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I was wondering if there is a vector field that satisfies the following condition:

$$\vec F=\nabla \times \vec F$$

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    $\begingroup$ Before anyone else points it out: $F=0$. $\endgroup$ – hjhjhj57 Jan 4 '15 at 3:38
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    $\begingroup$ Beat me by 7 seconds. $\endgroup$ – David K Jan 4 '15 at 3:38
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    $\begingroup$ Is there a vector that's perpendicular to itself (and not 0)? $\endgroup$ – GPerez Jan 4 '15 at 3:41
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    $\begingroup$ @GPerez This is a formal cross product, the curl of $F$ need not be orthogonal to $F$. $\endgroup$ – user147263 Jan 4 '15 at 3:48
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    $\begingroup$ @Behaviour Ah. I knew it was formal, but I thought it was orthogonal anyways (shame on me for extrapolating from what I encountered in most cases). Also I found an answer here $\endgroup$ – GPerez Jan 4 '15 at 3:51
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I'd like to offer a high-brow answer to this question. While I realize that this solution is likely to be above the OP's level, I'm hoping it'll be of some interest to others.

Claim: Let $u(x,y)$, $v(x,y)$, $w(x)$ be any real-analytic functions (defined on a neighborhood of $0$). Then there exists a unique real-analytic vector field $F = (F^1, F^2, F^3)$ that satisfies $\text{curl}\,F = F$ and $$\begin{align} F^1(x,0,0) & = w(x) \\ F^2(x,y,0) & = u(x,y) \\ F^3(x,y,0) & = v(x,y). \end{align}$$

We'll show this by applying the Cauchy-Kovalevskaya Theorem twice. Here I should acknowledge that I first saw this techinque applied to $\text{curl}\,F = F$ in an exercise in Robert Bryant's "Nine Lectures on Exterior Differential Systems."

Cauchy-Kovalevskaya Theorem: If $\mathbf{H}$ and $\phi$ are real-analytic functions near the origin, then there is a neighborhood of the origin on which there exists a unique real-analytic solution $\mathbf{g}(\mathbf{x}, t)$ to $$\begin{align} \frac{\partial \mathbf{g}}{\partial t}(\mathbf{x},t) & = \mathbf{H}(\mathbf{x},t, \mathbf{g}(\mathbf{x},t), \frac{\partial \mathbf{g}}{\partial x^i}) \\ \mathbf{g}(\mathbf{x}, 0) & = \phi(\mathbf{x}) \end{align}$$ A PDE system of this form will be called a "Cauchy problem."


Setup

The equation $\text{curl}\,F = F$ can be viewed as an overdetermined system of PDEs -- in fact, a system of 4 first-order quasilinear equations for 3 unknown functions. That is, if we write $F = (F^1, F^2, F^3)$, then the condition $\text{curl}\,F = F$ becomes: $$\begin{align} \frac{\partial F^2}{\partial z} - \frac{\partial F^3}{\partial y} & = F^1 \tag{1} \\ \frac{\partial F^3}{\partial x} - \frac{\partial F^1}{\partial z} & = F^2 \tag{2} \\ \frac{\partial F^1}{\partial y} - \frac{\partial F^2}{\partial x} & = F^3. \tag{3} \end{align}$$ The fourth equation is a hidden "integrability condition" of sorts: namely, any solution $F$ to $\text{curl}\,F = F$ must satisfy $\text{div}\,F = 0$. This gives us a fourth (first-order quasilinear) equation $$\frac{\partial F^1}{\partial x} + \frac{\partial F^2}{\partial y} + \frac{\partial F^3}{\partial z} = 0. \tag{4}$$


The Cauchy Problems

We can write this system as a sequence of two Cauchy problems as follows. Let $u(x,y)$, $v(x,y)$, and $w(x)$ be arbitrary real-analytic functions.

We first consider the problem of finding $g(x,y)$ such that $$\begin{align} \frac{\partial g}{\partial y} & = \frac{\partial u}{\partial x} + v(x,y) \\ g(x,0) & = w(x). \end{align}$$ By Cauchy-Kovalevskaya$^\dagger$, there exists a unique real-analytic solution $g(x,y)$.

Second, letting $g(x,y)$ be a solution as above, we consider the problem of finding $F^1, F^2, F^3$ such that $$\begin{align} \frac{\partial F^1}{\partial z} & = -F^2 + \frac{\partial F^3}{\partial x} \\ \frac{\partial F^2}{\partial z} & = F^1 + \frac{\partial F^3}{\partial y} \\ \frac{\partial F^3}{\partial z} & = -\frac{\partial F^1}{\partial x} - \frac{\partial F^2}{\partial y} \\ F^1(x,y,0) & = g(x,y) \\ F^2(x,y,0) & = u(x,y) \\ F^3(x,y,0) & = v(x,y) \end{align}$$ By Cauchy-Kovalevskaya again, there exists a unique real-analytic solution $F = (F^1, F^2, F^3)$.

