The answer to this question might seem obvious, but I'm struggling to find good, accessible sources on the calculus of variations generally, and this pesky detail in particular, so I thought I might as well try my luck and possibly help others struggling to get a firm hold of some of the basics.
My question is largely one about "proper" notation: Reading physics literature (I'm a physicist before a mathematician) and taking various classes, I've seen conflicting use of what the "$\delta$" in e.g. "$\delta F$" means.
Specifically, some sources (including https://cds.cern.ch/record/1383342/files/978-3-642-14090-7_BookBackMatter.pdf and Anthony Duncan's book "The Conceptual Framework of Quantum Field Theory") seems to label this "the variation of $F$", defined by
\begin{equation}
\tag{1}\label{1}\delta F[\rho] \equiv F[\rho + \delta\rho] – F[\rho],
\end{equation}
where $\delta\rho \mathop{\widehat{=}} \varepsilon\eta$, and, as far as I understand, $\eta$ belongs to the same space of functions as $\rho$, and $\varepsilon$ is a small number.
Other sources, however (for example Wikipedia: https://en.wikipedia.org/wiki/First_variation) label this "the first variation of $F$", defined by
\begin{equation}
\tag{2}\label{2}
\delta F[\rho]_{\eta} \equiv \lim_{\varepsilon \to 0} \frac{F[\rho + \varepsilon\eta] – F[\rho]}{\varepsilon} = \int \frac{\delta F[\rho]}{\delta\rho(x)}\, \eta(x)\, dx,
\end{equation}
where the last equation holds only when the functional derivative $\delta F/\delta\rho$ is defined.
Now, I do understand that these conflicting definitions often will be asymptotically equal, in the sense that
\begin{equation}
\tag{3}\label{3}
F[\rho + \varepsilon\eta] \simeq F[\rho] + \varepsilon\int \frac{\delta F[\rho]}{\delta\rho(x)}\, \eta(x)\, dx + \cdots,
\end{equation}
but I nonetheless wonder which might be seen as most pedantically "correct" …
If there's anything I might have misunderstood, feel free to point it out.
Best Answer
These definitions are not really conflicting, the first one is technically imprecise however.
The proper definition would be the "differentiation" definition.
In the first case, $\delta$ isn't what maps $F[\rho]$ to $F[\rho+\delta\rho]-F[\rho]$, but what maps $F[\rho]$ to $F[\rho+\delta\rho]-F[\rho]$ with second-order and higher terms neglected in the Taylor-expansion of $F[\rho+\delta\rho]$. A differentiation is essentially taking the first order coefficient of a Taylor-expansion, so technically these two are the same.
Of course in old-school physics literature, they will make it sound the Taylor-expansion definition is just some sort of "approximation", while the "differentiation" definition seems exact, but both essentially give the linear approximation to the change in the functional in the direction of the function $\delta\rho$, which is what differentiation does.