If $F$ is a vector field on $\mathbf R^3$, define the forms $$\omega_F^1=F^1dx+F^2dy+F^3dz,\\ \omega_F^2=F^1dy\wedge dz+F^2 zd\wedge dx+F^3 dx \wedge dy.$$ We see that $$df=\omega_{\operatorname{grad} f}^1,\\ d(\omega_F^1)=\omega^2_{\operatorname{curl} F},\\ d(\omega^2_F)=\operatorname {div} F dx\wedge dy\wedge dz.$$
- If $F$ is a vector field on a star shaped open set $A$ and $\operatorname {curl} F=0$, show that $F=\operatorname {grad} f$ for some function $f:A\to \mathbf R$.
- If $\operatorname {div} F=0$, show that $F=\operatorname {curl} G$ for some vector field $G$ on $A$.
What I have:
- $\operatorname{curl} F=0\implies \omega^2_{\operatorname{curl}F}\implies d(\omega^1_F)=0\implies\omega_F^1$ is closed.
- $\operatorname{div}F=0\implies \operatorname{div}F dx\wedge dy \wedge dz=0\implies d(\omega_F^2)=0\implies \omega^2_F$ is closed.
Now I obviously need to apply Poincare Lemma. The book from which I took this exercise (Spivak's Calculus on Manifolds) states it in this way: if $A\subset \mathbf R^n$ is an open set star-shaped w.r.t. $0$, then every closed form on $A$ is exact. It defines an exact form as follows: a form $\omega$ is exact if $\omega=d\eta$ for some $\eta$.
But the problem I have is that if understanding what is $\eta$. I guess it is also supposed to be a form. But of what degree? Back to the above, to conclude the proof of 1) I need to say that there is a function ($0$-form) $f$ s.t. $\omega^1_F=df$ (then the result will follow from the first display). For 2), I need to say that there is a 1-form $\omega^1_G$ s.t. $\omega^2_F=d(\omega^1_G)$. So what is the precise statement of the Poincare lemma? The way in which it's stated in Spivak (see above) doesn't guarantee that in case 1 the corresponding $\eta$ is a 0-form and in case 2 the corresponding $\eta$ is a 1-form. How am I supposed to know the degree of the form the existence of which the lemma ensures? In the above 2 cases I had to guess it since otherwise the problem is false.