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].(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Total derivative of integral seen as a functional, how?

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**