hint: Say x(t) solves x'(t) = F(x), with x(0) in U. If x(t) is not in U for all t > 0, there exists t_0 such that x(t_0) is in boundary of U, while x(t) is in U for all t between 0 and t_0. This forces x'(t_0) dot product with n(x(t_0)) to be >= 0.

4. Let 12 C R be open and F be a Cl vector field on 12. Let U CN be an open set whose closure U is a compact subset of 12, and whose boundary au is smooth. Let n: au R denote the outward pointing unit normal to av. Assume (3.82) n(x) F(x) 0, and deduce that one is forward complete on U, and also on U. 4. Let 12 C R be open and F be a Cl vector field on 12. Let U CN be an open set whose closure U is a compact subset of 12, and whose boundary au is smooth. Let n: au R denote the outward pointing unit normal to av. Assume (3.82) n(x) F(x) 0, and deduce that one is forward complete on U, and also on U