In the mathematical field of differential topology, the **Lie bracket of vector fields**, **Jacobi–Lie bracket**, or **commutator of vector fields** is a bilinear differential operator which assigns, to any two vector fields *X* and *Y* on a smooth manifold *M*, a third vector field denoted . It is the specialization of the Lie derivative to the case of Lie differentiation of a vector field. Indeed, equals the Lie derivative .

It plays an important role in differential geometry and differential topology, and is also fundamental in the geometric theory for nonlinear control systems (Isaiah 2009, pp. 20–21, nonholonomic systems; Khalil 2002, pp. 523–530, feedback linearization).

A generalization of the Lie bracket (to vector-valued differential forms) is the Frölicher–Nijenhuis bracket.

Read more about Lie Bracket Of Vector Fields: Definition, Properties, Examples, Applications

### Other articles related to "lie bracket of vector fields, lie bracket":

**Lie Bracket Of Vector Fields**- Applications

... The Jacobi–

**Lie bracket**is essential to proving small-time local controllability (STLC) for driftless affine control systems ...

### Famous quotes containing the words fields and/or lie:

“The need to exert power, when thwarted in the open *fields* of life, is the more likely to assert itself in trifles.”

—Charles Horton Cooley (1864–1929)

“The lakes are something which you are unprepared for; they *lie* up so high, exposed to the light, and the forest is diminished to a fine fringe on their edges, with here and there a blue mountain, like amethyst jewels set around some jewel of the first water,—so anterior, so superior, to all the changes that are to take place on their shores, even now civil and refined, and fair as they can ever be.”

—Henry David Thoreau (1817–1862)