@amsomniac (quick thoughts) you're taking a derivative, so you'll lose constant offsets, and then information is destroyed in the subtraction of derivatives. in Cartesian coordinates curl won't change if you make equal additions to both dFz/dy and dFy/dz (e.g. by adding (y k^ and z j^ to your vector field) and so on for all similar pairs...
@amsomniac that addition makes a weird shape! kind of a saddle? which would make curl invariant to saddles and translations