The commonly used $3\times 3$ Laplacian convolution kernel $\frac{1}{6}\begin{bmatrix}1 & 4 & 1\\4 & -20 & 4\\1 & 4 & 1\end{bmatrix}$ is only an approximation of rotational symmetry, as mentioned in the Wikipedia article on the Discrete Laplace operator.
Also, I found the first order Sobel operator to have an exactly rotational symmetric form without any proof.
So my questions: How can we compute the exact value of a rotational symmetric $n\times n$ convolution kernel for
- the $\nabla$ operator?
- the Laplacian operator?