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 [X, Y]. It is closely related to, and sometimes also known as, the Lie derivative. In particular, the bracket <math>[X,Y]</math> equals the Lie derivative <math>\mathcal{L}_X Y</math>.

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  • 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 [X, Y]. It is closely related to, and sometimes also known as, the Lie derivative. In particular, the bracket <math>[X,Y]</math> equals the Lie derivative <math>\mathcal{L}_X Y</math>. It plays an important role in differential geometry and differential topology, and is also fundamental in the geometric theory for nonlinear control systems. A generalization of the Lie bracket is the Frölicher–Nijenhuis bracket.
  • En topología diferencial, dados dos campos de vectores diferenciables X e Y sobre una variedad M, se define el corchete de Lie de los campos X e Y, notado <math>[X,Y]</math> como el único campo de vectores que cumple: Su expresión en un sistema de coordenadas asociado una carta local <math>x^\mu</math> será: <math>[X,Y]^i= \sum_{j=1}^n \left (X^j \frac {\partial Y^i}{\partial x^j} \right) - \left (Y^j \frac {\partial X^i}{\partial x^j} \right)</math> donde n es la dimensión de M. El corchete de Lie de dos campos constituye un caso particular de una operación más general: la derivada de Lie de un tensor cualquiera <math>\mathcal{L}_X T</math> a lo largo de la dirección que marque un campo X. Cuando T es un campo de vectores Y, recuperamos el corchete de Lie <math>\mathcal{L}_X Y = [X,Y]</math>.
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  • August 2008
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  • 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 [X, Y]. It is closely related to, and sometimes also known as, the Lie derivative. In particular, the bracket <math>[X,Y]</math> equals the Lie derivative <math>\mathcal{L}_X Y</math>.
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  • Lie bracket of vector fields
  • Corchete de Lie (campos de vectores)
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