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Rasalhague
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Roger Penrose, in The Road to Reality, introduces the idea of what he calls a "functional derivative", "denoted by using [itex]\delta[/itex] in place of [itex]\partial[/itex]; "Carrying out a functional derivative in practice is essentially just applying the same rules as for ordinary calculus" (Vintage 2005, p. 487). He cites expressions of the form
[tex]\frac{\delta \mathcal{L}}{\delta \Psi}[/tex]
presumably the partial derivative of the Lagrangian with respect to a function [itex]\Psi[/itex], specifically a tensor or spinor field on spacetime.
On p. 489, he uses the expression [itex]\delta S[/itex] to indicate that a quantity S has zero derivative with respect to all independent variables ("constituent fields").
On pp. 460-461, he describes [itex]\delta M[/itex] as the mass within an infinitesimal volume [itex]\delta V[/itex]. Is there any relationship between this notation and his later functional derivative delta? Is this, in some sense, an instance of a functional derivative, or just a coincidence of notation? When I first saw it, I guessed it might mean an integrand in general, and that he was using [itex]\delta V[/itex], say, for a volume element, rather than the traditional [itex]d V[/itex], so as to avoid the misleading impression that this was the exterior derivative of some covariant alternating tensor, V.
[tex]\frac{\delta \mathcal{L}}{\delta \Psi}[/tex]
presumably the partial derivative of the Lagrangian with respect to a function [itex]\Psi[/itex], specifically a tensor or spinor field on spacetime.
On p. 489, he uses the expression [itex]\delta S[/itex] to indicate that a quantity S has zero derivative with respect to all independent variables ("constituent fields").
On pp. 460-461, he describes [itex]\delta M[/itex] as the mass within an infinitesimal volume [itex]\delta V[/itex]. Is there any relationship between this notation and his later functional derivative delta? Is this, in some sense, an instance of a functional derivative, or just a coincidence of notation? When I first saw it, I guessed it might mean an integrand in general, and that he was using [itex]\delta V[/itex], say, for a volume element, rather than the traditional [itex]d V[/itex], so as to avoid the misleading impression that this was the exterior derivative of some covariant alternating tensor, V.