Notes:Differential (notation)

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Problem

I'm being bombarded with different notations for various forms of the differential, previously I've just made sure I can move between them, but for this project I need to commit, as such I'll enumerate a few here and propose some things.

Proposal

  • D, D - for directional derivatives, Dv(f)(a) is the directional derivative of f at a in the direction v
    • This suggests that Dv() is itself some sort of mapping, and that isn't far from the truth, we can define it on at least the class of maps that are at least once differentiable. (There are in fact more as we can have directional derivatives, but not be differentiable sometimes)
  • d, d - for total derivatives of both manifolds and Rn (recognising a distinction that doesn't really exist). d(f)(a) denotes the total derivative of f at a.

Other options

  • d|af is quite nice, as is df|a, these are also very clear. df(a) can be ambiguous (looks like the derivative of f(a)!), but neither of these are.

Problems

  • fx(a) vs x|af vs xf|a - I like the first one because f(a)x cannot be confused with it. However fx|a is also quite nice...