Let $ f:\mathbb R^n \to \mathbb R $ be in $ C^2 $ and $ R \in O(n) $ be an orthogonal transformation. My understanding of what invariance of the Laplacian under orthogonal transformations (e.g. rotations) means, is that
$ \triangle f = (\triangle f) \circ R. \qquad \qquad $ (1)
This is for instance how I understand the proof for the simple case of rotations in two dimensions by peek-a-boo given here. If $ x \in \mathbb R^n $ and $ R(x)=u $, then (1) means that $ \triangle f(x) = \triangle f(u) $.
However, often I find the following definition instead
$ (\triangle f) \circ R = \triangle (f \circ R). \qquad \qquad $ (2)
I can't figure out what the rhs of this actually means and whether or not (2) is equivalent to (1).
What is $ \big[ \triangle (f \circ R) \big](x), \quad x \in \mathbb R^n $?