De Rham Theory (part 2)
Continuing from last time we will now introduce the duality between forms and chains, or the Stokes theorem which is of a fundamental importance in math and physics. Recalling that ∂ represents the boundary, and d is the exterior derivative, George Stokes
proved the following identity:
We can introduce an inner product: between a cycle Ω and a cocycle ω: < Ω , ω > as the integral of ω over the domain Ω.
Stokes theorem then simply states: < ∂Ω , ω > = < Ω , dω > Why do we state it like this? Because the boundary of a boundary is zero: ∂∂ = 0 implies that dd = 0 as follows:
0 = < ∂∂Ω , ω > = < ∂Ω , dω > = < Ω , dd ω >
Recall that for boundaries we have an exact sequence (which is also called a chain complex). For the usual 3D space this means:
Then we have a De Rham co-chain complex (in co-chains the arrows point in the reversed direction):
and from dd = 0 we recognize the usual identities:
· Gradient of a curl is zero
· Curl of a divergence is zero.
Finally we arrive at De Rham Theorem.
From chain complexes we extract the Ker/Image homology group: Hp = Zp/Bp
From cochain complexes we the Ker/Image cohomology group:
DR= (α| d α = 0)/( α| d α = β)
De Rham theorem tells us that the two groups are isomorphic: Hp ~
This means that we can explore the topological properties of the space by looking at the solutions of differential equation on that space.
For example, an electric charge generates an electric field around it. The spatial distribution of the electric field can tell us the location of the charge. That is why the second cohomology group H2 DR can be interpreted as an electric charge! In general cohomological classes have a physical interpretation.
Next time we will venture into the related wonderful world of Hodge theory.