What is the name for a vector field that is both divergence-free and curl-free? Consider a smooth vector field $\mathbf u\colon\Omega\to\mathbb R^3$ defined on an open domain $\Omega\subseteq\mathbb R^3$ such that $\mathbf u$ has zero divergence and zero curl on $\Omega$, that is,
$$\begin{align}\nabla\cdot\mathbf u&=0,\\\nabla\times\mathbf u&=0.\end{align}$$
Is there a specific technical name for such a vector field?
Wikipedia calls it a Laplacian vector field, but


*

*it does not cite any references, and 

*it asserts that any such vector field is the gradient of a harmonic function, but this is only true if $\Omega$ is simply connected (counterexample: $\big(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2},0\big)$ on the region $x^2+y^2>0$),


so I'm disinclined to trust it. Can anyone provide references supporting this or any other name?
 A: In geometric calculus literature (see, for example, Doran and Lasenby), such a function is called monogenic.  Monogenic functions are generalizations of complex analytic (or holomorphic) functions.  This condition is strictly stronger than being harmonic--all monogenic functions are harmonic, but not all harmonic functions are monogenic.
The term monogenic is not restricted to vector fields, as well; a scalar field with zero gradient would also be referred to as monogenic.
You can also consult this page by Gull, Lasenby, and Doran.

Edit: Phrased in the language of geometric calculus, we define the vector derivative of a vector field $u$ as $\nabla u$, given by
$$\nabla u = \nabla \cdot u + \nabla \wedge u$$
When $\nabla u = 0$, then $\nabla^2 u = \nabla \wedge (\nabla \cdot u) + \nabla \cdot (\nabla \wedge u) = 0$ as well, fulfilling the harmonic condition.
A: Put musical isomorphism aside, I believe what Rahul Narain refers to is just harmonic $1$-form. 
In Hodge decomposition for $k$-forms $\omega$:
$$
\omega =\mathrm{d}\alpha +\delta \beta + \gamma
$$
where $\gamma$ is harmonic in that $(\mathrm{d}\delta + \delta\mathrm{d})\omega = 0$, and $\delta = (-1)^{nk+n+1} \star^{n-k+1}\mathrm{d}^{n-k}\star^k$.
In the 3-dimensional case. We have the cochain complex:
$$
\Lambda^0\ \stackrel{\mathrm{d}^0}{\longrightarrow}\ \Lambda^1 \ \stackrel{\mathrm{d}^1}{\longrightarrow}\ \Lambda^2\ \stackrel{\mathrm{d}^2}{\longrightarrow}\ \Lambda^3.
$$
Define $\delta  = \mathrm{d}^*_{k}: \Lambda^*_k\to\Lambda^*_{k-1}$ as the adjoint of $\mathrm{d}^{k-1}: \Lambda^{k-1}\to\Lambda^{k}$ with respect to the inner product. We can have somewhat a dual complex:
$$
\Lambda^*_3\ \stackrel{\mathrm{d}^*_3}{\longrightarrow}\ \Lambda^*_2 \ \stackrel{\mathrm{d}^*_2}{\longrightarrow}\ \Lambda^*_1\ \stackrel{\mathrm{d}^*_1}{\longrightarrow}\ \Lambda^*_0.
$$
For a harmonic 1-form $\gamma$: 
$$
(\mathrm{d}\delta + \delta\mathrm{d}) \gamma = (\mathrm{d}^0\mathrm{d}^*_1 + \mathrm{d}^*_2 \mathrm{d}^1)\gamma = 0. \tag{1}
$$
Note: $$\mathrm{d}^1 = \nabla \times, \quad\mathrm{d}^*_1 = (-1)\star^{0}\mathrm{d}^{2}\star^1 = -\nabla \cdot$$
(1) is:
$$
\nabla \times (\nabla \times \gamma) - \nabla (\nabla \cdot \gamma) = 0.\tag{2}
$$
We can say a curl-free and a divergence-free vector field is harmonic under musical isomorphism for
$$
 \nabla \times \gamma = 0\;\text{ and } \nabla \cdot \gamma = 0\Longrightarrow\nabla \times (\nabla \times \gamma) - \nabla (\nabla \cdot \gamma) = 0 .
$$
I am guessing the wikipedia page was using "Laplacian vector fields" in that (2)'s left side is actually the vector Laplace operator (or Laplace-Beltrami) acting on a vector field.
For references, we use this term a lot in computational geometry, a field which inherits a lot of terminologies from vector calculus, it is like almost a tradition that saying a vector field is harmonic means it is curl-free and divergence-free w/o citing anyone's book. 
A: There's no "canonical" name for such functions but E.M.Stein calls them Riesz systems in all of his books. V.P.Havin had a nice name for them (which I used too): "harmonic vector field". The reason is that for any vector field with zero curl and divergence (in any connected domain) the component functions turn out to be harmonic. This is true for any dimension, not just 3, with proper generalizations of the notions of curl and divergence, of course.
