Answer
$$0$$
Work Step by Step
Stoke's Theorem: Let us consider that $C$ be the closed curve which encloses surface $S$ for a vector field $F$. This also states that the line integral and the surface integral must be equal. That is,
$\oint_C F \ dx=\iint_S (\nabla \times F) \cdot n \ dS$
Here, $\nabla \times F=\begin{vmatrix} i&j&k \\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ 2xy \sin z & x^2 \sin z & x^2y \cos z \end{vmatrix}=(x^2 \cos z-x^2 \cos z, 2xy \cos z-2xy \cos z, 2x \sin z-2x \sin z =(0,0,0)$
and $n=(-z_x, -z_y,1) =(0,0,0)$
Thus, the surface integral can be computed as:
$\iint_S (\nabla \times F) \cdot n \ dS=\iint_S (0,0,0) \cdot (0,0,0) \ dS\\=0$
