- #1
shoehorn
- 420
- 2
Does there exist a chain rule for functional derivatives? For example, in ordinary univariate calculus, if we have some function y=y(x) then the chain rule tells us (loosely) that
<br /> \frac{d}{dy} = \frac{dx}{dy}\frac{d}{dx}.<br />
Now suppose that we have a functional F[f;x) of some function f(x). The functional derivative of F[f;x) is denoted
<br /> \frac{\delta F[f;x)}{\delta f(y)}.<br />
However, suppose that f is itself a functional of a function g(x). Can I then write
<br /> \frac{\delta}{\delta f} = <br /> \frac{\delta g}{\delta f} \frac{\delta}{\delta g}?<br />
<br /> \frac{d}{dy} = \frac{dx}{dy}\frac{d}{dx}.<br />
Now suppose that we have a functional F[f;x) of some function f(x). The functional derivative of F[f;x) is denoted
<br /> \frac{\delta F[f;x)}{\delta f(y)}.<br />
However, suppose that f is itself a functional of a function g(x). Can I then write
<br /> \frac{\delta}{\delta f} = <br /> \frac{\delta g}{\delta f} \frac{\delta}{\delta g}?<br />
