Is divergence theorem applicable for improper integrals?

Question

Consider the expression:

$$\lim\limits_{\delta \to 0} \iiint_{V-\delta} \nabla \cdot \mathbf{F}(x,y,z)\ dV \tag1$$

where $\delta$ is a small volume inside volume $V$

Now can we apply the divergence theorem and write $(1)$ as:

$$\iint_{\partial V} \mathbf{F}(x,y,z) \cdot \hat{n}\ dS +\lim\limits_{\delta \to 0} \iint_{\partial \delta} \mathbf{F}(x,y,z) \cdot \hat{n}\ dS$$

Answer 1

The divergence (Gauss-Green) theorem can be used to define the improper integral of the divergence of (weakly) singular vector fields $\mathbf{F}$ with isolated singular points $\mathbf{p}_o=(x_0,y_o,z_0)\in V$. Customarily, the definition goes as follows $$ \begin{split} \int\limits_{V}\nabla\cdot\mathbf{F}(x,y,z)\,\mathrm{d}V& \triangleq \lim_{R\to 0} \Bigg[\,\int\limits_{V\setminus B(\mathbf{p}_o,R)} \nabla\cdot\mathbf{F}(x,y,z)\, \mathrm{d}V - \int\limits_{\partial B(\mathbf{p}_o,R)} \mathbf{F}(x,y,z) \cdot \hat{n}\ dS\Bigg]\\ \\ &\triangleq \int\limits_{\partial V} \mathbf{F}(x,y,z) \cdot \hat{n}\ dS, \end{split}\label{1}\tag{1} $$ where, for the small volume $\delta$, a small ball $B(\mathbf{p}_o,R)$ with radius $R>0$ centered on the singular point of $\mathbf{F}$ is customarily chosen. The definition is clearly consistent if and only if the limits of the two integrals in formula \eqref{1} exist and are finite.

An example.
The most famous example of use of \eqref{1} as a definition is perhaps the calculation of the integral of the divergence of the following field: $$ \begin{split} \mathbf{F}(x,y,z)&=\nabla{\bigg[\sqrt{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2 }\,\bigg]^{-1}}\\ &=\nabla\frac{1}{|\;\mathbf{p}-\mathbf{p}_o|} \end{split} $$ where $\mathbf{p}=(x,y,z)\in V$. This vector field is, apart from a multiplicative constant, the gradient of the fundamental solution of the laplacian: therefore, the integral of the divergence of this vector field is zero in every domain $V\subset\Bbb R^3\setminus\mathbf{p}_o$ since $\nabla\cdot\mathbf{F}$ is zero in such domains. However, applying \eqref{1} we have $$ \begin{split} \int\limits_{V}\nabla\cdot\mathbf{F}(\mathbf{p})\,\mathrm{d}V&= -\lim_{R\to 0} \int\limits_{ \partial B(\mathbf{p},R)} \nabla\frac{1}{|\;\mathbf{p}-\mathbf{p}_o|} \cdot\hat{n}\, \mathrm{d}S \\ &= -\lim_{R\to 0} \int\limits_{ \partial B(\mathbf{p},R)} \frac{ \partial }{\partial \hat{n}} \frac{1}{|\;\mathbf{p}-\mathbf{p}_o|} \mathrm{d}S\\ &=-\lim_{R\to 0} \int\limits_{ \partial B(\mathbf{p},R)} \frac{ \partial }{\partial r} \frac{1}{r}\mathrm{d}S\\ &=\lim_{R\to 0} \frac{1}{R^2} \int\limits_{ \partial B(\mathbf{p},R)} \mathrm{d}S = 4\pi, \end{split} $$ and thus we can also define the flux of $\mathbf{F}$ throug $\partial V$.

Final notes

• All the above development is done under the hypothesis $\delta=B(\mathbf{p},R)$: however, formula \eqref{1} is valid for more general classes of small volumes $\delta$, provided that the singularity of $\mathbf{F}$ is sufficiently "weak".
• Let's precise the exact meaning of the locution "weak singularity" in the context of fields with a single isolated singularity. Being the field $\mathbf{F}$ singular, we can say that, near the singular point $\mathbf{p}_o\in V$, $$ |\mathbf{F}(\mathbf{p})|\le K{|\;\mathbf{p}-\mathbf{p}_o|^{-\alpha(|\mathbf{p}-\mathbf{p}_o)|}}\label{2}\tag{2} $$ where $\alpha:\Bbb R_+\to \Bbb R_+$ is a non negative function ($\alpha\ge0$). Then we have that $$ \lim_{R\to 0}\bigg|\int\limits_{ \partial B(\mathbf{p},R)}\mathbf{F}(\mathbf{p})\cdot\hat{n}\, \mathrm{d}S \Bigg|<\infty\iff \lim_{R\to 0}R^{-\alpha(R)+2}<\infty\iff \lim_{R\to 0} \alpha(R)\le 2 $$ Be it noted that this condition is stronger than the simple condition of local integrability of the field $\mathbf{F}$: this one implies that $$ \lim_{R\to 0} \alpha(R)<3 $$ in estimate \eqref{2}, thus there are locally integrable vector fields for which formula \eqref{1} is not applicable.
• As I stated clearly at the beginning of my answer, formula \eqref{1} is not really a divergence (Gauss-Green) theorem for singular vector fields: it is a definition which uses the standard theorem applied to non-singular regions of $F$ (by eventually cutting small volume pieces $\delta$) to extend its range of applicability to a class of singular vector fields. Therefore you will not find it stated as a theorem: however, in books on partial differential equations which do not use the theory of distributions, \eqref{1} is silently used in the proof of Green's formula. See for example Tikhonov and Samarskii [1], chapter IV, §2.1, pp. 316-318.

[1] A. N. Tikhonov and A. A. Samarskii (1990) [1963], "Equations of mathematical physics", New York: Dover Publications, pp. XVI+765 ISBN 0-486-66422-8, MR0165209, Zbl 0111.29008.