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Is it possible to get the value of:

\begin{equation} \underbrace{\left[\nabla\times\left[\nabla\times\left[\ldots\nabla\times\right.\right.\right.}_{\infty\text{-times taking curl operator}}\mathbf{V}\left.\left.\left.\right]\right]\ldots\right] = ? \end{equation}

For any possible values of vector $\mathbf{V}$.

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It seems a very strange question to ask. Why are you interested in this? Is it only idle curiosity, or do you have some application in mind? – Harald Hanche-Olsen Nov 10 '12 at 19:14
How do you mean the $\times$ operation? If it is vectorial product, how is it parenthesized? – Berci Nov 10 '12 at 19:14
@HaraldHanche-Olsen: That's now more looks like a curiosity, but based on the ideas of other vector relations: $\nabla\times(\nabla\phi) = 0$, $\nabla\cdot(\nabla\times\mathbf{A}) = 0$. So!, I think that this "relation" have some mathematical or physical meaning, but I have no idea from what point to start. – m0nhawk Nov 10 '12 at 19:26
@Berci: I've edited the question. I think now it's more understandable. – m0nhawk Nov 10 '12 at 19:29
Here is a paper that studies the eigenvalue problem $\nabla \times V = \lambda V$ on the ball, – Nick Alger Nov 10 '12 at 21:30
up vote 15 down vote accepted

For a general vector field $\mathbf{V}$, this sequence need not converge. Consider for example $\mathbf{V} = (e^{x-y}, e^{x-y}, 0)$. We have $\nabla \times \mathbf{V} = \mathbf{W} = (0,0,2 e^{x-y})$, and $\nabla \times \mathbf{W} = -2 \mathbf{V}$, and the cycle then repeats with a factor of $-2$.

It's worth noting that this operation is "unnatural" from the point of view of differential forms; curl is really the $d$ operator taking 1-forms to 2-forms; it's just that in $\mathbb{R}^3$, both 1-forms and 2-forms can be viewed as vector fields. But the result of the curl operator is a 2-form, which is not something that it makes sense to take the curl of. The other relations you cite in your comment don't suffer from this problem: grad takes 0-forms to 1-forms (so curl grad makes sense) and div takes 2-forms to 3-forms (so div curl makes sense), and they are both special cases of the fact that $d^2 = 0$.

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You can also think of curl as the $d^*$ operator taking 2-forms to 1-forms, so from a Hodge-theoretic standpoint this isn't that unnatural (though iterating curl an odd number of times is distinctly different from doing it an even number of times). – Micah Nov 10 '12 at 21:16
And what is the n-forms and what is $d^2 = 0$? – m0nhawk Nov 11 '12 at 17:51

The identity $$ \nabla \times (\nabla \times \mathbf{A})=\nabla(\nabla \cdot \mathbf{A})-\nabla^2 \mathbf{A} $$ is standard (where $\nabla^2$ denotes the component-wise Laplacian). Applying it repeatedly to $\nabla \times \mathbf{A}$ and using the fact that $\nabla \times (\nabla f)$ vanishes, we can see inductively that $$ (\nabla \times)^{2n+1}\mathbf{A}=(-1)^n\nabla \times (\nabla^2)^n \mathbf{A} \, . $$

So, when there's any hope of your thing converging, it'll be because the iterations of $-\nabla^2$ converge component-wise. Roughly speaking, this will happen when the Fourier transforms of your components are supported within some appropriate sphere, but getting the details right could be tricky (especially in cases where it includes some piece of the sphere's boundary).

A little more precisely, the equivalent of $-\nabla^2$ in the frequency domain is multiplication by $4\pi^2|\xi|^2$, so if $\operatorname{supp} \hat f \subset \{|\xi|<1/(2\pi)\}$ everything will converge to $0$. If $\operatorname{supp} \hat f \subset \{|\xi|\leq 1/(2\pi)\}$ everything should still converge; if $\hat f$ is a distribution with positive mass on the boundary of the sphere it should converge to that, but the analysis is icky enough that I don't want to say anything too definite...

You then have to worry about the even iterates, which you can get by replacing $\mathbf{A}$ with $\nabla \times\mathbf{A}$ everywhere in the above identity; there's no real reason why the even and odd iterates ought to have much to do with each other. On the other hand, in the nice case everything will converge to $0$ anyway, so this won't really matter.

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Nice answer, induction had not occurred to me at all. Pretty. – James S. Cook Nov 11 '12 at 1:10

Two applications of $\nabla$ yield $\nabla \times (\nabla \times F) = -\nabla^2 F + \nabla(\nabla \cdot F)$. Why? Well, setting $F = \sum_i F_i e_i$ where $e_i$ is the standard cartesian frame of $\mathbb{R}^3$ allows the formula: $$ (\nabla \times F)_k = \sum_{ij} \epsilon_{ijk} \partial_i F_j $$ Curling once more, $$ [\nabla \times (\nabla \times F)]_m = \sum_{kl}\epsilon_{klm}\partial_k\sum_{ij} \epsilon_{ijl} \partial_i F_j $$ But, the antismmetric symbol is constant and we can write this as $$ [\nabla \times (\nabla \times F)]_m = \sum_{ijkl}\epsilon_{klm}\epsilon_{ijl} \partial_k \partial_i F_j $$ A beautiful identity states:
$$ \sum_{l}\epsilon_{klm}\epsilon_{ijl} = -\sum_{l}\epsilon_{kml}\epsilon_{ijl} = -\delta_{ki}\delta_{mj}+\delta_{kj}\delta_{mi}$$ Hence, $$ [\nabla \times (\nabla \times F)]_m = \sum_{ijk}(-\delta_{ki}\delta_{mj}+\delta_{kj}\delta_{mi}) \partial_k \partial_i F_j = \sum_i [-\partial_i^2F_m+\partial_m(\partial_iF_i)]$$ and the claim follows since $m$ is arbitrary. Now, let's try for 3: $$ \nabla \times (\nabla \times (\nabla \times F)) = \nabla \times \bigl[-\nabla^2 F + \nabla(\nabla \cdot F)\bigr] =\nabla \times (-\nabla^2 F)$$ I used the curl of a gradient is zero. This need not be trivial, take $F = <x^2z,0,0>$ as an example. I suppose I could have shot for a four-folded curl by doubly applying the identity. $$ \nabla \times (\nabla \times (\nabla \times (\nabla \times F))) =?$$ Set $G = -\nabla^2 F $ since we know the gradient term vanishes, $$ \nabla \times (\nabla \times G) = -\nabla^2 G + \nabla \cdot G = \nabla^2(\nabla^2 F)-\nabla [\nabla \cdot (\nabla^2 F)]$$ So, there's the four-folded curl. Well, I see no reason this terminates. I guess you can give it a name. I propose we call (ordered as the edit indicates) $ \nabla \times \nabla \times \cdots \times \nabla = \top$

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