I don't see the situation where the DOF changes by cropping the image.
It is in how DOF is defined. DOF is the zone where sharpness is acceptable, usually defined by the
circle of confusion (which in turn comes from assumptions about the final image size, viewing distance, and human visual acuity).
Cropping magnifies blurriness. Uncropped, the near and far limits are slightly blurry, but acceptable. Enlarging those areas makes the blurriness greater and no longer acceptable. New boundaries need to be placed where the sharpness is just acceptable. So the DOF shrinks.
Don't believe me? Few do. There are DOF calculators online that can give you the answers. They will show you that Stephen is right about APSC being shallower and that David is wrong about them being the same.
BTW, there is an analogy in noise. It's baked in to the image, one might say, but cropping makes it appear worse. Does cropping or enlarging make the image noisier? How is that possible? But it appears to.
DOF was understood a hundred years ago. We have the equations and can calculate it to many decimal places. But today most still don't understand it.
But those DOF equations and calculators are approximations, actually. They assume that the lenses are thin, but they are not. DOF is affected by the
pupil magnification, which is ignored because it is an annoying complication. A simple flat lens has the same pupil size looking at it from the front or the back. But it is different with most complex modern lenses. Try it - take the lens off your camera and look at it from the front and then the back. Pupil magnification affects DOF, because DOF is the result of the exit pupil size.
Plus, viewing conditions vary all over the place. Smart phone pics look so sharp. The same pic can disappoint on a big monitor.
DOF equations and calculators are still useful because they give good approximate results that work in many situations, and can tell us about trends and principles.
