- #1

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## Main Question or Discussion Point

This is supposedly the chain rule with functional derivative:

[tex]

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

[/tex]

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

[tex]

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

[/tex]

but isn't the term

[tex]

\frac{\delta\phi(y)}{\delta\psi(x)} := ?

[/tex]

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

[tex]

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

[/tex]

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

[tex]

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

[/tex]

but isn't the term

[tex]

\frac{\delta\phi(y)}{\delta\psi(x)} := ?

[/tex]

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