Trouble understanding differential forms. A basic question: what does $w \times dw$ mean? After reading [1] and [2] I (kind of) understand what differential forms are, but  I am still having trouble understanding the following argument from [3,Lem.4.2]:
Let $\mathbb{T^3_n}$ be the 3-dimensional torus with period $2\pi n$ (meaning $\mathbb{T^3_n} \simeq [-\pi n, \pi n]^3$). And let $v \in H^1(\mathbb{T_n^3,\mathbb{C}})$ satisfy $\vert v \vert \geq \frac{1}{2}$ on $\mathbb{T_n^3}$.
We then may write $v= \vert v \vert w$ with $\vert w \vert = 1$. This seams clear.
Let $dw$ denote the differential form induced by $w$. The author then goes on to say that $d(w \times dw)=0$ and we therefore can perform the Hodge-de-Rham decomposition of $w \times dw$.
My (first basic) questions are

a) What is $w \times dw$ supposed to be? I couldn't find this notation for differential forms anywhere.
b) What is a good source to find information about the Hodge-de-Rham decomposition? 


[1] http://www.math.purdue.edu/~dvb/preprints/diffforms.pdf
[2] http://en.wikipedia.org/wiki/Differential_form
[3] Béthuel, F., P. Gravejat und J. C. Saut: Travelling waves for the Gross-
Pitaevskii equation. II. Comm. Math. Phys., 285(2):567–651, 2009.
 A: For (a), in certain traditions the symbol $\times$ is used (in the sense of the cross product of multivariable calculus) interchangeably with $\wedge$ for the wedge product.
For (b), if you just want to know what it is, Wikipedia is a good place to start. The most general version of the Hodge-de-Rham decomposition is quite complicated (stated for elliptic complexes in general), for the version that you seem to care about, random googling turns up this document which may be an okay place to start. Here's some lecture notes which gives the theory modulo analytical details.
It is hard to recommend a textbook for this, because most Hodge theory books deal with Kahler manifolds which is a bit stronger than what you need (as it appears). The best source I know (indeed, the one I would personally go to) is Morita's Geometry of Differential Forms (Google books link) where the Hodge decomposition is treated in Chapter 4. But be aware that his notation and language may be a bit different from the two resources that you linked to, so you may have to start mostly from the beginning of the book to really "get" it. 
