Proving Stokes' Theorem for Beginners

[Amsingh123]

How would one prove Stokes' Theorem? I'm 15. I learned about Stokes' Theorem recently and I have a decent understand of it, but I thought that it would be useful to know it's derivation. Thanks for your help, PF.

 

[leo.]

A good proof of Stokes' Theorem involves machinery of differential forms. Usually basic calculus do proofs of very special cases in three dimensions and the proofs usually doesn't reveal much of the idea behind. I would recommend you to take a look at Spivak's Calculus on Manifolds book, there's a very good exposure there using the notion of integrations over chains, which in my opinion simplify the work.

The pre reqs to read this book is analysis over the real line. Then for the beginning of the book that talks of the topology of euclidean spaces you'll probably like to see other books together like Munkres' Analysis on Manifolds.

I think it's the best way to reach a proof of Stokes' theorem.

 

[Amsingh123]

Ok, thanks for your help.

 

[SteamKing]

A good start is to understand Green's Theorem in the plane, and then generalize to higher dimensions.

These articles may help:

http://en.wikipedia.org/wiki/Stokes'_theorem

http://tutorial.math.lamar.edu/Classes/CalcIII/StokesTheorem.aspx

The references in the first article give some of the history behind this Theorem. It was actually discovered first by Lord Kelvin who included it in a letter to Stokes in 1850. In 1854, Stokes gave it as an examination question for the Smith's Prize. One of the students who took this exam and tied for first place was Clerk Maxwell.

 

[homeomorphic]

I like the physics/engineering approach to Stokes theorem. That yields a little more intuition. If you want a clean proof, then the place to look is differential forms, but that takes a little effort to learn (and if you understand differential forms well enough, you can see how it relates to the physics intuition). A good place to learn about differential forms is Mathematical Methods of Classical mechanics, by Vladimir Arnold--he's the only one I've seen so far who explains them in a way that meshes with the physics/engineering intuition.

The curl of a vector field is like an infinitesimal line integral. That's basically the idea. You want to integrate around a big loop, so you break it up into lots of little tiny loops. In the limit, the little tiny loop integrals approach the curl of the vector field dotted with a normal vector. So you add up the curl near each point to get the integral around the big loop.

You can draw a little picture and calculate what that infinitesimal loop integral is for a tiny square parallel to each coordinate plane. Those are the components of the curl vector field.

It's hard to explain very clearly without being able to draw pictures and stuff, but that's the general idea. A good book on electromagnetism, for example, should explain this in more detail.

 

[mathwonk]

fubini + fundamental theorem of calculus = stokes.