Introduction to the Modeling and Analysis of Complex Systems

13.2. FUNDAMENTALS OF VECTOR CALCULUS 233Actually, Fig. 13.4 may be a bit misleading; the particles can go in and out through anyedge in both directions. But we use the + sign for v x (x − h, y) and v y (x, y − h) and the −sign for v x (x + h, y) and v y (x, y + h) in Eq. (13.6), because the velocities are measuredusing the coordinate system where the rightward and upward directions are consideredpositive.Since N depends on the size of the area, we can divide it by the area ((2h) 2 in thiscase) to calculate the change in terms of the concentration of the particles, c = N/(2h) 2 ,which won’t depend on h:∂c∂t = 2hv x(x − h, y) + 2hv y (x, y − h) − 2hv x (x + h, y) − 2hv y (x, y + h)(2h) 2 (13.7)= v x(x − h, y) + v y (x, y − h) − v x (x + h, y) − v y (x, y + h)2hIf we make the size of the area really small (h → 0), this becomes the following:∂c∂t = lim v x (x − h, y) + v y (x, y − h) − v x (x + h, y) − v y (x, y + h)h→02h){(= lim − v x(x + h, y) − v x (x − h, y)h→0 2h= − ∂v x∂x − ∂v y∂y+(− v y(x, y + h) − v y (x, y − h)2h(13.8)(13.9))}(13.10)(13.11)= −∇ · v (13.12)In natural words, this means that the temporal change of a concentration of the stuffis given by a negative divergence of the vector field v that describes its movement. Ifthe divergence is positive, that means that the stuff is escaping from the local area. If thedivergence is negative, the stuff is flowing into the local area. The mathematical derivationabove confirms this intuitive understanding of divergence.Exercise 13.13-D cases.Confirm that the interpretation of divergence above also applies to