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

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SUMMARY

The discussion centers on the proof of the functional derivative formula in Density Functional Theory (DFT) as presented by Parr and Yang in their book "Density-Functional Theory of Atoms and Molecules." Specifically, the formula in question is Formula (A.33), which relates the partial derivative of a functional F with respect to a parameter λ to the functional derivative of F with respect to the function f(x). The mixed use of partial and functional derivatives raises questions about the rigor of the proof, prompting requests for clarification and detailed explanations.

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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 \lambda, that is F[f(x,\lambda)] then:
\frac{\partial F}{\partial \lambda} = \int \frac{\delta F}{\delta f(x)} \frac{\partial f(x)}{\partial \lambda} dx

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