A description/proof of that residue of derivative of normed vectors spaces map is differentiable at point of 2nd argument if original map is differentiable at corresponding point with derivative as minus original map derivative at 1st argument point plus original map derivative at corresponding point
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of derivative of normed vectors spaces map.
Target Context
- The reader will have a description and a proof of the proposition that the residue of the derivative of any normed vectors spaces map is differentiable at any point of the 2nd argument if the original map is differentiable at the corresponding point with derivative as minus the derivative of the original map at the 1st argument point plus the derivative of the original map at the corresponding point.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description
For any differentiable map, \(f: V_1 \to V_2\) such that \(f (v_{11} + v_{12}) = f (v_{11}) + (Df (v_{11})) (v_{12}) + r (v_{11}, v_{12})\), the residue, \(r (v_{11}, v_{12})\), is differentiable with respect to the 2nd argument at \(v_{12}\) with derivative as \(- Df (v_{11}) + Df (v_{11} + v_{12})\), if \(f\) is differentiable at \(v_{11} + v_{12}\).
2: Proof
Let us suppose that \(f\) is differentiable at \(v_{11} + v_{12}\) as well as at \(v_{11}\). $$f (v_{11} + v_{12} + v_{13}) = f (v_{11}) + (Df (v_{11})) (v_{12} + v_{13}) + r (v_{11}, v_{12} + v_{13}) = f (v_{11} + v_{12}) + (Df (v_{11} + v_{12})) (v_{13}) + r (v_{11} + v_{12}, v_{13})$$ $$ = f (v_{11}) + (Df (v_{11})) (v_{12}) + r (v_{11}, v_{12}) + (Df (v_{11} + v_{12})) (v_{13}) + r (v_{11} + v_{12}, v_{13}).$$ So, $$r (v_{11}, v_{12} + v_{13}) = r (v_{11}, v_{12}) + (- Df (v_{11}) + Df (v_{11} + v_{12})) (v_{13}) + r (v_{11} + v_{12}, v_{13}).$$ But $$lim_{v_{13} \Rightarrow 0} \frac{\Vert r (v_{11} + v_{12}, v_{13})\Vert}{\Vert v_{13}\Vert} = 0,$$ which means that \(r\) is differentiable at \(v_{12}\) with the derivative as \(- Df (v_{11}) + Df (v_{11} + v_{12})\).
