I'm trying to translate this theorem, below, into theorems about scalar and vector fields in $\mathbb R^3$:
Theorem: Let $A$ be a star-convex open set in $\mathbb R^n$. Let $\omega$ be a closed $k$-form on $A$. If $k > 1$ and if $\eta$ and $\eta_0$ are two $k-1$ forms on $A$, with $d\eta = d\eta_0 = \omega$, then $\eta - \eta_0 = d\vartheta$ for some $k-2$ form $\vartheta$ on $A$.
If $k = 1$, and if $f$ and $f_0$ are 2 $0$-forms on $A$ with $df = df_0 = \omega$, then $f = f_0 + c$, for some constant $c$.
my attempt: I tried considering the cases for $k = 1,2,3$ and not really seeing this geometrically