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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

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    This is done in Bott's book, for example.2012-04-26
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    Unfortunately, I don't have a copy of Bott's book, and won't be able to get a hold of it anytime soon. Just out of curiousity, what is the title of the book, when I do get a hold of it?2012-04-26
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    «Differential forms in algebraic topology», by Bott and Tu. In any case: your question is rather weird: it is not the kind of thing one thinks out of thin air: *why* do you want to do that translation?2012-04-26
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    Extra practice, and ability to understand important theorems clearly2012-04-26
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    But how do you know there *is* a translation? Is this homework? This question *must* have been asked in a context, otherwise it is unintelligible! To be honest, looking at your history of questions I sometimes get the feeling we are doing your homework... You do not seem to show much, if any, of your own work (and even rarely respond to hints) And you have not answered *any* question at all!2012-04-26
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    This is problems from a book and I havent posted my real thoughts since I have been rather lost and needed some sense of guidance/direction.2012-04-26
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    How did you know that you were not going to be able to get a hold of Bott's book before even knowing what the title was? :)2012-04-26
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    Since I don't have access to a library, or enough money to purchase it2012-04-26
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    Can you tell us what book it is that you *do* have that you got this problem from?2012-04-26
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    spivak's calculus2012-04-27

1 Answers 1

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  • $k=1$: 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*}

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    This looks complete, but we havent really done stuff with gradients. Can you please elaborate on this furhter?2012-04-26
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    On what exactly? You haven't done gradients, but differential forms?2012-04-26