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

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

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

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

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

---

TODO: This paragraph is repetitive, fix it

---

It is common to want to consider $\mathrm{d}f\vert$, which takes points in the domain of $f$ 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 $\mathrm{d}f\vert$ as a function that takes the domain of $f$ to a function (from the domain of $f$ to the co-domain of $f$). Then:

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

Overview

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

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

such that:

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

[Expand]

Alternate expressions:

1. $$\lim_{h\rightarrow 0}\left(\frac{f(a+h)-f(a)-\lambda(h)}{\Vert h\Vert_X}\right)\eq\mathbf{0}$$ - 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 $0$ (so $h$ may tend to it?) - additionally if the neighbourhood of $a$ is open in $X$ and $X$ 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.