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 $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$ is a function, and $a\in\mathbb{R}^n$ is any point.

$f$ differentiable at $a$

Also called: derivative of $f$ at $a$ - differential is not mentioned

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

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

Claims:

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

Proposed terminology and notation

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

Terminology and notation

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