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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. $\det(J(f))\not\equiv0$.

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    What about $f(x) = x^3$ with $M=N = \mathbb{R}$? – 2017-02-13
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    @copper.hat It is still ok, since $\det(J(f))|_{x=1}\neq0$. – 2017-02-13
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    Sorry, misread the question. – 2017-02-13
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    Perhaps Sard's theorem might be of use? – 2017-02-13
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    @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 $\det(J(f))\neq0$. – 2017-02-13
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    Perhaps you could show that if $\det(J(f)) = 0$ then $f$ cannot be surjective? – 2017-02-13

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