The Generalized Stokes Theorem

In a class on vector calculus, the students are presented with a few theorems such as the Fundamental Theorem of Line Integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem. The first one relates differentiation and integration in the following way. Take the gradient of a multivariable function $f:\mathbb{R}^n \to \mathbb{R}$ and consider a smooth path parametrized by $\vec{r}(t)$ with $t\in [a,b]$. Then the definite integral of the dot product $\nabla f \cdot d\vec{r} \, dt$, a quantity arising from a 1-dimensional object has a 0-dim interpretation: $\int^a_b \nabla f \cdot d \vec{r}\,dt = f(\vec{r}(b)) - f(\vec{r}(a))$. The proof of this really is just to use the usual Fundamental Theorem of Calculus.

Green’s theorem is a statement about an integral over some compact 2-dim region with boundary and relating that to an integral along the boundary. More precisely, let $R$ be the region and suppose is has smooth boundary which is a (possible disconnected) simple curve with positive orientation. Let $P,Q:\mathbb{R}^2 \to \mathbb{R}$ be once-differentiable functions. Then: $\iint_{R}\left({\frac {\partial Q}{\partial x}}-{\frac {\partial P}{\partial y}}\right)dx\,dy = \oint_{\partial R}(P dx+Q dy)$. So here, we equate some integral over a 2-dim region of differentiated functions with an integral over a 1-dim object of the undifferentiated functions.

The vector calculus Stokes’ theorem is similar and indeed, generalizes Green’s theorem. It’s about a surface with boundary in $\mathbb{R}^3$ and we take into consideration orientation. The derivative involved is the curl which is defined via the cross product of $\mathbb{R}^3$, a structure that doesn’t appear in any other dimension other than 7.

Lastly, the Divergence theorem relates an integral over a bounded region of $\mathbb{R}^3$ of the divergence of a vector field with an integral over the boundary of the region of the vector field dot product with an outward pointing unit normal vector field on the bounding surface.

Of course, these theorems are all unified by what I call the Stokes’ Theorem but what some call the generalized Stokes’ theorem. Given an $n$-manifold-with-boundary $M$ and an exact top form $d\omega$, we have $\int_M d\omega = \int_{\partial M} \omega$. The gradient, curl, and divergence are all special cases of the exterior derivative $d$ and one uses the Riemannian metric structure to move from differential forms to vector fields.

Lastly, to me, the generalized Stoke’s theorem is as follows. Consider a smooth submersion $\pi:X \to B$ of smooth manifolds where $X$ has boundary. Suppose $\pi|_{\partial X}$ is also a submersion. Then this means we have a fiber bundle and we may integrate out the fiber; i.e. have the pushforward $\pi_!$. Let $\alpha$ be any differential form on $X$. We have the following result: $d(\pi_! \alpha) = \pm \pi_!(d\alpha)\mp (\pi|_{\partial X})_! (\alpha)$.

In the special case that $B=pt$, then the pushforward gives a differential form on $\alpha$ which is always zero unless $\alpha$ is just a function. But then, $d$ of it is zero. So we just have $\pi_!(d\alpha)=(\pi|_{\partial X})_! (\alpha)$. When the entirety of $X$ is the fiber, then the integration of the fiber is just usual integration: $\int_X d\alpha = \int_{\partial X} \alpha$. This is the Stokes’ theorem from before.