I came across this problem in Intro to ED by Griffiths: "In two dimensions, show that the divergence transforms as a scalar under rotation". Now, I was able to prove that this statement is true, but something bothered me. Intuitively, I know that a scalar does not change under rotation, but how can this be showed rigorously? We cannot have the rotation matrix operate on a scalar, because that is not defined. Is it from this fact that it is not defined that we say a scalar is not changed by 2D-rotation? Or does it have to due with the magnitude of a vector not changing under rotation?
I know this is more of a math question and I have already asked this on math.stackexchange, but it seems to have been overlooked.