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Consider a surface integral over a closed surface $S$ in $\mathbb{R}^3$

$$ \iint_S V\vec{E} \cdot d\vec{S}$$

(I'm writing out $V$ and $\vec{E}$ to make connection with physics textbooks, and to show the motivation behind this question, where $\vec{E}$ is the electric field of a continuous charge distribution and $V$ is the corresponding potential. Let's think of the integrand as a function $V\vec{E} = \vec{F}: \mathbb{R}^3 \to \mathbb{R}^3$)

(What follows isn't even my main question, but it would be useful to setup in order to answer my main question on convergence seen further below) How do you mathematically specify that you want to expand the surface so that it encloses all of space? I would like an 'improper surface integral', in the same way that we have improper 1-dimensional integrals

$$ \int_a^\infty f(x) dx = \lim_{R\to \infty} \int_a^R f(x) dx$$ Is there any such notation or is

$$ \iint _{\text{surface enclosing all space}} \vec{F}\cdot d\vec{S}$$ the best you can do? Technically such a surface doesn't exist? You need some sort of limit to keep expanding the surface further and further?

To phrase this initial notation question in another way, consider a parameterization of the closed surface $g(\theta, \phi): \mathcal{D}\subset \mathbb{R}^2 \to \mathbb{R}^3$

In particular, consider a spherical coordinates parameterization of the surface $g(\theta, \phi) = r(\theta, \phi) \hat{r} + 0\hat{\theta} + 0\hat{\phi}$ where $r$ outputs a radial distance from the origin. How do I show that I want to "push" the output $r$, for every given $(\theta, \phi)$ pair, out to infinity so that my surface encloses all of space? There is no unique way to push this surface outwards. It seems like I'd have to give a sequence of functions that take me from the initial enclosing radial function $r_1$ to some final radial function $r_2$ which encloses all of space?

Question on convergence

Given that we have some sort of notation for a surface enclosing all of space, how do we evaluate if the surface integral converges or not over this limiting sequence? In particular, lets consider an integrand which physics textbooks claim make the integral $0$. They argue (in words, which I will try to translate into math) given

$$ \lim_{r\to\infty}\vec{F}(r,\theta,\phi) \sim \frac{1}{r^3}\hat{r}$$
i.e. integrand is asymptotically equivalent to the vector field $(1/r^3) \hat{r}$ at large distances, the surface integral over all space is $0$ since $d\vec{S}$ goes as $r^2$ ($d\vec{S} = r^2\sin\theta\; d\theta \;d\phi \;\hat{r}$). Is there mathematical rigor behind this statement? Considering my notation and thoughts above, it seems like the mathematical problem would be phrased as

$$ \lim_{\text{sequence of parameterizations} \;g_i} \iint_{g_i} \vec{F}(g(\theta,\phi)) \cdot |g_{\theta} \times g_{\phi}| \; d\theta \; d\phi $$ where

$$ g_{\theta} = \frac{\partial r}{\partial \theta}\hat{r} + r \hat{\theta}$$ $$ g_{\phi} = \frac{\partial r}{\partial \phi}\hat{r} + r \sin\theta\hat{\phi}$$

where I have used that $\partial \hat{r}/ \partial \theta = \hat{\theta}$ and $\partial \hat{r}/\partial \phi = \sin\theta \hat{\phi}$. The cross product yields

$$ g_{\theta} \times g_{\phi} = \hat{r}(r^2\sin\theta) + \hat{\theta}(-r\frac{\partial r}{\partial \theta}\sin\theta) + \hat{\phi}(-r\frac{\partial r}{\partial \phi}) $$

Therefore if we could somehow get the limit inside the integral, to turn $\vec{F}$ to $\hat{r}/r^3$, the dot product would result in the integral

$$ \lim_{\text{sequence of} g_i}\iint_{g_i} \frac{1}{r^3}r^2\sin\theta d\theta d\phi$$

(Can you make that replacement?) And I guess this now goes to $0$? So in summary, is my notation/thoughts on pushing a surface out to infinity so that it encloses all of $\mathbb{R}^3$ valid/the way to go? What is the mathematical proof behind improper surface integrals converging to $0$? The convergence should be defined so that it is independent of "path" right? Meaning, it doesn't matter what sequence of $g_i$'s I choose? I might as well start with a sphere radius $a$ and just let $a$ tend to infinity? Whatever value I get (say it converges), I hope that it's the same value for all other sequences of $g_i$'s.

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