Step 1 :Given a vector field \(F = (P, Q, R)\) and a closed surface \(S\) enclosing a volume \(V\).
Step 2 :The divergence theorem, also known as Gauss's theorem, states that the net outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface. In mathematical terms, it is expressed as: \[\iint_{S} F \cdot d S = \iiint_{V} \nabla \cdot F d V\]
Step 3 :The divergence of the vector field \(F = (P, Q, R)\) is zero, which means that the vector field is divergence-free or solenoidal.
Step 4 :This implies that the net outward flux of \(F\) across any closed surface \(S\) enclosing a volume \(V\) is also zero, according to the divergence theorem.
Step 5 :Therefore, the surface integral \(\iint_{S} F \cdot d S\) is zero.
Step 6 :Final Answer: \(\boxed{0}\)