= Solution
For a sufficiently smooth <vector field> on a volume $V$ and its closed boundary $S$, the <divergence theorem> states
$$
\boxed{\int_S\mathbf v\cdot\mathbf n\,dS
=\int_V\boldsymbol\nabla\cdot\mathbf v\,dV,}
$$
with the <unit normal> directed outwards. Here
$$
\boldsymbol\nabla\cdot\mathbf v
=\partial_x(-\alpha yz)+\partial_y(\alpha xz)+\partial_z0=0.
$$
Hence \b[the net volume <flux> through every <closed surface> in the liquid is zero]. This verifies <incompressible flow> directly. It does not require the <velocity> to vanish pointwise on an arbitrary <closed surface>; inward and outward contributions can cancel.
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