What are the gradient, divergence and curl of the three-dimensional delta function?

The three-dimensional delta function is defined as follows:

$$\delta(\mathbf{r}-\mathbf{r'})= 0 \;\; \mathrm{for} \;\;\mathbf{r}\neq\mathbf{r'}$$ $$\delta(\mathbf{r}-\mathbf{r'})= \infty \;\; \mathrm{for} \;\;\mathbf{r}=\mathbf{r'}$$

$$\int_V\delta(\mathbf{r}-\mathbf{r'})\;dV= 1 .$$

By definition also: $$\int_V f(\mathbf{r})\delta(\mathbf{r}-\mathbf{r'})\;dV= f(\mathbf{r'})$$.

I am wondering what the vector calculus operators are for the delta function. For example:

Curl:

$\nabla \times \delta(\mathbf{r}-\mathbf{r'})$

Divergence:

$\nabla \cdot \delta(\mathbf{r}-\mathbf{r'})$

$\nabla \delta(\mathbf{r}-\mathbf{r'})$