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In variational calculus how many ways are there to define a variation, i.e. can it only be $$ \delta F(x) = \bar{F}(x) - F(x) \mbox{ , where } \bar{F}(x) = F(x + \delta x)$$ or is there another form?

basically I want to know the definition of the variation $\delta$ used in;

HOBSON, EFSTATHIOU and LASENBY ''General Relativity, An Introduction for Physicists''

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What do you mean by another form? What you did write is correct. – Occupy Gezi Mar 16 '13 at 21:35
is there any other way to write it? just out of interest – user63407 Mar 16 '13 at 21:41
It depends on the notation of author; but your notation is the most common one. Sometimes (in unusual ways) $\Delta$ or d can be used. – Occupy Gezi Mar 16 '13 at 21:59

If $J : C([a,b]) \to \mathbf R$ ($C([a,b])$ is taken just for example's sake) is a functional and if $y \in C([a,b]),$ the function (you consider in your question) $$ \Delta J(y)[h]=J(y+h)-J(y) $$ is commonly called the increment or (less often) change of $J$ corresponding to an increment $h \in C([a,b]).$ Thus $\Delta J(y) : C([a,b]) \to \mathbf R.$ In the book you've referred to in your question $\Delta J$ is always referred to as variation, which might be not a good idea, since in the calculus of variation the word "variation" is commonly referred to Gateaux or Frechet derivative of a functional.

The Gateaux derivative $\delta J(y) : C([a,b]) \to \mathbf R$ of $J$ above at $y \in C([a,b])$ is defined via the condition $$ \delta J(y)[h]=\lim_{t \to 0} \frac 1t (J(y+th)-J(y)) $$ where $h \in C([a,b])$ (and $t$ is a real number).

The Frechet derivative of $J$ at $y$ is a linear functional $\varphi(y) : C([a,b]) \to \mathbf R$ such that $$ \Delta J(y)[h]= \varphi(y)[h]+r(h) $$ where the remainder $r(h)$ must satisfy the condition $$ \lim_{\|h\| \to 0} \frac{|r(h)|}{\|h\|}=0 $$ where $\|\cdot\|$ is the standard (max-norm) on $C([a,b]).$ Thus \begin{equation*} \tag{$*$} \Delta J(y)[h] \approx \varphi(y)[h] \end{equation*} for "small" functions $h \in C([a,b]).$ The Frechet derivative of $J$ at $y$ is denoted either by $J'(y),$ or by $\delta J(y)$ (since if $J$ is Frechet differentiable at $y$ it is Gateaux differentiable at $y$).

It differs from book to book of course but, most commonly, if the Frechet derivative of $J$ at $y$ exists it is called the (first) variation of $J$ at $y.$ So you see it would be most awkward to read Eq. $(*)$ as "the variation of $J$ at $y$ is approximately equal to the variation of $J$ at $y$"; but when you use another word to refer to $\Delta J(y)$ the whole thing would definitely sound better.

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