= Solution
<Stokes theorem> equates the circulation of a continuously differentiable <vector field> around an oriented boundary to the flux of its <curl> across a spanning oriented <surface>:
$$
\oint_{\partial S}\mathbf A\cdot d\mathbf r=\int_S(\nabla\times\mathbf A)\cdot\hat{\mathbf n}\,dS.
$$
The boundary direction follows the <right-hand rule> relative to the <normal vector>. For the polynomial <vector field>, only the $z$ component of the <curl> can be nonzero, and
$$
\boxed{\nabla\times\mathbf A=\left(\frac{\partial(2xy)}{\partial x}-\frac{\partial(x^2-y^2)}{\partial y}\right)\hat{\mathbf z}=4y\hat{\mathbf z}.}
$$
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