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I am trying to compute the first variation of the functional $$\mathcal F(\rho) = \int_{\Omega} R(x;\rho) d\rho(x)$$ where $R$ is some function of $x$ that also depends on $\rho$. Here $\rho$ is a ...

In Chapter 8 of the book Gradient Flows In Metric Spaces and in the Space of Probability Measures by Ambrosio et al., the tangent space to the space of distributions on $X$ (let's say $X=\mathbb{R}^d$)...