# Functional Derivative with Difficult Chain Rule

I am trying to evaluate the functional derivatives $$\dfrac{\delta F[\phi(\textbf{r})]}{\delta n_{1}}$$ and $$\dfrac{\delta F[\phi(\textbf{r})]}{\delta n_{2}}$$ where

$$\begin{gather} F[\phi_1(\textbf{r}), \phi_2(\textbf{r})] = \int d \textbf{r} \bigl [ f (\phi_{1}(\textbf{r}), \phi_{2}(\textbf{r}), \nabla\phi_{1}(\textbf{r}), \nabla\phi_{2}(\textbf{r}) ) \bigr ] \\ n_{1}[\phi_1(\textbf{r})] = \int d\textbf{r} \, g_{1} (\phi_{1}(\textbf{r})) \\ n_{2}[\phi_2(\textbf{r})] = \int d\textbf{r} \, g_{2} (\phi_{2}(\textbf{r})) \end{gather}$$

I have explicit expressions for $$f$$, $$g_1$$ and $$g_2$$. I believe the appropriate chain rule for functional derivatives is given by

$$$$\frac{\delta F[\phi_1, \phi_2]}{\delta n_{1}} = \int d\textbf{r} \left ( \frac{\delta F}{\delta \phi_1} \frac{\delta \phi_{1}}{\delta n_1} + \frac{\delta F}{\delta \phi_2} \frac{\delta \phi_{2}}{\delta n_1}\right) \textrm{.}$$$$

But the chain rule is only useful if I know $$\phi_1[n_1]$$, when in fact I have the inverse. How can I evaluate these derivatives?

• When you write $g_k(\phi_k)$ do you mean $g_k \circ \phi_k$? Are the integration limits dependent on $r$? Are the various functions sufficiently smooth? – copper.hat May 30 '20 at 7:41
• Yes, $g_k \circ \phi_k$. The limits of integration do not depend on r. One can assume natural/periodic boundary conditions. The functions are smooth. – Doug May 30 '20 at 14:16