<s>[INST] two short doubts about the inverse function in a point the function is $F(x,y,z)=(y^2+z^2, z^2+x^2, x^2+y^2)$ the point is (-1,1,-1)
task: find the local inverse of F in that point.
I have already proved that F is actually invertible there. then i solved the system:
$$
\begin{cases}
y^2+z^2=a\\
z^2+x^2=b\\
x^2+y^2=c
\end{cases}
$$
which gave me
$$
x=\pm \sqrt{\frac{c-a+b}{2}}\\
y=\pm \sqrt{\frac{a-b+c}{2}}\\
z=\pm \sqrt{\frac{b-c+a}{2}}\\
$$
The doubt I have is: how should I choose the signs? I thought of looking at the values of x,y,z in the point, which means x,z=negative y=positive
so i picked
$$
x= -\sqrt{\frac{c-a+b}{2}}\\
y= \sqrt{\frac{a-b+c}{2}}\\
z= -\sqrt{\frac{b-c+a}{2}}\\
$$
finding the inverse G:
$$ G(a,b,c)=(-\sqrt{\frac{c-a+b}{2}},\sqrt{\frac{a-b+c}{2}},-\sqrt{\frac{b-c+a}{2}})$$
the second doubt is:
should i leave it written that way or is it better:
$G(y^2+z^2, z^2+x^2, x^2+y^2)=(-x,y,-z)$ ? (with the same doubt about the signs?)
thank you in advance
 [/INST] First doubt: you are right
Secon doubt: the first way you have written is good, the other does not make sense.
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