Is there a smooth covector field on $\displaystyle {\mathbb S}^2 $ that is exact and vanishes at exactly one point?

I think the answer is no...

in case of exact field we have $\displaystyle w = df $ for some smooth function on $\displaystyle {\mathbb S}^2 $... it vanishes at some point $\displaystyle p $ when partial derivatives of $\displaystyle f $ at p are equal to zero (in some chart containing $\displaystyle p $).

..is it correct to say that if we consider stereographic coordinates on $\displaystyle {\mathbb S}^2 $ then for any point $\displaystyle p $ where $\displaystyle w=df $ vanishes it will also vanish at point $\displaystyle -p $ ??