I.10.10 Flux of the vector field

Calculate the flux of the vector field \( \vec{a} \) through the full surface of the cone directly and using the Gauss-Ostrogradsky formula: \[ \vec{a}=(y-x) \vec{\imath}+y \vec{\jmath}+2 z \vec{k}, \quad S:\left\{\begin{array}{l} z^{2}=x^{2}+y^{2} \\ z=1 \end{array}\right. \]