# Chain rule with functional derivative

• jostpuur
In summary, the conversation discusses the chain rule with functional derivatives, which is used to calculate the derivative of a functional with respect to a function. This can be understood by mapping each value of the function to a set of functions, and then using the resulting functional to calculate the derivative.
jostpuur
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?

It could be I understood this. If $\phi$ depends on $\psi$ somehow, that means that every $\psi$ can be mapped into a set of functions $\{\phi\}$, $\psi\mapsto\phi_{\psi}$, then with a fixed y there's the natural functional G, $G(\psi) = \phi_{\psi}(y)$.

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## 1. What is the chain rule with functional derivative?

The chain rule with functional derivative is a mathematical tool used to calculate the derivative of a function that depends on another function, known as a functional. It is commonly used in the field of functional analysis, which involves the study of functions that operate on other functions.

## 2. How does the chain rule with functional derivative work?

The chain rule with functional derivative states that the derivative of a function that depends on another function can be calculated by multiplying the derivative of the outer function with the derivative of the inner function. This is similar to the standard chain rule for ordinary derivatives.

## 3. What is the purpose of using the chain rule with functional derivative?

The chain rule with functional derivative allows for the calculation of derivatives of complex functions that depend on other functions. It is particularly useful in solving problems in functional analysis and in other areas of mathematics and physics where functions of functions are involved.

## 4. Can the chain rule with functional derivative be applied to any type of function?

Yes, the chain rule with functional derivative can be applied to any type of continuously differentiable function. This includes both scalar functions and vector functions, as well as functions defined on infinite-dimensional spaces.

## 5. Are there any limitations or special considerations when using the chain rule with functional derivative?

One limitation of the chain rule with functional derivative is that it cannot be applied to functions that are not continuously differentiable. Additionally, when using the chain rule with functional derivative, one must be careful to correctly identify the inner and outer functions in order to calculate the derivative accurately.

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