Local flow (${\Phi }^t$)

A local flow of a smooth vector field $V$ is a smooth map $\Phi :D\to M$, defined on an open neighborhood of $\{0\}\times M$, such that ${\Phi }^0(x)=x$, ${\partial }_t{\Phi }^t(x)=V({\Phi }^t(x))$, and ${\Phi }^{s+t}={\Phi }^s\circ {\Phi }^t$ whenever both sides are defined.

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 $x(t)=x_0/(1-t{x}_0)$ has finite-time blowup for $x_0>0$.

On ${\mathbb{R}}^2$, the vector fields $X=y{\partial }_x$ and $Y=(x^2/2){\partial }_y$ are complete, since their local flows are $(x+ty,y)$ and $(x,y+t{x}^2/2)$. Their Lie bracket of vector fields is $-(x^2/2){\partial }_x+xy{\partial }_y$. Its integral curve of a vector field starting at $(2,0)$ is $(2/(1+t),0)$, which cannot continue through $t=-1$. The sum has an integral curve of a vector fieldthat cannot continue through $t=2\sqrt{3}$. Thus neither operation preserves complete vector fields on a general noncompact smooth manifold. This contrasts with a compact manifold without boundary, where every smooth vector field is complete.

Near a point where a smooth vector field is nonzero, there are coordinates in which it is $\partial /\partial x_1$. Flow a transverse coordinate slice by the local flow; the inverse function theorem makes the resulting map a coordinate chart. This is the first step of the coordinate proof of the Frobenius theorem.