2022-03-27

50: 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

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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

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About: vectors space

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Starting Context



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


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There is a list of propositions discussed so far in this site.


Main Body


1: Description


For any differentiable map, f:V1V2 such that f(v11+v12)=f(v11)+(Df(v11))(v12)+r(v11,v12), the residue, r(v11,v12), is differentiable with respect to the 2nd argument at v12 with derivative as Df(v11)+Df(v11+v12), if f is differentiable at v11+v12.


2: Proof


Let us suppose that f is differentiable at v11+v12 as well as at v11. f(v11+v12+v13)=f(v11)+(Df(v11))(v12+v13)+r(v11,v12+v13)=f(v11+v12)+(Df(v11+v12))(v13)+r(v11+v12,v13) =f(v11)+(Df(v11))(v12)+r(v11,v12)+(Df(v11+v12))(v13)+r(v11+v12,v13). So, r(v11,v12+v13)=r(v11,v12)+(Df(v11)+Df(v11+v12))(v13)+r(v11+v12,v13). But limv130r(v11+v12,v13)v13=0, which means that r is differentiable at v12 with the derivative as Df(v11)+Df(v11+v12).


References


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