I am reading the Many-body quantum theory in condensed matter physics by Henrik Bruus. I am stuck at Eq. (4.51): The unitary transformation is given by $$U = \exp \left(i \int d\vec{r} \rho(\vec{r}) \phi(\vec{r}) \right)\quad ,$$ where $\rho(\vec{r}) = \Psi^\dagger(\vec{r})\Psi(\vec{r})$ and $$\tilde{\Psi}(\vec{r}) = U\Psi(\vec{r})U^{-1} \quad ,$$ where $\Psi(r)$ is the position operator. Then we have $$\tilde{\Psi}(\vec{r}) = \Psi(\vec{r})\exp\left(-i\phi(\vec{r})\right) \quad .$$ According to the book, this can be derived from a functional differential equation \begin{align}\frac{\delta}{\delta \phi(\vec{r'})}\tilde{\Psi}(\vec{r}) = i\tilde{\Psi}(\vec{r})\delta(\vec{r} - \vec{r'}) \quad .\end{align} I tried to multiply both side with $\phi(\vec{r})$ and integrate. but it just derives the above functional differential equation. I found an easy derivation from Deriving the unitary transformation for phase shifts. I understand this, but I want to understand Bruus' derivation.
How can I derive the third equation from the fourth one?
