Variational calculus 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''

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. 
