The Generalized Stokes’ Theorem

Joseph Mellor
20 min readMay 16, 2023

If there were one theorem to rule them all, it would be the Generalized Stokes’ Theorem.

This is stop 19 on the Road to Quantum Mechanics.

Over the past few articles, we’ve been setting up the fundamental concepts behind differential forms:

Furthermore, we came up with

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.

Green’s Identities

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.

--

--

Joseph Mellor

BS in Physics, Math, and CS with a minor in High-Performance Computing. You can find all my articles at https://josephmellor.xyz/articles/.