What is the proof for the functional derivative formula in DFT?

In summary, a functional derivative is a mathematical concept used in functional analysis to find the rate of change of a functional with respect to its input variables. It is calculated using the Euler-Lagrange equation and is significant in optimizing functionals and solving variational problems. In quantum mechanics, it is used to calculate probabilities and approximate the ground state of a system. Functional derivatives have a wide range of real-world applications in fields such as physics, economics, and engineering.
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Hi,

in their book ''Density-Functional Theory of Atoms and Molecules'' Parr and Yang state in Appendix A, Formula (A.33)

If F ist a functional that depends on a parameter [itex]\lambda[/itex], that is [itex]F[f(x,\lambda)][/itex] then:
[tex]\frac{\partial F}{\partial \lambda} = \int \frac{\delta F}{\delta f(x)} \frac{\partial f(x)}{\partial \lambda} dx[/tex]

Does anyone know a rigorous proof? (What bothers me a bit is the mixed appearance of the partial derivative [itex]\partial[/itex] and the functional derivative [itex]\delta[/itex])
 
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1. What is a functional derivative?

A functional derivative is a mathematical concept used in the field of functional analysis to find the rate of change of a functional with respect to the variations of its input variables. It is an extension of the concept of a derivative in traditional calculus, where the input variables are typically scalars or vectors. In functional analysis, the input variables can be functions, hence the name "functional" derivative.

2. How is a functional derivative calculated?

To calculate a functional derivative, one must use the Euler-Lagrange equation, which is a differential equation that relates the functional derivative of a functional to the original functional. This equation can be solved to find the functional derivative.

3. What is the significance of a functional derivative?

The functional derivative is an important tool in functional analysis as it allows for the optimization of functionals. It is used in various fields such as physics, economics, and engineering to find the minimum or maximum of a functional. It also helps in solving variational problems, which involve finding a function that minimizes a given functional.

4. How is a functional derivative used in quantum mechanics?

In quantum mechanics, the wave function of a particle is described by a functional, which depends on the position and time variables. The functional derivative of this wave function is used to calculate the probability density of finding the particle at a specific position at a given time. It is also used in the variational principle, which is used to find approximations to the ground state of a quantum system.

5. What are some real-world applications of functional derivatives?

Functional derivatives have a wide range of applications in various fields. In physics, they are used in quantum mechanics, statistical mechanics, and fluid dynamics. In economics, they are used in game theory, optimal control, and portfolio optimization. In engineering, they are used in optimization and control theory. They are also used in image processing, signal processing, and machine learning.

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