(Solved): 10. Use the divergence theorem to calculate surface integral \( \iint_{S} \overrightarrow{\mathbf{F ...

10. Use the divergence theorem to calculate surface integral \( \iint_{S} \overrightarrow{\mathbf{F}} \cdot d S \), where \( \overrightarrow{\mathbf{F}}(x, y, z)=\left(e^{2} \hat{\mathbf{i}}+\left(y+\sin \left(z^{2}\right)\right) \hat{\mathbf{j}}+(z-1) \hat{\mathbf{k}}\right. \) and \( S \) is upper hemisphere \( x^{2}+y^{2}+z^{2}=1, z \geq 0 \), oriented upward. Answer