解

由斯托克斯公式得

\[

\begin{aligned}

I &= \iint_{\sum}

\begin{vmatrix}

dydz & dzdx & dxdy \\

\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\

yz^{2}-\cos z & 2xz^{2} & 2xyz + x\sin z

\end{vmatrix} \\

&= \iint_{\sum} -2xzdydz + z^{2}dxdy.

\end{aligned}

\]

补面$\sum_{1}:x = 0$，取后侧，$\sum_{2}:y = 0$，取左侧，$\sum_{3}:z = 0$，取下侧， 所以$\sum + \sum_{1}+\sum_{2}+\sum_{3}$围成了封闭的曲面，取外侧，所围空间几何体为$\Omega$，由高斯公式可得

\[

\begin{aligned}

I &= \underset{\sum_{4}}{\iint} -2xzdydz + z^{2}dxdy - 0 \\

&= \underset{\Omega}{\iiint} (-2z + 0 + 2z) \, dV \\

&= 0.

\end{aligned}

\]