Let F: X Rn Rn be a continuous vector field. Let(a, b) be an interval inRthat contains

Chapter 3, Problem 31

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Let F: X Rn Rn be a continuous vector field. Let(a, b) be an interval inRthat contains 0. (Think of(a, b) as a time interval.) A flow of F is a differentiable function : X (a, b) Rn of n + 1 variables such that t (x, t) = F((x, t)); (x, 0) = x. Intuitively, we think of (x, t) as the point at time t on the flow line of F that passes through x at time 0. (See Figure 3.37.) Thus, the flow of F is, in a sense, the collection of all flow lines of F. Exercises 2631 concern flows of vector fields. (x, 0) = x (x, t) F( ( x, t)) Figure 3.37 The flow of the vector field FDerive the equation of first variation for a flow of a vector field. That is, if F is a vector field of class C1 with flow of class C2, show that t Dx(x, t) = DF((x, t))Dx(x, t). Here the expression Dx(x, t) means to differentiate with respect to the variables x1, x2,..., xn, that is, by holding t fixed.