# Intuition Behind Generalized Stokes Theorem

Consider the Generalized Stokes Theorem:

$$\begin{equation} \int_Md\omega = \int_{\partial{M}} \omega \end{equation}$$ Here, $$\omega$$ is a k-form defined on $$R^n$$, and $$d\omega$$ (a k+1 form defined on $$R^n$$) is the exterior derivative of $$\omega$$. Let M be a smooth k+1-manifold in $$R^n$$ and $$\partial{M}$$ (the boundary of M) be a smooth k manifold.

I know that the above theorem is simply a generalization of well-known vector calculus theorems. However, I am looking for the intuition behind the Generalized Stokes Theorem itself.

I started off by defining the exterior derivative at a point p in $$R^n$$ as: $$\begin{equation} d\omega_p =\lim_{|vol|\to 0}\frac{\int_{\partial{vol}} \omega}{|vol|} \end{equation}$$

In this case, "$$vol$$" represents a k+1 "parallelpiped" in $$R^n$$ that contains point p (with $$|vol|$$ being its "volume"). $$\partial{vol}$$ represents the boundary of this k+1 "parallelpiped", a k "parallelpiped" itself.

With this definition (assuming it is correct), can we say that $$\omega$$ represents an infinitesimal "flux" element through $$\partial{vol}$$ which would imply that $$d\omega_p$$ is simply the "flux density" at a point p?

If the above is true, can we take the idea that (when applying the Generalized Stokes Theorem) the interior "fluxes" through each $$\partial{vol}$$ within M cancel out leaving us with the total "flux" out of $$\partial{M}$$ as the intuition behind the Generalized Stokes Theorem?

Any help is much appreciated.

• The divergence operator can be discovered by computing the flux of a vector field over the surface of a tiny cube. (Physicists often present this argument.) The divergence theorem then becomes intuitive, by thinking of a volume as being chopped up into tiny cubes. I suspect that, analogously, the $d$ operator can be discovered by computing the integral of a differential form over the boundary of a tiny parallelopiped. The generalized Stokes theorem would then become truly intuitive by thinking of a manifold as being chopped up into tiny parallelopipeds. Aug 8, 2019 at 18:10
• I felt that defining dw in the way that I did made the most sense considering the definition of divergence. Hopefully, the analogy holds. Aug 8, 2019 at 18:19
• How does your definition of exterior derivative make sense? $d\omega$ is a $(k+1)$-form, and yet you wrote down a real number on the right-hand side. Indeed, it only makes sense to integrate $\omega$ over the boundary of a volume when $\omega$ is an $(n-1)$-form. Aug 8, 2019 at 18:33
• That is a good point. Maybe we could refine the definition of dw by using a similar "flux density at a point" definition but making sure both sides of the equation are (k+1) forms? Aug 8, 2019 at 19:07
• @Ted Shifrin Could we not just specify that dw is evaluated at the k+1 tangent vectors that parametrize the (k+1) parallelpiped? Both sides of the equation would have the same rank then, right? Aug 9, 2019 at 22:31

Yes, this is very good intuition for the theorem. All of this can be made precise.

In fact, it is made precise on pages 188-190 of the second edition of Arnold's Mathematical Methods of Classical Mechanics. There Arnold gives the following theorem, where $$\omega$$ is a given $$k$$-form on an $$n$$-dimensional manifold $$M$$.

First, the geometric setup. The idea is to construct a $$(k+1)$$-form by computing its value on a given list of $$k+1$$ vectors.

So let $$\xi_1,\ldots,\xi_{k+1}$$ be tangent vectors in $$T_xM$$, where $$x\in M$$. We can pull these back to $$\mathbb{R}^n$$ in the following way. Choose a coordinate system $$\phi:U\to\mathbb{R}^n$$ for $$x\in U$$ with $$\phi(x)=0$$. The preimages of the $$\xi_i$$ under the differential of $$\phi^{-1}$$ are tangent vectors $$\xi_i^*$$ in $$T_0\mathbb{R}^n$$. But we can naturally identify this tangent space with $$\mathbb{R}^n$$ itself. Let $$\Pi^{*}$$ be the parallelepiped in $$\mathbb{R}^n$$ spanned by these vectors. The map $$\phi^{-1}$$ carries this linear parallelepiped onto a "curvilinear parallelepiped" $$\Pi$$ in $$M$$ (this is like what you call "vol"), the boundary of which is a $$k$$-chain, $$\partial\Pi$$ (which is like what you call the boundary of "vol"). Define $$F$$ to be the integral of the given $$k$$-form $$\omega$$ on the boundary of this curvilinear parallelepiped:

$$F(\xi_1,\ldots,\xi_{k+1})=\int_{\partial\Pi}\omega$$

Theorem. There is a unique $$(k+1)$$-form $$\Omega$$ on $$T_xM$$ which is the principal $$(k+1)$$-linear part at zero of $$F$$, i.e.

$$F(\epsilon\xi_1,\ldots,\epsilon\xi_{k+1})=\epsilon^{k+1}\Omega(\xi_1,\ldots,\xi_{k+1})+o(e^{k+1})$$

as $$\epsilon\to0$$. The form $$\Omega$$ does not depend on the choice of coordinates. And this unique form $$\Omega$$ is precisely $$d\omega$$ (in the usual calculation).

In this sense, $$d\omega$$ is indeed a kind of "flux density" of $$\omega$$.

The proof of the generalized Stokes theorem then follows, as littleO suggests in a comment, by making precise the idea of chopping up the manifold into little parallelepipeds. One just keeps track of all the orientations as one integrates the flux density over these little regions, $$\int_{M}d\omega$$. When you do so carefully, the interior fluxes cancel, leaving only the flux along the boundary, $$\int_{\partial M}\omega$$.