= Solution
The field is the <gradient> of a globally defined polynomial:
$$
\phi(x,y,z)=2xy+y^2z,\qquad
\nabla\phi=(2y,2x+2yz,y^2)=\mathbf v.
$$
It is therefore a <conservative vector field> on all of $\mathbb R^3$. The <fundamental theorem for line integrals> gives
$$
\boxed{\oint_C\mathbf v\cdot d\mathbf l=\phi(C_{\rm end})-\phi(C_{\rm start})=0}
$$
for every closed piecewise smooth loop. Constructing the global <vector-field potential> avoids relying on a zero-<curl> test without checking the topology of the domain.
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