I just thought of an interesting example of a surface integral. Perhaps someone with a background in differential forms/integration of manifolds can help with this. A general multivariable calculus perspective would be useful, too.
Can surface integrals be defined over the entirety of $\mathbb{R}^3$? Both in terms of vector and scalar fields. Intuitively, at least with regard to flux integrals, this should work. If we allow the surface integral to enclose $\mathbb{R}^3$, then the Divergence Theorem can show, for example, that if $\nabla \cdot \vec{F} = \frac{1}{(1+x^2)(1+y^2)(1+z^2)}$, then $\iiint_{\mathbb{R}^3} \nabla \cdot \vec{F} dV = \pi ^3.$ Can this be interpreted as, for example, doing a surface integral over a sphere of radius $R$ and letting $R \to \infty$? This is also an example of using the divergence theorem for a flux integral, but can we do the same thing over a scalar field? Is there a function $f(x,y,z)$ such that $\oint_S f dS$ is a finite value, with $S$ being a sphere of radius $R$ as $R \to \infty$? And if we can interpret a flux integral as flux through a surface, does this surface integral mean…… a flux coming out of the whole of $\mathbb{R}^3$? And if it's a scalar field, and we interpret $f(x,y,z)$ as a mass-density function, is this the mass of the exterior of $\mathbb{R}^3$…..?
Is someone able to give a little bit of an analysis on this? Thanks.
EDIT: I realize now I've misused the term "defined on $\mathbb{R}^3$" with regard to surface integrals. What I mean is that a surface integral to have a surface that encloses $\mathbb{R}^3$. Have the interior be $\mathbb{R}^3$.
Best Answer
Let $B_R$ be the ball centered on the origin of radius $R$ and $S_R$ its boundary (the sphere)
Then by divergence theorem :
$$\iiint_{B_R} \nabla \cdot \vec F ~d\vec V = \iint_{S_R} \vec F~ d\vec S.$$
Then we make $R \to \infty$. If $\nabla \cdot \vec F$ is integrable, then dominated convergence theorem ensures that taking this limit is the same as integrating on the whole space $\mathbb{R}^3$ in the first place and this is how far you can go. An eventual surface enclosing the whole space can be seen, in this context, as a limit of the integral over larger and larger ones.
From a "flux interpretation", the integrability ensures that most of the contribution to the flux is reachable with bounded surfaces. Indeed you will able to approximate the integral with an arbitrary accuracy using the appropriate bounded surface. Formally speaking it means that an underlying physical system will lose/radiate or receive energy. For example in Electromagnetics this would be connected to the power radiated by an antenna.