Lie bracket of vector fields
In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns, to any two vector fields X and Y on a smooth manifold M, a third vector field denoted [X, Y].
Conceptually, the Lie bracket [X,Y] is the derivative of Y along the flow generated by X. A generalization of the Lie bracket is the Lie derivative, which allows differentiation of any tensor field along the flow generated by X. The Lie bracket [X,Y] equals the Lie derivative of the vector Y (which is a tensor field) along X, and is sometimes denoted (read "the Lie derivative of Y along X").
The Lie bracket is an R-bilinear operation and turns the set of all vector fields on the manifold M into an (infinite-dimensional) Lie algebra.
The Lie bracket plays an important role in differential geometry and differential topology, for instance in the Frobenius theorem, 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).
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