First of all I dont know what is exact equivalent expression of "regular" in english.
question was:
let $\ell :R^2\mapsto R^3\quad (x_1,x_2)\mapsto \ell(x_1,x_2)=(x_1x_2+3,x_1^2-x_2,x_1+x_2^3) $
is $\ell$ regular on $R^2?$
we need to show that rank of jacobian matrix $J_\ell$ must be equal to 2 every points on $R^2$
$J_\ell=\begin{pmatrix}
x_2 & x_1 \\
2x_1 & -1 \\
1 & 3x_2^2 \\
\end{pmatrix}$
how do we continue?
Best Answer
For starters the rank is $\geq 1$ as there always exist nonzero elements. The rank is also $\leq 2$, due to the shape of the matrix.
Suppose there were a point $(x_1,x_2)$ where the rank of the Jacobian is $1$. In that case the columns of the Jacobian would be linearly dependent. This will imply that the $2 \times 2$ minors $$\det \begin{pmatrix} x_2 & x_1 \\ 2x_1 & -1 \end{pmatrix},\;\det \begin{pmatrix} x_2 & x_1 \\ 1 & 3x_2^2 \end{pmatrix}$$ are both zero. Is it possible?