Why can we think the divergence and the curl as some kind of "derivative"?

Question

My question arises from think about why the divergence theorem and the stoke's theorem are considered generalizations of the fundamental theorem of calculus.

Answer 1

The FTC says (under some conditions) $f(b)-f(a)=\int_a^bf'(t)\,dt$. Or writing it more suggestively, we can say $f(b)-f(a)=\int_{[a,b]}df$. Or even more suggestively, we can write this as \begin{align} \int_{[a,b]}df = \int_{\partial [a,b]}f= \int_{\{a,b\}}f \end{align} What this is saying is that the integral over a region $[a,b]$ of the derivative $df=f'\,dt$ is somehow related to the "integral" of the original $f$ over the boundary of the interval, namely the two points $\{a,b\}$ (the minus sign has to do with orientation). In this 1-dimensional case, this is pretty overkill, but it sets the stage for higher dimensions.

Morally, this is what Stokes' theorem is about. The general case looks like \begin{align} \int_Md\omega = \int_{\partial M}\omega.\tag{$*$} \end{align} i.e on the LHS we have the integral of "some kind of derivative" $d\omega$ of an object $\omega$ over a region $M$. Stokes theorem then relates this to the integral of the original thing, $\omega$, but this time the integral is over the boundary $\partial M$.

---

In the classical vector calculus settings, we have the formulas \begin{align} \int_{\Omega}\text{div}(\mathbf{F})\,dV &= \int_{\partial \Omega}\mathbf{F}\cdot \mathbf{n}\,dA \tag{divergence theorem}\\ \int_{S}\text{curl}(\mathbf{F})\cdot \mathbf{n}\,dA &= \int_{\partial S}\mathbf{F} \cdot\mathbf{t}\,dl\tag{classical Stokes} \end{align} where $\mathbf{F}$ is a vector field on $\Bbb{R}^3$, $\Omega$ is some region in $\Bbb{R}^3$, and $S$ is some "surface" in $\Bbb{R}^3$. Again, the divergence and curl are given by some formulas involving partial derivatives. Morally speaking, this has the same meaning as above. The LHS is the integral of "some kind of derivative" over a region. The RHS is the integral of the "original" thing over the boundary.

So Stokes theorem roughly says "integration allows us to relate derivatives inside a region to the behavior on the boundary", and this is a direct generalization of the Fundamental theorem of calculus. In fact (*), which is called the generalized Stokes theorem encompasses all these theorems:

• The FTC is the special case of a $1$-dimensional region of integration (the interval $[a,b]$) and $0$-dimensional boundary (the two boundary points $a,b$).
• The classical Stokes theorem is a special case of $(*)$ with a $2$-dimensional surface with a $1$-dimensional boundary.
• The classical divergence theorem is a special case of a $3$-dimensional volume with a $2$-dimensional boundary.

Answer 2

Both of them give information about the way a function is changing locally.

"Locally" means:

• knowing the value of the function at a point is not enough to know the value of the "derivative" at that point, but
• knowing the value of the function everywhere in some open neighborhood of the point is enough, no matter how small that open neighborhood is.