Definition
Each vector field X on a smooth manifold M may be regarded as a differential operator acting on smooth functions on M. Indeed, each vector field X becomes a derivation on the smooth functions C∞(M) when we define X(f) to be the element of C∞(M) whose value at a point p is the directional derivative of f at p in the direction X(p).
The space of derivations of C∞(M) is a Lie algebra under the operation . This Lie algebra structure can be transferred to the set of vector fields on M as follows.
The Jacobi–Lie bracket or simply Lie bracket, of two vector fields X and Y is the vector field such that
Such a vector field exists because the right hand side is a derivation of C∞(M), and the vector space of such derivations is known to be isomorphic to the space of vector fields on M by the map sending a vector field X to the derivation .
To make the connection to the Lie derivative, let be the 1-parameter group of diffeomorphisms (or flow) generated by the vector field . The differential of each diffeomorphism maps the vector field Y to a new vector field . To pull-back the vector field one applies the differential of the inverse, . The Lie bracket is defined by
In particular, is the Lie derivative of the vector field with respect to . Conceptually, the Lie bracket is the derivative of in the `direction' generated by .
Though neither definition of the Lie bracket depends on a choice of coordinates, in practice one often wants to compute the bracket with respect to a coordinate system. Let be a set of local coordinate functions, and let denote the associated local frame. Then
(Here we use the Einstein summation convention)
Read more about this topic: Lie Bracket Of Vector Fields
Famous quotes containing the word definition:
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)