Prove $\iiint_{\mathbb{R}^3}\left ( \frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}+\frac{\partial f}{\partial z} \right) dxdydz=0$ Assume $f(x,y,z)$ is continuously differentiable on $\mathbb R^3$, and both the integral
$$
\iiint_{\mathbb{R}^3}{\left| f\left( x,y,z \right) \right|\text{d}}x\text{d}y\text{d}z
$$
and
$$
\iiint_{\mathbb{R}^3}{\left( \frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}+\frac{\partial f}{\partial z} \right) \text{d}}x\text{d}y\text{d}z
$$
exists. Prove that
$$\iiint_{\mathbb{R}^3}{\left( \frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}+\frac{\partial f}{\partial z} \right) \text{d}}x\text{d}y\text{d}z=0$$
I don't have any ideas about the problem. Can anyone help?
 A: Let $\mathcal{C}_r = [-r,r]^3$ denote the cube of side length $2r$ centered at $0$. Then
\begin{align*}
\left| \int_{\mathcal{C}_r} \frac{\partial f}{\partial x} \, \mathrm{d}x\mathrm{d}y\mathrm{d}z \right|
&= \left| \int_{[-r,r]^2} \left( f(r, y, z) - f(-r, y, z) \right) \, \mathrm{d}y\mathrm{d}z \right| \\
&\leq \int_{\partial\mathcal{C}_r \cap \{x=\pm r\}} \left| f(\mathbf{x}) \right| \, \sigma(\mathrm{d}\mathbf{x}),
\end{align*}
where $\sigma$ is the surface measure on $\partial\mathcal{C}_r$. Since similar results hold along the other coordinate directions, we have
\begin{align*}
\left| \int_{\mathcal{C}_r} \left(\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}+\frac{\partial f}{\partial z}\right) \, \mathrm{d}x\mathrm{d}y\mathrm{d}z \right|
&\leq f(r), \qquad\text{where} \quad f(r) := \int_{\partial\mathcal{C}_r} \left| f(\mathbf{x}) \right| \, \sigma(\mathrm{d}\mathbf{x}).
\end{align*}
However, since
$$ \int_{0}^{\infty} f(r) \, \mathrm{d}r
= \int_{\mathbb{R}^3} \left| f(\mathbf{x}) \right| \, \mathrm{d}x\mathrm{d}y\mathrm{d}z
< \infty, $$
we have $\liminf_{r\to\infty} f(r) = 0$ and hence there exists a sequence $(r_n)_{n\geq 1}$ with $r_n \to \infty$ such that $f(r_n) \to 0$ as $n\to\infty$. From this, we obtain
$$ \left| \int_{\mathbb{R}^3} \left(\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}+\frac{\partial f}{\partial z}\right) \, \mathrm{d}x\mathrm{d}y\mathrm{d}z \right|
= \lim_{n\to\infty} \left| \int_{\mathcal{C}_{r_n}} \left(\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}+\frac{\partial f}{\partial z}\right) \, \mathrm{d}x\mathrm{d}y\mathrm{d}z \right|
\leq 0 $$
and therefore the desired conclusion follows.
