= Complete vector field
A smooth <vector field> is complete if its maximal <integral curves of a vector field> exist for every real time. Its <local flow> is then a global one-parameter group of <diffeomorphisms>; uniqueness of <ordinary differential equations> proves the group law and the inverse-time identity. On a <compact> manifold without boundary, every smooth vector field is complete: <compactness> gives a uniform positive local existence interval through every point, so a curve can be successively extended past any proposed finite endpoint. On $\mathbb R$, $x^2\partial_x$ is not complete because its solution $x(t)=x_0/(1-tx_0)$ has finite-time blowup for $x_0>0$.
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