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:
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)