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The Divergence As the Rate of Change in Area Or Volume

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The Divergence As the Rate of Change in Area Or Volume The divergence as the rate of change in area or volume Here we give a very brief sketch of the following fact, which should be familiar to you from multivariable calculus: If a region D of the plane moves with velocity given by the vector field V (x; y) = (f(x; y); g(x; y)); then the instantaneous rate of change in the area of D is the double integral of the divergence of V over D: ZZ ZZ rate of change in area = r · V dA = fx + gy dA: D D (An analogous result is true in three dimensions, with volume replacing area.) To see why this should be so, imagine that we have cut D up into a lot of little pieces Pi, each of which is rectangular with width Wi and height Hi, so that the area of Pi is Ai ≡ HiWi. If we follow the vector field for a short time ∆t, each piece Pi should be “roughly rectangular”. Its width would be changed by the the amount of horizontal stretching that is induced by the vector field, which is difference in the horizontal displacements of its two sides. Thisin turn is approximated by the product of three terms: the rate of change in the horizontal @f component of velocity per unit of displacement ( @x ), the horizontal distance across the box (Wi), and the time step ∆t. Thus ! @f ∆W ≈ W ∆t: i @x i See the figure below; the partial derivative measures how quickly f changes as you move horizontally, and Wi is the horizontal distance across along Pi, so the product fxWi measures the difference in the horizontal components of V at the two ends of Wi. Now if we follow the flow of the vector field V for a short time ∆t, the total change in width is approximated by the product of fx, Wi, and ∆t. Similarly, its height would be changed by the product of the analogous three factors, namely ! @g ∆H ≈ H ∆t: i @y i 1 Thus the change in the area of one piece is ! @f @g ∆A ≈ H ∆W + W ∆H ≈ + H W ∆t: i i i i i @x @y i i Summing over i we see that " ! # X X @f @g ∆A = ∆Ai ≈ + HiWi ∆t i i @x @y ZZ ≈ ∆t · fx + gy dA; D (here we have used the fact that the sum in the large square brackets is a Riemann sum for the double integral in the last line). This is saying that for small times ∆t the change in area is roughly given by ∆t times the integral of the divergence, which is another way of saying that the instantaneous rate of change in the area is the integral of the divergence over D. (Of course, a lot of careful work has to be done to ensure that all of the rough approximations done in the outline above can be verified.) The importance of all this in differential equations is simple: in a region where the divergence of the vector field V (x; y) is negative, area shrinks as you follow the solutions of the differential equation (x0(t); y0(t)) = V (x; y), and in a region where the divergence is positive, areas expand as they follow solutions of the differential equation. Knowing this can be useful in a situation where you can determine that some solutions in the plane are spiraling around an equilibrium: spiraling in corresponds to shrinking area and hence to negative divergence, while spiraling out corresponds to expanding area and to positive divergence. For general vector fields V , one does not expect to find large regions where the divergence is 0, but there is an important class of examples where the divergence is zero everywhere. These are called the conservative, or divergence-free vector fields. In two dimensions this class coincides with the class of Hamiltonian vector fields discussed in section 5.3 of the text. Note that for a divergence-free vector field in the plane, all areas are preserved as you follow solutions. In particular, there cannot be any spiraling in nor spiraling out, so if any orbit moves around an equilibrium, it must exactly close up on itself and be a periodic orbit. 2.
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