TITLE: Differential forms: The authors of a paper define $d(u\times du)$, but what is $u \times du$ supposed to mean?
QUESTION [9 upvotes]: I'm reading [1] recently and have another question about a remark in this paper. I tried to solve it myself (see below) but did not succeed. It could be just a notation problem.
The Setup:

Let $u \in H^1(\Omega,\mathbb{C})$ where $\Omega = [-\pi,\pi]^3 \subset \mathbb{R}^3$.
In [1, Rmk.4.2] the authors denote by $Ju$ the 2-form \begin{equation} Ju \equiv \frac{1}{2} d(u \times du) = \sum_{1 \leq i < j \leq 3} (\partial_i u \times \partial_j u) dx_i \wedge dx_j. \end{equation}
Furthermore they define \begin{equation} \zeta_1(x)= - x_2 dx_1 \wedge dx_2 - x_3 dx_1 \wedge dx_3. \end{equation}
From this definition it follows that \begin{equation} \star \zeta_1 = -x_2 dx_3 + x_3 dx_2 \end{equation} which I checked. Here $\star$ denotes the Hodge-star-operator.
I then calculated \begin{equation} d(\star \zeta) = -2dx_2 \wedge dx_3 \tag{G1}\end{equation} which I hope is correct.
Now the authors state that \begin{equation} (u \times du) \wedge d(\star \zeta_1) = 2 \langle i \partial_1 u, u \rangle dx_1 \wedge \ldots \wedge dx_3 \tag{G2}  \end{equation}

My Question:

What is $u \times du$ supposed to be?

My Attempt:
I tried to find out myself, but I discovered the following difficulty: 

Let $\omega=\sum_{i=1}^3 \omega^i dx_i$ be the 1-Form $\omega=u \times du$. Then \begin{equation} \frac{1}{2} d\omega = \frac{1}{2} \sum_{1 \leq i< j \leq 3} \omega^i_{x_j} dx_j \wedge dx_i\stackrel{!}{=} Ju = \sum_{1\leq i < j\leq 3} (\partial_i u \times \partial_j u) dx_i \wedge dx_j.  \end{equation} This suggests that $\omega_{x_j}^i = -2 (\partial_i u \times \partial_j u)$.
On the other hand because of $(G1)$ equation $(G2)$ becomes \begin{equation} \omega \wedge d(\star \zeta) = -2\omega^1dx_1 \wedge dx_2 \wedge dx_3 \stackrel{!}{=} 2 \langle i \partial_1 u, u \rangle dx_1 \wedge dx_2 \wedge dx_3, \end{equation} which suggests $\omega^1=-\langle i \partial_1 u, u \rangle$.

So my question reduces to

How are these two equations compatible? What is the $\times$ supposed to mean?


[1] Béthuel, F., P. Gravejat und J. C. Saut: Travelling waves for the Gross-
Pitaevskii equation. II. Comm. Math. Phys., 285(2):567–651, 2009.

REPLY [7 votes]: I believe this is all resolved if you have one typo. If $u$ maps $\Omega$ to $\mathbb R^3$, then you are taking the cross product of a vector in  $\mathbb R^3$ with an $\mathbb R^3$-valued $1$-form. This is totally consistent with their formulas.
So, we both win.:) They are interpreting $\mathbb C \cong \mathbb R^2$ and defining the cross product of two vectors $u,v\in\mathbb R^2$ as the real number $(u\times v)\cdot e_3\in\mathbb R$. In particular, 
$$u\times du = \big(u_2(\partial_1u_1) - u_1(\partial_1 u_2)\big)dx_1 \pmod{dx_2,dx_3}\,.$$
All their formulas are consistent with this, recalling that they've defined $\langle u,v\rangle = \text{Re}(u\bar v)$, i.e., the real dot product of the vectors in $\mathbb C \cong \mathbb R^2$.