Proving $F(x)=F(a)+\sum_{\mu=1}^{n}(x^\mu -a^\mu )H_\mu (x)$

Where $F:R^n \rightarrow R$, $a=(a^1,...,a^\mu)$ and also $x=(x^1,...,x^\mu)$.

Additionally, $H_\mu (a)=\frac{\partial F}{\partial x^\mu}|_{x=a}$

Hi there, this is a trouble on General Relativity from Robert Wald. I'm attempting to address it for a couple of hrs yet still, it does not look that hard. I'm rather sure the basic theory of calculus is an excellent start:

$F(x)-F(a)=∫_{a}^{x}F′(s)ds$

Then, with $ s=t(x-a)+a$ we have

$F(x)-F(a)=(x-a)∫_{0}^{1}F′[t(x-a)+a]dt$

Which exposes a wonderful resemblance! Yet I do not recognize just how to generalise from there. I was checking out Stokes theory to see if there is any kind of link.

Every single time I get stuck on a trouble for greater than one hr I recognize it has to be something noticeable, Weird

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2022-06-07 14:30:42
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Answers: 1

I'm sorry for my lack of knowledge, possibly I'm not obtaining something, yet. isn't it simply the Taylor development of a function, approximately the hand order?

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2022-06-07 14:52:08
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