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Total derivative of integral seen as a functional, how?

  1. Mar 27, 2012 #1
    To be specific, with total derivative I mean the linear map that best approximates a given function [itex]f[/itex] at a given point. For [itex]f:ℝ\toℝ[/itex] we have [itex]D(f,x_0):ℝ\toℝ[/itex], i.e. [itex]D(f,x_0)(h) \in ℝ[/itex]. Often it is also denoted as just [itex]\delta f[/itex].

    Now in physics, in particular in the area of the Lagrangian, I find the following. Let [itex]S_{a,b}(f) = \int_a^b f(x)dx[/itex] a functional that maps functions [itex]f[/itex] to the real line. Then [itex]D(S_{a,b},f) = \delta S_{a,b}[/itex] should be well defined given any necessary smoothness conditions. In particular [itex]D(S_{a,b},f)[/itex] maps functions [itex]h[/itex] of the same type of [itex]f[/itex] to real numbers. Because the integral is linear, so my hunch, its best linear approximation should be itself. Yet in a physics course, equation 1.5, first line, I find what I understand to be

    [tex]\delta \int_a^b f(x) dx = \int_a^b \delta f dx[/tex]

    Can anyone explain how the algebraic types on the left and on the right would match up? My interpretation is, that on the left I have a the total derivative of a functional, which itself should be a functional, written explicitly as [itex]D(S_{a,b},f)[/itex]. On the right I have the integral over, hmm, the total derivative of [itex]f[/itex], where I don't see how this could be a functional?

    Any hints appreciated.
  2. jcsd
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