If there were one theorem to rule them all, it would be the Generalized Stokes’ Theorem.
Over the past few articles, we’ve been setting up the fundamental concepts behind differential forms:
- the raising operator ♯,
- the lowering operator ♭,
- the wedge product ∧,
- the Hodge star ⋆,
- and the exterior derivative d.
Furthermore, we came up with
- An algebraic/topological definition of a region and its boundary.
- A formal definition of a manifold that allows us to do Calculus (including integration) on curved surfaces.
With these tools and some time on our hands, differential forms makes everything in Vector Calculus trivial.
Although we’ve shown some examples of this fact in previous articles, we haven’t touched any theorems involving integration in Vector Calculus. In this article, we’re going to prove these theorems by proving the Generalized Stokes Theorem.
Check Your Understanding
I said that you can prove almost everything in Vector Calculus with differential forms, so let’s do it.
Use differential forms to prove Green’s first and second identities.
Integration By Parts
Use differential forms to prove the following integration by parts theorems.
Maxwell’s Integral Equations
Use the Generalized Stokes’ Theorem to convert Maxwell’s equations into their integral form.