# Thread: Smooth covector field on S^2

1. ## Smooth covector field on S^2

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 $\displaystyle -p$ ??

2. Originally Posted by Different
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 $\displaystyle p$).

Not quite sure what you mean, but if $\displaystyle w = df$ for $\displaystyle f\in C^{\infty}({\mathbb S}^2)$, compactness of the sphere implies $\displaystyle f$ must attain two extrema (at least). So there are two points where $\displaystyle \omega=df=0$.

..is it correct to say

No.