Wedge Product in Coordinates

Given a manifold \(M\), the wedge product is a map that constructs higher order forms

\[\wedge:\quad\Lambda^{k}\times\Lambda^{l}\to\Lambda^{k+l}\]

The wedge product has the following properties:

  • \(\alpha\wedge\beta\) is associative: \(\alpha\wedge(\beta\wedge\gamma)=(\alpha\wedge\beta)\wedge\gamma\)

  • \(\alpha\wedge\beta\) is bilinear in \(\alpha\) and \(\beta\):

    \[(a\alpha_{1}+b\alpha_{2})\wedge\beta\]
    \[\alpha\wedge(c\beta_{1}+d\beta_{2})\]
  • \(\alpha\wedge\beta\) is anticommutative: \(\alpha\wedge\beta=(-1)^{kl}\beta\wedge\alpha\), where \(\alpha\) is a \(k\)-form and \(\beta\) is an \(l\)-form.

In the discrete setting we will only be able to preserve some of these continuous properties. Namely, the bilinearity and anticommutativity will be preserved exactly, whereas the associativity will be satisfied only in the limit where the mesh size tends to zero (\(h\to0\)) and will not be exact.

The wedge product is a an operator that is independent of the metric, i.e. it is a homomorphism under a pull-back:

\[\varphi^{*}(\alpha\wedge\beta)=(\varphi^{*}\alpha)\wedge(\varphi^{*}\beta)\]

Consider the 2D case

\[\begin{split}\begin{aligned} \alpha & =\alpha\\ \beta & =\beta_{x}dx+\beta_{y}dy\\ \alpha\wedge\beta & =\alpha\beta_{x}dx+\alpha\beta_{y}dy\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned} \alpha & =\alpha_{x}dx+\alpha_{y}dy\\ \beta & =\beta_{x}dx+\beta_{y}dy\\ \alpha\wedge\beta & =\left(\alpha_{x}\beta_{y}-\beta_{x}\alpha_{y}\right)dx\wedge dy\end{aligned}\end{split}\]

The 3D case, on the other hand, will be

\[\begin{split}\begin{aligned} \alpha & =\alpha\\ \beta & =\beta_{x}dx+\beta_{y}dy+\beta_{z}dz\\ \alpha\wedge\beta & =\alpha\beta_{x}dx+\alpha\beta_{y}dy+\alpha\beta_{z}dz\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned} \alpha= & \alpha_{x}dx+\alpha_{y}dy+\alpha_{z}dz\\ \beta= & \beta_{x}dx+\beta_{y}dy+\beta_{z}dz\\ \alpha\wedge\beta= & (\alpha_{x}\beta_{y}-\alpha_{y}\beta_{x})dx\wedge dy+\\ + & (\alpha_{y}\beta_{z}-\alpha_{z}\beta_{y})dy\wedge dz+\\ + & (\alpha_{z}\beta_{x}-\alpha_{x}\beta_{z})dz\wedge dx\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned} \alpha & =\alpha_{x}dx+\alpha_{y}dy+\alpha_{z}dz\\ \beta & =\beta_{xy}dx\wedge dy+\beta_{yz}dy\wedge dz+\beta_{zx}dz\wedge dx\\ \alpha\wedge\beta & =(\alpha_{x}\beta_{yz}+\alpha_{y}\beta_{zx}+\alpha_{z}\beta_{xy})dx\wedge dy\wedge dz\end{aligned}\end{split}\]