Could any one help me to solve these two problems on vector fields
Any $\mathbb{C}^{\infty}$ vector field on a compact manifold is complete.
Is every vector field on $\mathbb{R}$ complete?
Could any one help me to solve these two problems on vector fields
Any $\mathbb{C}^{\infty}$ vector field on a compact manifold is complete.
Is every vector field on $\mathbb{R}$ complete?
The 1st is true, not the 2nd. The flow associated to $\dot{x}=x^2$ is given by $\phi_t(x)=x/(1-tx)$, and obviously does not exist for every $t$.
For the first, use the neighbourhood definition of compactness and do a proof by contradiction. The second is not true and you only have to find a suitable counter example.