设函数 $f(x, y, z)$ 在单位球 $B=\left\{(x, y, z) \mid x^2+y^2+z^2 \leq 1\right\}$ 上连续可微，且当 $(x, y, z)$ 满足 $x^2+y^2+z^2=1$ 时，$f(x, y, z)=0$ 。证明：

$$
\begin{aligned}
& \text { (I) } \iiint_B\left(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}+z \frac{\partial f}{\partial z}\right) \mathrm{d} x \mathrm{~d} y \mathrm{~d} z \\
& =-3 \iiint_B f(x, y, z) \mathrm{d} x \mathrm{~d} y \mathrm{~d} z \\
& \text { (II) }\left|\iiint_B f(x, y, z) \mathrm{d} x \mathrm{~d} y \mathrm{~d} z\right| \\
& \leq \frac{\pi}{3} \max _{(x, y, z) \in B}\|\nabla f(x, y, z)\| \text {, 其中 } \nabla f=\left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right) \text { 。 }
\end{aligned}
$$