How to solve integrations of following types:
- $$ I_1 = \int d\mathbf{r} [ \mathbf{\nabla_r}f(\mathbf{r})]\cdot [ \mathbf{\nabla_{r'}}g(\mathbf{r'})]\{\mathbf{\nabla_r} \mathbf{\nabla_{r'}}[\delta(\mathbf{r}-\mathbf{r'})]\} $$
- $$ I_2 = \int d\mathbf{r} [ \mathbf{\nabla_r}f(\mathbf{r})]\cdot [ \mathbf{\nabla_{r'}}g(\mathbf{r'})]\{\mathbf{\nabla_r }[\delta(\mathbf{r}-\mathbf{r'})\cdot \mathbf{A(r)}]\} $$
here $\mathbf{r}$ and $\mathbf{r'}$ are two different vectors in 3D space, $f(\mathbf{r})$ and $g(\mathbf{r'})$ two scalar functions, and $\mathbf{A(r)}$ is a vector field.
I know that for an integral with one derivative of delta function, one can use distributional derivatives as $\int dx f(x) \delta^{(n)}(x-x_0)=(-1)^nf^{(n)}|_{x=x_0}$. But I am confused how to translate this rule for two derivatives of delta functions at different points and a derivative of delta function which dot product of some other vector field.