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Question on a particular integral property

  1. Feb 22, 2015 #1
    I've been reading Wald's book on general relativity and in one of the questions at the end of chapter 2 he gives a hint which says to make use the following integral identity (for a smooth function in): [tex]F(x)=F(a)+\int_{0}^{1}F'(t(x-a)+a)dt[/tex]
    Is this result true simply because [tex]\int_{0}^{1}F'(t(x-a)+a)dt=F(t(x-a)+a)\bigg\vert_{t=0}^{t=1}=F(x)-F(a)[/tex]
    And so [tex]F(x)=F(a)+\int_{0}^{1}F'(t(x-a)+a)dt=F(a)+\left(F(x)-F(a)\right)[/tex]
    Or is there a deeper explanation to it?
    I get that one can arrive at this expression by starting from the fundamental theorem of calculus and making a change of variables, but I'm unsure of what he's trying to say with it?!
     
  2. jcsd
  3. Feb 22, 2015 #2

    HallsofIvy

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    Nothing deeper- that's basically why it is true. As to "what he's trying to say with it", I suspect that depends on what he is try to do.
     
  4. Feb 22, 2015 #3
    I should mention sloppy notation here. The formula is true if ##F'(t(x-a) + a)## is understood as derivative of the function ##\phi##, ##\phi(t) = F(t(x-a) + a)## (evaluated at ##t##). However, in the standard notation ##F'(t(x-a) + a)## means the derivative of ##F## evaluated at the point ##t(x-a) + a## , and in this case the formula is false (consider say ##F(x) =x##, ##a=0##).
     
  5. Feb 22, 2015 #4
    Yes, I admit it is a bit sloppy, I was just copying over the notation that Wald gives it in. I'm assuming he meant [itex] F'(t(x-a) +a)=\frac{dF(t(x-a)+a)}{dt}[/itex]?!

    He gives it in a question in which he asks the reader to prove by induction that for any smooth function [itex] F : \mathbb{R} ^{n} \rightarrow \mathbb{R} ^{n} [/itex] one can express it as the following power series (i. e. a Taylor series) [itex] F(x) =F(a) +\sum_{i=1}^{\infty}(x^{i}-a^{i})H_{i}(x)[/itex], where [itex] x=(x^{1},\ldots,x^{n}) [/itex] and [itex] a=(a^{1},\ldots,a^{n}) [/itex], and in particular, [itex] \frac{\partial F} {\partial x^{i}} \bigg\vert_{x=a} =H_{i} (a) [/itex]. He gives a hint that for [itex] n=1[/itex], you should use the "known" identity I gave in my first post. I can do the first step in the induction process, by starting from the fundamental theorem of calculus and making a change of variables, but I'm not entirely sure what this is showing. I'm also unsure how to proceed after making the induction step?!
     
  6. Feb 22, 2015 #5
    You just should apply the formula $$\phi(1) = \phi(0) +\int_0^1 \phi'(t) dt$$ (the fundamental theorem of calculus) to $$\phi(t) = F(a+(x-a)t). $$
     
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