>>11130161>>11130176Also I think that differential starts to make better sense when one considers mappings , not just into . For example is some, non-linear in general, transformation of the plane. Here the derivative, as a slope of something, doesn't make sense all. It's also not a good idea to try to understand this map via its graph which now lives in . The differential, as the best linear approximation, makes perfect sense:
Look at pic related. Blue grid is the original non-linear transformation (before and after), i.e. a map . Red grid is a linear transformation which approximates the blue one near the green point. This is the differential.
