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Chain rule with functional derivative

  1. Feb 25, 2008 #1
    This is supposedly the chain rule with functional derivative:

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

    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

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

    but isn't the term

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

    now a derivative of a function with respect to another function? :confused:
  2. jcsd
  3. Feb 26, 2008 #2
    It could be I understood this. If [itex]\phi[/itex] depends on [itex]\psi[/itex] somehow, that means that every [itex]\psi[/itex] can be mapped into a set of functions [itex]\{\phi\}[/itex], [itex]\psi\mapsto\phi_{\psi}[/itex], then with a fixed y there's the natural functional G, [itex]G(\psi) = \phi_{\psi}(y)[/itex].
    Last edited: Feb 26, 2008
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