I have an acoustic research problem that leads to the following integral formulation:
\begin{align}
\int\_{-\infty}^{\infty}\int\_{-\infty}^{\infty}p(\mathbf{y},\tau)\frac{\partial}{\partial y\_i}\left(n\_i \delta(f) |\nabla f|\right) g(\mathbf{x} - \mathbf{y}, t-\tau) d^3\mathbf{y} d\tau,
\end{align}
 
where $p$ is a scalar function, $n\_i=\nabla f / |\nabla f|$ is the normal vector of my control surface, and $\delta(f)$ is the dirac delta with my level set function, $f$, the zero set of which describes the control surface. $g$ is a numerical green's function. If legal, I would like to use the vector property,
\begin{align}
\nabla \cdot (\psi \mathbf{v}) = \nabla \psi \cdot \mathbf{v} + \psi \nabla \cdot \mathbf{v},
\end{align}
to convert the integrand to a form such as 
\begin{align}
\int\_{-\infty}^{\infty}\int\_{-\infty}^{\infty}p(\mathbf{y},\tau)\left[ \frac{\partial n\_i}{\partial y\_i} \left(\delta(f) |\nabla f|\right) + n\_i\frac{\partial}{\partial y\_i} \left(\delta(f) |\nabla f|\right) \right] g(\mathbf{x} - \mathbf{y}, t-\tau) d^3\mathbf{y} d\tau.
\end{align}
However, since $\delta(f)$ is a distribution, I am not quite sure if   
(1) this is a correct manipulation;   

(2) how to deal with the second term in the bracket?   
  

I want to convert this to a surface integral to evaluate my boundary sources, and the first term gives me just that. Thanks in advance.



acoustic dipole volume integral w/ dirac delta?
First do an integration by parts:


$$\int F(\mathbf{y})\frac{\partial}{\partial y\_i}\left(n\_i\delta(f)|\nabla f|\right)\,d^n\mathbf{y}=
-\int \delta(f)|\nabla f|n\_i\frac{\partial}{\partial y\_i}F(\mathbf{y})\,d^n\mathbf{y}.$$


Then use this [identity](https://en.wikipedia.org/wiki/Dirac_delta_function#Properties_in_n_dimensions),


$$\int A(\mathbf{y})\delta(f(\mathbf{y}))d^n \mathbf{y}=\int\_{S}\frac{A(\mathbf{y})}{|\nabla f|}\,d^{n-1}\mathbf{y},$$


with $S$ the surface defined by $f(\mathbf{y})=0$, to arrive at


$$\int F(\mathbf{y})\frac{\partial}{\partial y\_i}\left(n\_i\delta(f)|\nabla f|\right)\,d^n\mathbf{y}=
-\int\_S n\_i\frac{\partial}{\partial y\_i}F(\mathbf{y})\,d^{n-1}\mathbf{y}.$$


