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$f:\mathbb{R}^3 \rightarrow \mathbb{R}^3$

$f(x,y,z)=(xy,yz,xz)$

a) Determine set $S$ which contains all $(x,y,z) \in \mathbb{R}^3$ where $f$ is locally invertible.

b) At what points $c\in S$ there exists locally differentiable inverse of $f$? Calculate Jacobian determinant of inverse function at $f(c)$.

c) Is restriction $f|_S:S \rightarrow f(S)$ bijection?

Function $f$ is class $C^1$ so for every point where Jacobian matrix is regular, $f$ is locally invertible.

$$J_f(x,y)=\begin{vmatrix} y & x & 0 \\ 0 & z & y \\ z & 0 & x \\ \end{vmatrix}= 2xyz $$

Function $f$ has locally differentiable inverse at $ \{ (x,y,z): xyz \neq 0 \}$.

It is easy now to calculate Jacobian determinant of inverse function. Can somebody help me with a) and b)? In my book we define local invertibility with regularity of jacobian matrix.

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    Thus, you have to exclude all the coordinate planes ($x=0$, $y=0$, $z=0$) that's all... – 2017-02-12
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    Yea, thats part of b) question. But I have to find set $S$ where $f$ is locally invertible. Doesn't necessarily have to do with jacobian matrix – 2017-02-12
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    For part a) it is more trickier indeed. You may take advantage to embed you problem in the (projective) framework of the so-called Veronese map $v: \ (x,y,z)\mapsto(x^2,y^2,z^2,xy,yz,zx)$ – 2017-02-12
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    Yea, we haven't done that. But is this valid. – 2017-02-12

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