Let F: X ⊆ Rn → Rn be a continuous vector field. Let (a, b) be an interval in R...

Let F: X ⊆ Rⁿ → Rⁿ be a continuous vector field. Let (a, b) be an interval in R that contains 0. (Think of (a, b) as a “time interval.”) A flow of F is a differentiable function φ: X × (a, b) → Rⁿ of n + 1 variables such that

Intuitively, we think of φ(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 26–31 concern flows of vector fields.

Derive 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 φ of class C², show that

Here the expression “D_xφ(x, t)” means to differentiate φ with respect to the variables x₁, x₂, . . . , x_n, that is, by holding t fixed.