shoehorn
- 420
- 2
Does there exist a chain rule for functional derivatives? For example, in ordinary univariate calculus, if we have some function [itex]y=y(x)[/itex] then the chain rule tells us (loosely) that
[tex] \frac{d}{dy} = \frac{dx}{dy}\frac{d}{dx}.[/tex]
Now suppose that we have a functional [itex]F[f;x)[/itex] of some function [itex]f(x)[/itex]. The functional derivative of [itex]F[f;x)[/itex] is denoted
[tex] \frac{\delta F[f;x)}{\delta f(y)}.[/tex]
However, suppose that [itex]f[/itex] is itself a functional of a function [itex]g(x)[/itex]. Can I then write
[tex] \frac{\delta}{\delta f} = <br /> \frac{\delta g}{\delta f} \frac{\delta}{\delta g}?[/tex]
[tex] \frac{d}{dy} = \frac{dx}{dy}\frac{d}{dx}.[/tex]
Now suppose that we have a functional [itex]F[f;x)[/itex] of some function [itex]f(x)[/itex]. The functional derivative of [itex]F[f;x)[/itex] is denoted
[tex] \frac{\delta F[f;x)}{\delta f(y)}.[/tex]
However, suppose that [itex]f[/itex] is itself a functional of a function [itex]g(x)[/itex]. Can I then write
[tex] \frac{\delta}{\delta f} = <br /> \frac{\delta g}{\delta f} \frac{\delta}{\delta g}?[/tex]