Let $f:M\to N$ be a surjective smooth map between two manifolds with the same dimension.
Q: Why the determinate of Jacobi matrix $J(f)$ can not always vanish , i.e. $\det(J(f))\not\equiv0$.
Asked
Active
Viewed 34 times
0
-
What about $f(x) = x^3$ with $M=N = \mathbb{R}$? – copper.hat Feb 13 '17 at 03:51
-
@copper.hat It is still ok, since $\det(J(f))|_{x=1}\neq0$. – DLIN Feb 13 '17 at 04:23
-
Sorry, misread the question. – copper.hat Feb 13 '17 at 04:28
-
Perhaps Sard's theorem might be of use? – copper.hat Feb 13 '17 at 04:30
-
@copper.hat Sard just says that the non ciritical point is dense, but even if $\left(\frac{\partial f}{\partial x}\right)\neq 0$, can we say $\det(J(f))\neq0$. – DLIN Feb 13 '17 at 04:41
-
Perhaps you could show that if $\det(J(f)) = 0$ then $f$ cannot be surjective? – copper.hat Feb 13 '17 at 04:50