Doctrine:Differentiation notation & terminology

Problem

Almost everyone has a different view of how to write derivatives. I have compiled these as "the best of the best" and tweaked it a little to enhance readability and consistency of the notation.

Notation & terminology

Derivative

• [ilmath]\mathrm{d}f\vert_a[/ilmath] - the derivative of [ilmath]f[/ilmath] at [ilmath]a[/ilmath]. A linear map from the domain of [ilmath]f[/ilmath] to the co-domain of [ilmath]f[/ilmath]
  • This means [ilmath]f[/ilmath] is differentiable at [ilmath]a[/ilmath] and [ilmath]\mathrm{d}f\vert_a[/ilmath] is its derivative; not differential, that is something else.

We use the [ilmath]\mathrm{d} [/ilmath] and the [ilmath]\vert[/ilmath] as brackets. Everything between is what we're taking the derivative of.

• Example: The chain rule - [ilmath]\mathrm{d}(g\circ f)\big\vert_{a}\eq\mathrm{d}g\big\vert_{f(a)}\circ\mathrm{d}f\big\vert_a[/ilmath]
  • As opposed to: [ilmath]D(f\circ g)(a)=Dg(f(a))\circ Df(a)[/ilmath]

The notation employed by Munkres and Spivak in [ilmath]Df(a)[/ilmath] makes it hard to tell if it is referring to the derivative of [ilmath]f(a)[/ilmath] considered as a function (which may be constant, or [ilmath]f[/ilmath] might map [ilmath]a[/ilmath] to a function that can be differentiated itself!) and if so where. This is especially true when looking at functions of functions.

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TODO: This paragraph is repetitive, fix it

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It is common to want to consider [ilmath]\mathrm{d}f\vert[/ilmath], which takes points in the domain of [ilmath]f[/ilmath] to their derivatives at that point. This is a slight abuse of notation however in our notation it is not a big leap to see [ilmath]\mathrm{d}f\vert[/ilmath] as a function that takes the domain of [ilmath]f[/ilmath] to a function (from the domain of [ilmath]f[/ilmath] to the co-domain of [ilmath]f[/ilmath]). Then:

• [ilmath]\mathrm{d}f\vert(a)[/ilmath] simply reads like another way of saying [ilmath]\mathrm{d}f\vert_a[/ilmath].

Overview

Let [ilmath](X,\Vert\cdot\Vert_X)[/ilmath] and [ilmath](Y,\Vert\cdot\Vert_Y)[/ilmath] be normed spaces and let [ilmath]f:X\rightarrow Y[/ilmath] be a function. Then:

• [ilmath]f[/ilmath] is differentiable at [ilmath]a\in X[/ilmath] if:
  1. There exists a neighbourhood of [ilmath]a[/ilmath], [ilmath]N[/ilmath] and
  2. There exists a linear map, [ilmath]\lambda:X\rightarrow Y[/ilmath]

such that:

• [math]\lim_{h\rightarrow 0}\left(\frac{\Vert f(a+h)-f(a)-\lambda(h)\Vert_Y}{\Vert h\Vert_X}\right)\eq 0[/math] - Caution:There are other expressions for this limit that may be equivalent. This has not been decided/confirmed yet

Alternate expressions:

1. [math]\lim_{h\rightarrow 0}\left(\frac{f(a+h)-f(a)-\lambda(h)}{\Vert h\Vert_X}\right)\eq\mathbf{0} [/math] - Caution:unconfirmed

Caveats/Pending questions

1. Is this limit equivalent to the definition of differentiable that is least constraining?
2. Are there normed subspaces? If so must it contain [ilmath]0[/ilmath] (so [ilmath]h[/ilmath] may tend to it?) - additionally if the neighbourhood of [ilmath]a[/ilmath] is open in [ilmath]X[/ilmath] and [ilmath]X[/ilmath] is simply a topological subspace the requirements become blurred.
   • A normed subspace is almost certainly a vector subspace in its own right, so this shouldn't be a problem. However I mention this to remind myself, and as food for thought.