Prove that $\iint_S \text{curl }\textbf{F} \cdot d\textbf{S} = 0$ where $S$ is a sphere.

Question

Prove without using the divergence theorem.

The proof using the divergence theorem is very obvious, but I need the proof which does not rely on the divergence theorem.

Thanks in advance.

Answer 1

$$ \nabla\times\vec{F}\cdot{\rm }{\rm d}\vec{\rm S} = \nabla\cdot\left(\vec{F}\times{\rm d}\vec{\rm S}\right) $$