what happened to the rest of the variables? I mean based on the original equations i don't see how the $x$, $y$, $z$, $w$ vanish when isolating one of them.
I know it is asking alot to look at the document and such but the context is better stated through that way. Thanks