How can I simplify
$\nabla \cdot \left(\vec{f}(x)\delta_S(x)\right)$ where $\nabla \cdot$ is 3D divergence operator and $\vec{f}$ is a 3D vector valued function. The delta function $\delta_S(x)$ is delta measure defined on a surface in the volume.
I tried
$=\left(\nabla \cdot \vec{f}(x)\right)\delta_S(x)+ \vec{f}(x)\cdot \nabla\delta_S(x)$
Is this true $\vec{f}(x)\cdot \nabla\delta_S(x) = -\nabla \cdot \vec{f}\delta_S(x)$?
How do I simplify this? Am I going in the right direction?