A simple functional derivative

In summary, a functional derivative is a derivative of a functional with respect to its argument, which is typically a function. It differs from a regular derivative in that it results in a new function and is used to find the minima or maxima of a functional. The calculation of a functional derivative involves taking the limit of a difference quotient and it has various applications in fields such as optimization, quantum mechanics, and fluid dynamics.
  • #1
sprik
1
0
Hi!
I am doing some numerical calculations recently. I need to calculate the functional derivative. eg. functional :
[tex]n(\rho)=\int dr'r'\rho(r')f(r,r')[/tex]
it need to calculate:
[tex]\frac{\delta n(r)}{\delta\rho(r')}[/tex]

I think the derivative is r'f(r,r'). Is this right?

Thanks very much!
 
Last edited:
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  • #2
I calculate it like this:
[tex]\frac{\delta n(r)}{\delta \rho(r^{\prime \prime})}=\int dr^{\prime} r^{\prime}\delta (r^{\prime}-r^{\prime \prime })f(r, r^{\prime})=r^{\prime \prime}f(r,r^{\prime \prime})[/tex]
So, I think your result is right.
 

Related to A simple functional derivative

1. What is a functional derivative?

A functional derivative is a mathematical concept used in the field of calculus of variations. It is defined as the derivative of a functional with respect to its argument, which is typically a function rather than a single variable. In other words, it represents the change in the output of a functional caused by a small change in the input function.

2. How is a functional derivative different from a regular derivative?

A functional derivative differs from a regular derivative in that it is taken with respect to a function rather than a single variable. In addition, while a regular derivative results in a single value, a functional derivative results in a new function.

3. What is the purpose of using a functional derivative?

The main purpose of using a functional derivative is to find the minima or maxima of a functional. This is useful in many fields such as physics, engineering, and economics, where finding the optimal solution is important.

4. How is a functional derivative calculated?

The calculation of a functional derivative involves taking the limit of a difference quotient as the change in the input function approaches zero. This process is similar to finding a regular derivative, but involves integrating over the entire domain of the function instead of just a single point.

5. What are some applications of functional derivatives?

Functional derivatives have many applications in various fields, such as optimization and control theory, quantum mechanics, and economics. They are also used in the study of fluid dynamics, elasticity, and statistical mechanics. Additionally, functional derivatives are important in the development of variational methods for solving partial differential equations.

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