Chain rule with functional derivative

  • Thread starter jostpuur
  • Start date
  • #1
2,111
16

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? :confused:
 

Answers and Replies

  • #2
2,111
16
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:

Related Threads for: Chain rule with functional derivative

  • Last Post
Replies
6
Views
3K
Replies
6
Views
4K
  • Last Post
Replies
7
Views
2K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
6
Views
16K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
1K
Top