Notes:Differential notation and terminology

Sources

There are two sources used:

1. This and
2. This

Work required

1. Find out what differential is
2. Get Munkres' story
3. Get's view

Definitions

Here [ilmath]f:\mathbb{R}^n\rightarrow\mathbb{R}^m[/ilmath] is a function, and [ilmath]a\in\mathbb{R}^n[/ilmath] is any point.

[ilmath]f[/ilmath] differentiable at [ilmath]a[/ilmath]

Also called: derivative of [ilmath]f[/ilmath] at [ilmath]a[/ilmath] - differential is not mentioned

[ilmath]f[/ilmath] is differentiable at [ilmath]a[/ilmath] if there is a linear map [ilmath]\lambda:\mathbb{R}^n\rightarrow\mathbb{R}^m[/ilmath] such that:

• [math]\lim_{h\rightarrow 0}\left(\frac{\Vert f(a+h)-f(a)-\lambda(h)\Vert}{\Vert h\Vert}\right)=0[/math]
  • Notice that no direction of [ilmath]h\rightarrow 0[/ilmath] is given. So presumably for all paths tending towards zero, I wonder if there is a way to use sequences here.
  • The [ilmath]h\rightarrow 0[/ilmath] bit coupled with the use of norms suggests we might be able to use balls centred at zero for [ilmath]h[/ilmath], then look at the limit of those getting smaller
  • The norms are on [ilmath]\mathbb{R}^m[/ilmath] for the numerator and [ilmath]\mathbb{R}^n[/ilmath] for the denominator, and these need not be the usual norms (source - prior reading)

Claims:

1. If [ilmath]f[/ilmath] is differentiable at [ilmath]a[/ilmath] then the linear transformation, [ilmath]\lambda:\mathbb{R}^n\rightarrow\mathbb{R}^m[/ilmath] is unique

Proposed terminology and notation

• 1. [ilmath]df\vert_a[/ilmath] for the derivative of [ilmath]f[/ilmath] at [ilmath]a[/ilmath]
  2. [ilmath]d(g\circ f)\big\vert_a\eq dg\big\vert_{f(a)}\circ df\big\vert_a[/ilmath] - the chain rule. Note: [ilmath]dg\circ f\big\vert_a[/ilmath] would do but brackets make it easier to read

Terminology and notation

• Spivak:
  1. Differentiable at [ilmath]a[/ilmath] if has derivative
  2. [ilmath]Df(a)[/ilmath] for the derivative of [ilmath]f[/ilmath] at [ilmath]a[/ilmath]
  3. [ilmath]D(g\circ f)(a)\eq D(g(f(a))\circ Df(a)[/ilmath] - Chain rule