= Solution
For a <smooth> <vector field> $X$, solve the initial-value problem $d\phi_t(p)/dt=X(\phi_t(p))$, $\phi_0(p)=p$. Local existence, uniqueness and smooth dependence for <ordinary differential equations> give a <local flow>. Uniqueness yields $\phi_{t+s}(p)=\phi_t(\phi_s(p))$ whenever both sides are defined, and $\phi_{-t}$ is the inverse of $\phi_t$ on the corresponding domains. Thus each local flow map is a <diffeomorphism> between open subsets.
\b[A group of diffeomorphisms of the entire manifold for every real $t$ requires a <complete vector field>.] This follows, for example, on a <compact> manifold without boundary: local existence times can be made uniform on a finite cover, and repeated continuation prevents finite-time escape. Without completeness, the source's group is only local. The vector field $x^2\partial_x$ on $\mathbb R$ has flow $x\mapsto x/(1-tx)$ and blows up at $t=1/x$ for $x>0$, so smoothness alone cannot imply a global one-parameter group.
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