I.10.35 Flux of the vector field

Condition: Find the flow of the vector field \( \vec{a} \) through the closed surface \( S \) (external normal), using the Ostrogradsky-Gauss formula. Having chosen a side of the surface, directly find the flow of the vector field \( \vec{a} \) through the surface \( S_{1} \), which is part of the surface \( S \) and defined by the given equations. \[ \vec{a}=2 x^{2} y \vec{\imath}+x y^{2} \vec{\jmath}+(z-2) \vec{k} \] \( S:\left\{\begin{array}{c}z=2-x^{2}-y^{2} \\ z \geq 0\end{array}, \quad S_{1}: z=2-x^{2}-y^{2}\right. \).