Notes:Differential (notation)

Solution

There are many ways to write the differential, most of them stem from the fact they want to "pretend" that they're a function on some domain, for example:

• [ilmath]d_p[/ilmath] I have seen for differential at a point, then used for things like [ilmath]d_pf[/ilmath] and [ilmath]d_pg[/ilmath]
• [ilmath]df[/ilmath], then used for [ilmath]df(p)[/ilmath] and [ilmath]df(q)[/ilmath] for the differential of [ilmath]f[/ilmath] at [ilmath]p[/ilmath] and [ilmath]q[/ilmath]

A "fully formed" differential looks like this:

• [ilmath]d(f)(p)(a)[/ilmath] - as repeated application of functions (debatable, see the bullet point below)
  1. [ilmath]d(f)[/ilmath] - the differential of [ilmath]f[/ilmath], a function that takes the domain to linear maps.
  2. [ilmath]d(f)(p)[/ilmath] - the differential of [ilmath]f[/ilmath] at [ilmath]p[/ilmath] - a linear map
  3. [ilmath]d(f)(p)(a)[/ilmath] - that linear map evaluated in the direction of [ilmath]a[/ilmath].
• [ilmath]d(f)(p)[/ilmath] is perhaps a better "fully formed" differential, this is a linear map.
  • Now some authors want things like [ilmath][d(f)(p)]_{i,j} [/ilmath]
    • Writing [ilmath][d(\cdot)(p)]_{i,j} [/ilmath] for the [ilmath]i,j[/ilmath]^th entry of the derivative at [ilmath]p[/ilmath] is very cumbersome and I've seen some authors write:
      • [ilmath]d_{i,j}\vert_p[/ilmath] (ie, for: [ilmath]d_{i,j}\vert_pf[/ilmath] for the [ilmath]i,j[/ilmath]^th entry of the matrix for the differential of [ilmath]f[/ilmath] at [ilmath]p[/ilmath])

It is fair to assume as a result that at some point every permutation [ilmath]f[/ilmath], [ilmath]p[/ilmath] and [ilmath]a[/ilmath] in [ilmath]d(f)(p)(a)[/ilmath] will be wanted (getting the [ilmath]i,j[/ilmath]^th term can be done by clever choice of [ilmath]a[/ilmath]), thus we can conclude:

• There is no "true" definition of [ilmath]d[/ilmath] as whichever order of parameters you write it as a function of, it cannot satisfy all 3. Without silly things like [ilmath][d(\cdot)(p)]_{i,j} [/ilmath] anyway.

So all I can do is be clear and consistent and careful about my choice of notation in each case, and avoid the ambiguities described in the old section below.

Notations

I shall use [ilmath]d[/ilmath] and [ilmath]\vert[/ilmath] like "brackets", that is [ilmath]d[/ilmath] opens and [ilmath]\vert[/ilmath] closes. So:

• [ilmath]d(\ldots f\ldots)\vert_p[/ilmath] (which is a nice way of writing [ilmath]d(\ldots f\ldots)\vert(p)[/ilmath] - syntatic sugar). Examples:
  1. [ilmath]df\vert_p[/ilmath] - the derivative of [ilmath]f[/ilmath] at [ilmath]p[/ilmath]
  2. [ilmath]df(a)\vert_p[/ilmath] - the derivative of whatever [ilmath]f(a)[/ilmath] is (presumably a function, or a constant) taken at [ilmath]p[/ilmath]

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Old section

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

• [ilmath]D[/ilmath], [ilmath]\mathrm{D} [/ilmath] - for directional derivatives, [ilmath]D_v(f)(a)[/ilmath] is the directional derivative of [ilmath]f[/ilmath] at [ilmath]a[/ilmath] in the direction [ilmath]v[/ilmath]
  • This suggests that [ilmath]D_v(\cdot)[/ilmath] 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)
• [ilmath]d[/ilmath], [ilmath]\mathrm{d} [/ilmath] - for total derivatives of both manifolds and [ilmath]\mathbb{R}^n[/ilmath] (recognising a distinction that doesn't really exist). [ilmath]d(f)(a)[/ilmath] denotes the total derivative of [ilmath]f[/ilmath] at [ilmath]a[/ilmath].

Other options

• [ilmath]d\big\vert_af[/ilmath] is quite nice, as is [ilmath]df\big\vert_a[/ilmath], these are also very clear. [ilmath]df(a)[/ilmath] can be ambiguous (looks like the derivative of [ilmath]f(a)[/ilmath]!), but neither of these are.

Problems

• [ilmath]\frac{\partial f}{\partial x}(a)[/ilmath] vs [ilmath]\frac{\partial}{\partial x}\big\vert_af[/ilmath] vs [ilmath]\frac{\partial}{\partial x}f\big\vert_a[/ilmath] - I like the first one because [ilmath]\frac{\partial f(a)}{\partial x} [/ilmath] cannot be confused with it. However [ilmath]\frac{\partial f}{\partial x}\big\vert_a[/ilmath] is also quite nice...