Lie derivative

Let M be a smooth manifold, X a vector fieldMathworldPlanetmath on M, and T a tensor on M. Then the Lie derivativeMathworldPlanetmathPlanetmath XT of T along X is a tensor of the same rank as T defined as


where ρ is the flow of X, and ρt* is pullback by ρt.

The Lie derivative is a notion of directional derivativeMathworldPlanetmathPlanetmath for tensors. Intuitively, this is the change in T in the direction of X.

If X and Y are vector fields, then XY=[X,Y], the standard Lie bracket of vector fields.

Title Lie derivative
Canonical name LieDerivative
Date of creation 2013-03-22 13:14:10
Last modified on 2013-03-22 13:14:10
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 6
Author rspuzio (6075)
Entry type Definition
Classification msc 53-00
Related topic LeibnizNotationForVectorFields
Related topic CartanCalculus