Wedge Product in Coordinates¶
Given a manifold , the wedge product is a map that constructs higher order forms
The wedge product has the following properties:
is bilinear in and :
is anticommutative: , where is a -form and is an -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 () 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:
Consider the 2D case
The 3D case, on the other hand, will be