 Surface delta distribution

For a smooth hypersurface $S$, its surface delta is the distribution $⟨{\delta }_S,\phi ⟩={\int }_S\phi dS$. Its normalization uses geometric surface measure rather than a defining function. On the unit circle, ${\delta }_{S^1}$ pairs by integrating $\phi (\cos t,\sin t)$ over $0\le t\le 2\pi$.