It is relatively easy to show that the Laplacian

$$ \Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} $$

Is the unique second order linear differential operator that is invariant under rotations in the sense that

$$ \Delta (f(R\mathbf{x})) = (\Delta f)(R\mathbf{x}). $$

The way I remember proving this was to write down a general

$$ D = \sum_i a_i \frac{\partial}{\partial x_i} + \sum_{ij} b_{ij} \frac{\partial^2}{\partial x_i \partial x_j}, $$

and then demand the invariance property. A multiple of the Laplacian will then fall out.

I am wondering if there is a way to general all such operators (up to a certain degree). Are they all powers of the Laplacian? What happens for the vector Laplacian?

constant coefficientdifferential operators? If so, then the first fundamental theorem of invariant theory for the orthogonal group ${\rm O}_n$ shows that indeed every constant coefficient differential operator on $K^n$ ($K$ a field) is a polynomial in $\Delta$. There is an analogous description, due to Roger Howe, for the case of ${\rm O}_n$-invariant polynomial coefficient differential operators. $\endgroup$