Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 481Then we haveSince⎧⎨ γ ∗ (dx) = cos θdr − r sin θdθγ ∗ (dy) = sin θdr + r cos θdθ⎩γ ∗ (dx ∧ dy) = r (dr ∧ dθ).dω = (∂G/∂y + ∂F/∂x) dx ∧ dy = div (F) dx ∧ dy,thenγ ∗ (dω) = div (F) r (dr ∧ dθ).On the other hand,γ ∗ ω = (F sin θ − G cos θ)dr + (F r cos θ + G r sin θ) dθ, (3.301)(all F , G, and F to be evaluated at (r cos θ, r sin θ)). Therefore∫ ∫ 2π ∫ 1dω = div(F) r dr dθ;γ00this is ∫ B 1div (F) dA. On the other hand, by Stokes’ Theorem ∫ ∫γ dω =ω, which is a curve integral of the 1–form (3.301) around the boundary∂γof the rectangle [0, 2π] × [0, 1]. This curve integral is a sum of four termscorresponding to the four sides of the rectangle. Two of these (correspondingto the sides θ = 0 and θ = 2π) cancel, and the term corresponding tothe side where r = 0 vanishes because of the r in r (dr∧dθ), so only the sidewith r = 1 remains, and its contribution is, with the correct orientation,∫ 2π∫(F (cos θ, sin θ) cos θ + G(cos θ, sin θ) sin θ) dθ = F · n ds ,0S 1where n is the outward unit normal of the unit circle. This expression isthe flux of F over the unit circle, which thus equals the divergence integralcalculated above.3.17.2.5 Time Derivatives of Expanding SpheresWe now combine vector calculus with the calculus of the basic ball– andsphere–distributions, to get the following result [Kock and Reyes (2003)]:In R n (for any n), we have, for any t,ddt St = t · ∆(B t ),