Use the divergence theorem to calculate the surface integral $\iint_{S} F \cdot d S$; that is, calculate the flux of $F$ across $S$.

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: \(\boxed{0}\)