6 1. Differential Equations and Their Solutions for the moment, but mathematically it makes no difference, and as we shall see later, even when working with real-world, three-dimensional problems, it is often important to make use of higher-dimensional spaces. On the other hand, an “instant of time” will always be repre- sented by a single real number t. (There are mathematical situations that do require multi-dimensional time, but we shall not meet them here.) Thus, knowing the wind velocity at every point of space and at all instants of time means that we have a function V that as- sociates to each (x, t) in Rn × R a vector V (x, t) in Rn, the wind velocity at x at time t. We will denote the n components of V (x, t) by V1(x, t),... , Vn(x, t). (We will always assume that V is at least continuous and usually that it is even continuously differentiable.) How should we model the path taken by a smoke particle? An ideal smoke particle is characterized by the fact that it “goes with the flow”, i.e., it is carried along by the wind. That means that if x(t) = (x1(t),... , xn(t)) is its location at a time t, then its velocity at time t will be the wind velocity at that point and time, namely V (x(t),t). But the velocity of the particle at time t is x (t) = (x 1 (t),... , x n (t)), where primes denote differentiation with respect to t, i.e., x = dx dt = ( dx1 dt , . . . , dxn dt ). So the path of a smoke particle will be a differentiable curve x(t) in Rn that satisfies the differential equation x (t) = V (x(t),t), or dx dt = V (x, t). If we write this in components, it reads dxi dt = Vi(x1(t),... , xn(t),t), for i = 1,... , n, and for this reason it is often called a system of differential equations. Finally, if at a time t0 we observe that the smoke particle is at the point x0 in Rn, then the “initial condition” x(t0) = x0 is also satisfied. The page devoted to Chapter 1 in the Web Companion contains a QuickTime movie showing the wind field of a time-dependent two- dimensional system and the path traced out by a “smoke particle”. Figure 1.1 shows the direction field and a few such solution curves for an interesting and important one-dimensional ODE called the logistic equation.
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