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A local flow of $v$ is a smooth family ${\Phi }^t$ defined near $\{0\}\times X$ such that ${\Phi }^0=\mathrm{id}$, ${\partial }_t{\Phi }^t(x)=v({\Phi }^t(x))$, and ${\Phi }^{s+t}={\Phi }^s\circ {\Phi }^t$ whenever defined. The Lie derivative of a differential form isCartan's magic formula is

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.

For the local flow of a smooth compactly supported vector field and any scalar BV space function $w$, the change of variables formula and polar decomposition of a vector measure giveThe two cofactor matrices are $I\pm tB+O(t^2)$ uniformly. On $|{\sigma }_w|=1$, the linear terms in their Euclidean norm expansions cancel. Integration bounds the remainder by $C{t}^2|Dw|(\Omega )$. The identity includes all components of the decomposition of a BV derivative; it does not require smoothness of $w$ or its jump interfaces.

For smooth vector fields $X,Y$, the local flows commute wherever both compositions are defined exactly when $[X,Y]=0$. The derivative of the pullback of $Y$ by the $X$ flow is the pullback of $[X,Y]$. A zero bracket therefore makes $Y$ invariant under that flow; uniqueness of solutions to its ordinary differential equation gives commutation. Conversely commutation implies invariance, whose derivative gives the zero bracket. Neither field need have a complete flow.

A vector field preserves a chosen volume form $\omega$ when its local flow pulls $\omega$ back to itself. Infinitesimally this is ${\mathcal{L}}_X\omega =0$, using the Lie derivative. The condition depends on the chosen volume form.