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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:

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

Gradient:

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

Any insight or references are appreciated. Please note that I am a geophysicist by training and it has been a few years since I have taken any vector calculus classes!

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