I.10.19 Flux of the vector field

\( \underline{\text { Condition: }} \) Calculate the field flux \( \vec{F} \) through the closed surface \( S \) directly and using the Gauss-Ostrogradsky formula (external normal): \[ \vec{F}=b x \vec{\imath}+c y \vec{\jmath}+z \vec{k}, \quad S:\left\{\begin{array}{c} (a-z)^{2}=x^{2}+y^{2} \\ z=0,(z \leq 0 \leq a) \end{array}\right. \] \[ a=6, b=-1, c=1 \]