Math 6510 - Homework 9
Due at 4 PM on 11/24/04
1. Define f:R2−→ R2by f(x, y) = (1,3y2/3). Let α1(t) = (t, 0) and α2(t) = (t, t3). Show that
for i= 1,2
α0
i(t) = f(αi(t))
αi(0) = (0,0)
Why does this not contradict the uniqueness for ODE’s that we proved in class?
2. Let vtbe a one parameter family of vector fields on a compact manifold X. Define a vector
field won Y=X×Rby
w(x, t) = vt(x),∂
∂t .
Show that there exists a flow
Ψ : Y×R−→ Y
defined for all time (even though Yis not compact). Let
Φ : X×R−→ X
be defined by Φ(x, t) = π(Ψ(x, t, t)) where π:Y−→ Xis the projection of Yonto its first
factor. Set αx(t) = Φ(x, t) and show that
(dαx)t0
∂
∂t =vt0(αx(t0))
αx(0) = x
In this case Φ is the flow of the time dependent vector field vt.
3. Let Xbe a manifold, U⊂Xan open subset and
Φ : U×(−b, b)−→ X
the flow of a vector field von X. Let φt(x) = Φ(x, t). Show that
φt+s=φt◦φs
where defined. (I did this in class in a slapdash way. You should write down a more careful
proof.)
1