I have a question about the definition of the functional derivative from Wikipedia:
Functional derivative: Given a manifold $M$ representing (continuous/smooth) functions $\rho$ (with certain boundary conditions etc.), and a functional $F$ defined as $$F\colon M \rightarrow \mathbb{R} \quad \mbox{or} \quad F\colon M \rightarrow \mathbb{C} \, ,$$ the functional derivative $F[\rho]$, denoted $\frac{\delta F}{\delta \rho}$, is defined by: \begin{align} \int \frac{\delta F}{\delta\rho}(x) \phi(x) \; dx &= \lim_{\varepsilon\to 0}\frac{F[\rho+\varepsilon \phi]-F[\rho]}{\varepsilon} \\ &= \left [ \frac{d}{d\epsilon}F[\rho+\epsilon \phi]\right ]_{\epsilon=0}, \end{align} where $\phi$ is an arbitrary function.
Since the functional is defined on a manifold, I am wondering how one can consider the sum $\rho + \epsilon \phi$ and evaluate $F(\rho + \epsilon \phi)$?