Functional Derivatives: Overview & Tips

I
Thread starter The black vegetable
Start date Nov 28, 2019
Tags

Derivatives Functional Quantum fields

In summary, functional derivatives are mathematical tools used in functional analysis to calculate the rate of change of a functional with respect to its argument. They play a crucial role in many scientific fields and are calculated using the Euler-Lagrange equation. Some applications of functional derivatives include describing physical systems, optimizing utility functions, and designing structures. Tips for using functional derivatives effectively include having a strong understanding of the theory, using correct notation, practicing, double-checking calculations, and utilizing software tools.

Nov 28, 2019

The black vegetable

TL;DR Summary: Can someone please explain how you take a functional derivative of a generating function. I have included a basic example of how I understand it, if it's not the case can someone explain explicitly for a dummy how this is performed. My question is in picture form, as it's much easier for me to do it this way.
Many thanks

In the last sentence I mean you do include constant terms like I have done when taking the product above?

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Dec 1, 2019

A. Neumaier

Science Advisor

Insights Author

Replace ##J## by ##J+\epsilon dJ##, expand, drop terms involving higher powers of ##\epsilon##. The coefficient of ##\epsilon## is the integral of the functional derivative ##dZ(J)/dJ(x)## multiplied by ##dJ(x)##.

What are functional derivatives?

Functional derivatives are a mathematical tool used in functional analysis to calculate the rate of change of a functional with respect to its independent variables.

Why are functional derivatives important in science?

Functional derivatives are important because they allow scientists to study complex systems with many variables and understand how small changes in those variables affect the overall system.

How are functional derivatives calculated?

Functional derivatives are calculated by taking the limit of a difference quotient, where the difference between two functionals is divided by the difference between their independent variables and then taking the limit as that difference approaches zero.

What are some common applications of functional derivatives?

Functional derivatives have many applications in physics, engineering, and other fields. They are commonly used in the study of fluid dynamics, quantum mechanics, and optimization problems.

What are some tips for working with functional derivatives?

Some tips for working with functional derivatives include understanding the underlying mathematical principles, carefully defining the functional and its independent variables, and using the chain rule and other mathematical techniques to simplify calculations.