TITLE: Support of Pullback of Differential form
QUESTION [3 upvotes]: This is a dumb question, but I'm learning about differntial forms, and it seems to me that if $f:N^n\to M^n$ is a diffeomorphism and $\omega$ a smooth $n$ form on $M^n$, then $\omega$ vanishes at $p\in M^n$ $\iff$ the pullback $f^*\omega$ vanishes at $f^{-1}(p)$. This implies that if $\omega$ has compact support in $M^n\iff$  $f^{-1}\omega$ has compact support in $N^n$.
Question 1: Is the above correct?
Question 2: Does "$\omega$ vanishes at $p\in M^n$ $\iff$ the pullback $f^*\omega$ vanishes at $f^{-1}(p)$If $f$" hold if $f$ is just a surjective smooth map, instead of diffeomorphism?

REPLY [4 votes]: Q1: Always true.
Q2: True if you replace "surjective smooth map" with "submersion", otherwise only one direction of the implication holds. 
Remember the definition of the pullback: $$f^* \omega (X_1, X_2, \ldots) = \omega(Df (X_1), Df(X_2), \ldots).$$ One direction is obvious: if $\omega = 0$, then the RHS is always zero, and thus $f^* \omega(X_1,\ldots) = 0$ for all possible inputs $X_i$; i.e. $f^* \omega = 0$. 
Conversely if $\omega( Df(X_1), \ldots) = 0$ for all $X_i$, then if $D f$ is surjective this is the same as $\omega(X_1,\ldots) = 0$ for all $X_i$, so $\omega = 0$.
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