Translating theorem on closed/exact forms 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
 A: *

*$k=1$: So $\omega$ corresponds to a vector field $v_\omega\colon A \to \mathbb R^3$ via $\omega = v_{\omega,1}dx^1 + v_{\omega, 2}dx^2 + v_{\omega,3}dx^3$ and your theorem reads: If $f, f_0\colon A \to \mathbb R^3$ are such that $\nabla f = \nabla f_0 = \omega$, then $f = f_0 + c$ for some constant $c$.

*$k=2$: Here $\omega$ corrensponds to a vector field $w_\omega\colon A \to \mathbb R^3$ via $\omega = w_{\omega, 1}dx^1 \wedge dx^2 + w_{\omega,2} dx^2\wedge dx^3 + w_{\omega, 3} dx^3 \wedge dx^1$. Your theorem reads: If $v_{\eta}$, $v_{\eta,0}$ are such that $\nabla \times v_{\eta} = \nabla \times v_{\eta_0} = w_\omega$, then $v_\eta - v_{\eta_0} = \nabla f$ for some $f \colon A \to \mathbb R^3$.

*$k=3$ Here $\omega$ corresponds to a $f_\omega\colon A \to \mathbb R^3$ via $\omega = f_\omega dx^1 \wedge dx^2 \wedge dx^3$ and your theorem reads: If $w_\eta, w_{\eta_0}\colon A \to \mathbb R^3$ are such that $\nabla \cdot w_\eta = \nabla \cdot w_{\eta_0} = f_\omega$, that $w_\eta - w_{\eta_0} = \nabla \times v$ for some $v\colon A \to \mathbb R^3$.


Note that for $\omega = \sum_i v_idx^i$ we have
\begin{align*}
   d\omega &= \sum_i dv_i\wedge dx^i\\\
           &= \sum_{i,j} \partial_jv_i dx^j \wedge dx^i\\\
           &= (\partial_1 v^2 - \partial_2v^1)dx^1 \wedge dx^2 
             + (\partial_3 v^1 - \partial_1v^3)dx^3 \wedge dx^1
             + (\partial_2 v^3 - \partial_3v^2)dx^2 \wedge dx^3
\end{align*}
