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2. Consider a domain D in configuration space (Fig. 5). The decrease of the mass it contains, which is – ∫D ∂tf, equals outgoing mass. The latter is the flux through the boundary S of the vector field {v, γ } f, which is the speed, not of a particle in physical space, but of the representative point {x, v} in configuration space. By Ostrogradskii, ∂tf + div({v, γ } f) = 0. Since γ does not depend on v, div({v, γ }) = 0. So div({v, γ } f) = {v, γ } · ∇f ≡ v · ∇x f + γ · ∇vf. ) If γ depends on v, an additional term f div vγ will appear on the left-hand side of (34).

Depending on the local shape of the boundary, it may still be possible to define a gradient at a boundary point, by taking directional derivatives. How to do that is clear in the case of a smooth boundary (on each line through a boundary point, there is a half-line going inwards). But it’s more problematic at a corner, at the tip of a cusp, etc. This is why the concept of smoothness over a region (not only inside it), including the boundary, is delicate. To avoid ambiguities about it, I’ll say that a function f is smooth over a region R (which may itself be very irregular, devoid of a smooth boundary) if there is a domain D containing R in which some extension of f (cf.

Schäfer: “On the uniqueness of the vector for energy flow density in electromagnetic fields”, Am. J. , 54, 3 (1986), pp. 279–280. K. B. Langdon: Plasma Physics via Computer Simulation, McGraw-Hill (New York), 1985. K. Birkeland: “Ueber die Strahlung electromagnetischer Energie im Raume”, Annalen der Physik, 52 (1894), pp. 357–380. A. Bossavit: “On local computation of the force field in deformable bodies”, Int. J. Applied Electromagnetics in Materials, 2, 4 (1992), pp. 333–343. B. Cabrera : “First results from a superconducting detector for moving magnetic monopoles”, Phys.

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Computational electromagnetism by Bossavit A.

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