I.10.11 Flux of the vector field

Calculate the flux of the vector field \( \vec{a}=(x-y) \vec{\imath}+ \) \( (2 x+y) \vec{\jmath}+\left(x^{2}+2 z+4\right) \vec{k} \quad \) through a closed surface bounded by the surfaces \( S_{1}, S_{2} \) directly and using the Gauss-Ostrogradsky formula: \[ S_{1}: x^{2}+y^{2}=z+2, \quad S_{2}: z=0 \text {. } \]