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Total differential of general function mapping

  1. Apr 13, 2012 #1
    I am looking for an explanation and derivation of a total differential of a 2nd order function, i.e. a function that maps one function to another.

    To be more specific, lets say I have a function [itex]l:ℝ^n\to ℝ[/itex] that I use to define a 2nd order function [itex]L:(ℝ^k\to ℝ^n) \to (ℝ^k\to ℝ)[/itex] as [itex]L(f) := l\circ f[/itex] for every [itex]f:ℝ^k\to ℝ^n[/itex].

    Given that I have sufficiently useful norms defined on a function space [itex]ℝ^i\to ℝ^j[/itex], I assume that the total differential, [itex]D(L,f)[/itex] is well defined and is, for each "point" [itex]f:ℝ^k\to ℝ^n[/itex] of the function space a linear function (map) with the same algebraic type as [itex]L[/itex], i.e. [itex]D(L,f):(ℝ^k\to ℝ^n) \to (ℝ^k\to ℝ)[/itex].

    My hunch is that [itex]D(L,f)[/itex] can be expressed in terms of the total differentials of [itex]l[/itex] and [itex]f[/itex], if they are uniformly continuous, but I just cannot derive the solution.

    Can someone confirm, that the total differential of [itex]L[/itex] is a well defined concept?
    What is the solution in terms of [itex]l[/itex] and [itex]f[/itex]?
     
  2. jcsd
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