Chain rule with functional derivative

[jostpuur]

This is supposedly the chain rule with functional derivative:

<br /> \frac{\delta F}{\delta\psi(x)} = \int dy\; \frac{\delta F}{\delta\phi(y)}\frac{\delta\phi(y)}{\delta\psi(x)}<br />

I have difficulty understanding what everything in this identity means. The functional derivative is usually a derivative of a functional with respect to some function, like in the term

<br /> \frac{\delta F}{\delta\phi(y)} := \lim_{\epsilon\to 0} \frac{1}{\epsilon}\big( F(\phi + \epsilon \delta_y) - F(\phi)\big),<br />

but isn't the term

<br /> \frac{\delta\phi(y)}{\delta\psi(x)} := ?<br />

now a derivative of a function with respect to another function?

 

[jostpuur]

It could be I understood this. If \phi depends on \psi somehow, that means that every \psi can be mapped into a set of functions \{\phi\}, \psi\mapsto\phi_{\psi}, then with a fixed y there's the natural functional G, G(\psi) = \phi_{\psi}(y).