Cosidering the usual central field in physics:
\begin{equation} \mathbf{V}=\frac{\hat{r}}{\mathbf{r}^2}. \end{equation}
I can get that divergence and curl (and therefore vorticity) are zero when $r\neq0$, and that divergence is $4\pi\delta(\mathbf{r})$ using divergence definition:
\begin{equation} \int \nabla \cdot \mathbf{V} d\mathbf{r}^3=\int\mathbf{V}\cdot d\mathbf{A}=4\pi, \end{equation}
For a sphere centered around $r=0$. Also i can understand why curl is zero as there is a point singularity and not a line one.
However, how can i show how is the shear tensor in the presence of the singularity? Is there a sort of generalized integral theorem for the full gradient tensor of a field?
Particulary for this case, the shear tensor can be computed as:
\begin{eqnarray*} \sigma_{ij}&=&\left (\delta_{ij}\frac{1}{\lvert \mathbf{r}\rvert^3}-\frac{3x_ix_j}{\lvert \mathbf{r}\rvert^5}\right )-\frac{4}{3}\pi\delta(\mathbf{r})\delta_{ij}, \\ \end{eqnarray*}
But I'm not sure how to treat the singularity at $r=0$ for the shear.
I hope i was clear. Thank you so much to everyone for your response!