Why is it that there are so many instances in analysis, both real and complex, in which the values of a function on the interior of some domain are completely determined by the values which it takes on the boundary?
I know that this has something to do with the general version of Stokes's theorem, but I'm not advanced enough to understand this yet — does anyone have a (semi) intuitive explanation for this kind of phenomenon?
Best Answer
Well it might be easier to start with the version of stokes theorem you probably know best, the fundamental theorem of calculus: $\int_a^b df = f(b) - f(a)$ (when applicable).
A sketch of (a) proof is that $\int_a^b df = \lim_{N \to \infty} \Sigma_1^N (x_i - x_{i-1})df(y_i)$, where $y_i \in [x_{i-1}, x_i]$. By the mean value theorem, we can choose $y_i \in [x_{i-1}, x_i]$ such that $f(x_i) -f(x_{i-1}) = (x_i - x_{i-1})df(y_i)$, and so in the sum you cancel a lot. But you can't continue this process by virtue past the edges, the buck stops there.
Somehow, in some of these things, you can think of the problem has being pushed off to the sides, because a little tiny "cell" and their interactions in the interior have certain nice properties that lets you do so, but the edge lacks this.
Another example is the residue theorem; nothing exciting happens except at the poles.
This might be a bit tangental, is in de Rham cohomology, contractible sets, the building blocks of manifolds are boring, but how they patch together is not. Interesting cohomology is a emergent phenomenon.