The mathematics of K-conserving functional differentiation, with K being theintegral of some invertible function of the functional variable, is clarified.The most general form for constrained functional derivatives is derived fromthe requirement that two functionals that are equal over a restricted domainhave equal derivatives over that domain. It is shown that the K-conservingderivative formula is the one that yields no effect of K-conservation on thedifferentiation of K-independent functionals, which gives the basis for itsgeneralization for multiple constraints. Connections with the derivative withrespect to the shape of the functional variable and with the shape-conservingderivative, together with their use in the density-functional theory ofmany-electron systems, are discussed. Yielding an intuitive interpretation ofK-conserving functional derivatives, it is also shown that K-conservingderivatives emerge as directional derivatives along K-conserving paths, whichis achieved via a generalization of the Gateaux derivative for that kind ofpaths. These results constitute the background for the practical application ofK-conserving differentiation.
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