Ivancevic_Applied-Diff-Geom

Technical Preliminaries: Tensors, Actions and Functors 79which is the absolute derivative of a product, and therefore expands into˙ρdv + ρ ˙ dv = 0. (2.34)Now, as the fluid density ρ = ρ(x k , t) is a function of both time t andspatial coordinates x k , for k = 1, 2, 3, that is, a scalar–field, its total timederivative ˙ρ, figuring in (2.34), is defined by˙ρ = ∂ t ρ + ∂ x kρ ∂ t x k ≡ ∂ t ρ + ρ ;k u k , (2.35)or, in vector form ˙ρ = ∂ t ρ + grad(ρ) · u,where u k = u k (x k , t) ≡ u is the velocity vector–field of the fluid.Regarding dv, ˙ the other term figuring in (2.34), we start by expanding anelementary volume dv along the sides {dx i (p) , dxj (q) , dxk (r)} of an elementaryparallelepiped, asdv = 1 3! δpqr ijk dxi (p) dxj (q) dxk (r), (i, j, k, p, q, r = 1, 2, 3)so that its absolute derivative becomesdv˙= 1 ˙2! δpqr dx i (p)dx j (q) dxk (r)ijk= 1 2! ui ;lδ pqrijk dxl (p) dxj (q) dxk (r) (using ˙ dx i (p) = u i ;ldx l (p) ),which finally simplifies into˙dv = u k ;kdv ≡ div(u) dv. (2.36)Substituting (2.35) and (2.36) into (2.34) gives˙ρdv ≡ ( ∂ t ρ + ρ ;k u k) dv + ρu k ;kdv = 0. (2.37)As we are dealing with arbitrary fluid particles, dv ≠ 0, so from (2.37)follows∂ t ρ + ρ ;k u k + ρu k ;k ≡ ∂ t ρ + (ρu k ) ;k = 0. (2.38)Equation (2.38) is the covariant form of the continuity equation, which instandard vector notation becomes (2.32), i.e., ∂ t ρ + div(ρu) = 0.