= Solution
The map $G$ is $C^1$ on the open ball $B$. Its <Jacobian matrix> is
$$
DG(x)=\begin{pmatrix}(x_3+3)^{-1}&(x_3+3)^{-1}&-(x_1+x_2)(x_3+3)^{-2}\\0&1&0\\0&0&1\end{pmatrix}.
$$
At the origin $G(0)=0$ and $\boxed{\det DG(0)=1/3\ne0}$. The <inverse function theorem> therefore supplies the required <open sets> $V_1\subset B$ and $V_2\subset\mathbb R^3$, both containing zero, with $G:V_1\to V_2$ a <diffeomorphism>. The explicit inverse formula on its image,
$$
G^{-1}(y_1,y_2,y_3)=(y_1(y_3+3)-y_2,\ y_2,\ y_3),
$$
also checks the local conclusion directly.
Back to article page
