(D) For an irrotational vector field $\mathbf{F} = \nabla \phi$ across a closed surface $S$, $\iint_S (\nabla \times \mathbf{F}) \cdot \hat{n} \, dS = 0$
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
All four options are fundamental theorems of vector analysis:
1. **Option A**: Green's Theorem relates a line integral around a simple closed plane curve $C$ to a double integral over the bounded planar region $R$ (True).
2. **Option B**: Stokes' Theorem equates the circulation of $\mathbf{F}$ along boundary curve $C$ to the surface flux of its curl (True).
3. **Option C**: Gauss's Divergence Theorem relates outward surface flux to the volume integral of divergence (True).
4. **Option D**: Since $\nabla \times (\nabla \phi) = \mathbf{0}$, the integrand is identically zero, so the surface integral vanishes (True).