Strong derivative
From Maths
| Strong derivative | |
| [math]\lim_{h\rightarrow 0}\left(\frac{\big\Vert f(x_0+h)-f(x_0)-df\vert_{x_0}\big\Vert_Y}{\Vert h\Vert_X}\right)[/math] For two normed spaces [ilmath](X,\Vert\cdot\Vert_X)[/ilmath] and [ilmath](Y,\Vert\cdot\Vert_Y)[/ilmath] and a mapping [ilmath]f:U\rightarrow Y[/ilmath] for [ilmath]U[/ilmath] open in [ilmath]X[/ilmath] [ilmath]df\vert_{x_0}:X\rightarrow Y[/ilmath] a linear map called the "derivative of [ilmath]f[/ilmath] at [ilmath]x_0[/ilmath]" |
Definition
The strong derivative (AKA the Fréchet derivative) has several definitions, however they are all equivalent, as will be shown. In all cases we are given:
- Two normed vector spaces, [ilmath](X,\Vert\cdot\Vert_X)[/ilmath] and [ilmath](Y,\Vert\cdot\Vert_Y)[/ilmath]
- A mapping, [ilmath]f:U\rightarrow Y[/ilmath] where [ilmath]U[/ilmath] is an open set of [ilmath]X[/ilmath]
- Some point [ilmath]x_0\in U[/ilmath] (the point we are differentiating at)
Definition 1
If there exists a linear map [ilmath]L_{x_0}:X\rightarrow Y[/ilmath] such that:
- [math]f(x+h)-f(x)=L_{x_0}(h)+r(x_0;h)[/math] where [math]\lim_{h\rightarrow 0}\left(\frac{\Vert r(x_0;h)\Vert_Y}{\Vert h\Vert_X}\right)=0[/math]
Definition 2
If there exists a linear map [ilmath]L_{x_0}:X\rightarrow Y[/ilmath] such that:
- [math]\lim_{h\rightarrow 0}\left(\frac{f(x_0+h)-f(x_0)-L_{x_0}(h)}{\Vert h\Vert_X}\right)=0_Y[/math]
TODO: Check this, I've just been sick, so I'm going to save my work and lie down
Todo
TODO: Find reference for and add "total derivative" to list of AKA names, see also derivative (analysis) and do the same thing there