By construction, this solution $(F^1, F^2, F^3)$ satisfies equations (1), (2) and (4). One can check (exercise!$^*$) that this $F$ necessarily satisfies equation (3) as well. This completes the proof. $\lozenge$


End notes

$^\dagger$ Admittedly overkill here

$^*$ Hint for Exercise: Consider $E(x,y,z) := \frac{\partial F^1}{\partial y} - \frac{\partial F^2}{\partial x} - F^3$. Check that $E(x,y,0) = 0$ and $\frac{\partial E}{\partial z} = 0$, and then apply uniqueness in Cauchy-Kovalevskaya.

Note that this exercise shows why we needed the first Cauchy problem at all: that is, we couldn't just choose $F^1(x,y,0) = g(x,y)$ completely arbitrarily!

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    $\begingroup$ Wow! That's pretty cool! +1! $\endgroup$ – Robert Lewis Jan 4 '15 at 6:14
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    $\begingroup$ I see now that the linked paper by GPerez and David K essentially carries out this proof in Section 3. Interestingly, Section 2 shows that any differentiable vector field with $\text{curl}\,F = F$ is automatically real-analytic -- essentially because of $F = \text{curl}(\text{curl}\,F) = -\Delta F$ and elliptic regularity -- and so the real-analyticity restrictions above are not as strong as they seem :-) $\endgroup$ – Jesse Madnick Jan 4 '15 at 9:02
  • $\begingroup$ Interesting . . . elliptic regularity was the first thing I noticed, but I didn't have time to do much with it or and related approaches and then you posted your nice answer! Cheers! $\endgroup$ – Robert Lewis Jan 4 '15 at 9:21
  • $\begingroup$ I'm not familiar with Cauchy problems so forgive the naivete: is there a standard way to solve these things? The way Duhamels formula can be implemented for arbitrarily high order linear ODE, do Cauchy problems admit some general solution? $\endgroup$ – frogeyedpeas Jan 4 '15 at 14:29
  • $\begingroup$ Thank You for the detailed explanation. $\endgroup$ – user204299 Jan 5 '15 at 1:11
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To give a final answer, I've found that, in the standard unit coordinates, the field $$F(x,y,z) = (-\cos z, \sin z, 0)$$ is equal to its curl. Again, brought to you by this paper.

P.S. My first attempt revealed that Wolfram Alpha is a wise-guy.

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    $\begingroup$ Oh good, Wolfram Alpha can also solve $\mathrm{P=NP}$ in the standard manner. $\endgroup$ – Rahul Jan 4 '15 at 5:36
  • $\begingroup$ What if you tried to get W.A. to solve for $f$? By hacking the "input interpretation"? $\endgroup$ – Robert Lewis Jan 4 '15 at 6:11
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    $\begingroup$ @RobertLewis: That doesn't really help. $\endgroup$ – Ilmari Karonen Jan 4 '15 at 12:18
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    $\begingroup$ @Rahul It says $P = 0$, which I take to be $\emptyset$. That explains why my computer is so dang slow. $\endgroup$ – GPerez Jan 4 '15 at 13:20
  • $\begingroup$ @IlmariKaronen: But your link does not do what Robert Lewis asked for; it's still trying to solve for curl f. $\endgroup$ – ruakh Jan 4 '15 at 17:27
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if $f =(\sin z, \cos z, 0)\ $ then $\nabla \times f = f$

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Jesse Madnick's answer notwithstanding one might be interested in nontrivial explicit solutions from which more complicated solutions can be generated by transportation and superposition.

I'm looking for solutions that do not depend on the third variable – "plane waves", so to speak. When $F=(F_1,F_2,F_3)$ is such a solution then $${\rm curl}\,F=(F_{3.2},-F_{3.1}, F_{2.1}-F_{1.2})\ .$$ We write $F_3=:u$. The condition ${\rm curl}\,F=F$ then leads to $F_1=u_y$, $F_2=-u_x$, and above all to $$u+\Delta u=0\ .\tag{1}$$ The well known equation $(1)$ can be treated using separation of variables. From a geometrical standpoint it would be natural to try $u=R(r)\>\Phi(\phi)$, but this leads to Bessel's differential equation $r^2 R''+r R'+(r^2-k^2)R=0$, $\>k\in{\mathbb N}$, whose solutions are non elementary.

When we write $u=X(x)\>Y(y)$ instead we are lead to $$X''+\lambda X=0,\qquad Y''+(1-\lambda) Y=0\ ;$$ here $\lambda\in{\mathbb R}$ is arbitrary. When $0<\lambda<1$ we obtain solutions $u$ which are periodic in $x$ and in $y$. For other values of $\lambda$ the resulting $u$ is periodic in one variable, and a hyperbolic function in the other. The values $\lambda=0$ and $\lambda=1$ are special; here one of $X(\cdot)$ and $Y(\cdot)$ is linear, the other periodic with period $2\pi$.

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