Invariance of the Discrete Wedge Product¶

Consider a manifold $\text{[math]}$ and its polyhedrization in the context of discrete exterior calculus, and let $\text{[math]}$ denote the simplices and $\text{[math]}$ the corresponding basis functions, such that the pairing given by integration is natural

Given the projection $\text{[math]}$ and reconstruction operators $\text{[math]}$ ,

the discrete wedge product is defined as a three tensor

where

Consider now a second manifold $\text{[math]}$ and a diffeomorphism

which induces a new polyhedrization with the corresponding simplices $\text{[math]}$ and basis functions $\text{[math]}$ .

Given a simplex $\text{[math]}$ and a form $\text{[math]}$ , the following identities hold for push-forwards and pull-backs:

(1)¶

(2)¶

The discrete wedge product on this new manifold will be elementwise equal to the wedge product on the original manifold. To see why this is the case:

So we have shown that the discrete wedge product is indeed invariant under diffeomorphisms.

This result is consistent with theory in differential geometry, where the wedge product is thought of as a pureley topological operator that is independent of the metric.