(Solved):   (b) Use the Divergence Theorem to evaluate \[ \iint_{S} \overrightarrow{\mathrm{F}} \cdot ...

 

(b) Use the Divergence Theorem to evaluate \[ \iint_{S} \overrightarrow{\mathrm{F}} \cdot d \vec{S} \] where \( \overrightarrow{\mathbf{F}}=y x^{2} \vec{i}+\left(x y^{2}-3 z^{4}\right) \vec{j}+\left(x^{3}+y^{2}\right) \vec{k} \) and \( S \) is the surface of the sphere of radius 4 with \( z \leq 0 \) and \( y \leq 0 \). \( [14] \) \[ x x^{2}+y^{2}+z^{2}=4 \]