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.

If φ is a flow of the vector field F, explain why φ(φ(x, t), s) = φ(x, s + t). (Hint: Relate the value of the flow φ at (x, t) to the flow line of F through x. You may assume the fact that the flow line of a continuous vector field at a given point and time is determined uniquely.)