The Lie derivative may be defined in several equivalent ways. To keep things simple, we begin by defining the Lie derivative acting on scalar functions and vector fields, before moving on to the definition for general tensors.
The (Lie) derivative of a function Defining the derivative of a function f\colon M \to {\mathbb R} on a manifold takes care because the
difference quotient \textstyle (f(x+h)-f(x))/h cannot be determined while the displacement x+h is undefined. The Lie derivative of a function f\colon M\to {\mathbb R} with respect to a
vector field X at a point p \in M is the function :(\mathcal{L}_X f) (p) = {d \over dt} \biggr|_{t=0} \bigl(f \circ \Phi^t_X\bigr)(p) = \lim_{t\to 0} \frac{f\bigl(\Phi^t_X(p)\bigr) - f\bigl(p\bigr)}{t} where \Phi^t_X(p) is the point to which the
flow defined by the vector field X maps the point p at time instant t. In the vicinity of t=0, \Phi^t_X(p) is the unique solution of the system : \frac{d}{dt}\biggr|_t \Phi^t_X(p) = X\bigl(\Phi^t_X(p)\bigr) of first-order autonomous (i.e. time-independent) differential equations, with \Phi^0_X(p) = p. Setting \mathcal{L}_X f = \nabla_X f identifies the Lie derivative of a function with the
directional derivative, which is also denoted by X(f):= \mathcal{L}_X f = \nabla_X f.
The Lie derivative of a vector field If
X and
Y are both vector fields, then the Lie derivative of
Y with respect to
X is also known as the
Lie bracket of
X and
Y, and is sometimes denoted [X,Y]. There are several approaches to defining the Lie bracket, all of which are equivalent. We list two definitions here, corresponding to the two definitions of a vector field given above: {{unordered list : \mathcal{L}_X Y (p) = [X,Y](p) = \partial_X Y(p) - \partial_Y X(p), where \partial_X and \partial_Y denote the operations of taking the
directional derivatives with respect to
X and
Y, respectively. Here we are treating a vector in
n-dimensional space as an
n-
tuple, so that its directional derivative is simply the tuple consisting of the directional derivatives of its coordinates. Although the final expression \partial_X Y(p) - \partial_Y X(p) appearing in this definition does not depend on the choice of local coordinates, the individual terms \partial_X Y(p) and \partial_Y X(p) do depend on the choice of coordinates. : [X,Y]: C^\infty(M) \rightarrow C^\infty(M) : [X,Y](f) = X(Y(f)) - Y(X(f)) is a derivation of order zero of the algebra of smooth functions of
M, i.e. this operator is a vector field according to the second definition. }}
The Lie derivative of a tensor field Definition in terms of flows The Lie derivative is the speed with which the tensor field changes under the space deformation caused by the flow. Formally, given a differentiable (time-independent) vector field X on a smooth manifold M, let \Phi^t_X : M \to M be the corresponding local flow. Since \Phi^t_X is a local diffeomorphism for each t, it gives rise to a
pullback of tensor fields. For covariant tensors, this is just the multi-linear extension of the
pullback map \left(\Phi^t_X\right)^*_p : T^*_{\Phi^t_X(p)}M \to T^*_{p}M, \qquad \left(\left(\Phi^t_X\right)^*_p \alpha\right) (Y) = \alpha\bigl(T_p \Phi^t_X(Y)\bigr), \quad \alpha \in T^*_{\Phi^t_X(p)}M, Y \in T_{p}M For contravariant tensors, one extends the inverse :\left(T_p\Phi^t_X\right)^{-1} : T_{\Phi^t_X(p)}M \to T_{p}M of the
differential . For every t, there is, consequently, a tensor field (\Phi^t_X)^* T of the same type as . If T is an (r,0)- or (0,s)-type tensor field, then the Lie derivative {\cal L}_XT of T along a vector field X is defined at point p \in M to be :{\cal L}_X T(p) = \frac{d}{dt}\biggl|_{t=0} \left(\bigl(\Phi^t_X\bigr)^* T\right)_p = \frac{d}{dt}\biggl|_{t=0}\bigl(\Phi^t_X\bigr)^*_p T_{\Phi^t_X(p)} = \lim_{t \to 0}\frac{\bigl(\Phi^t_X\bigr)^*T_{\Phi^t_X(p)} - T_p}{t}. The resulting tensor field {\cal L}_X T is of the same type as T. More generally, for every smooth 1-parameter family \Phi_t of diffeomorphisms that integrate a vector field X in the sense that {d \over dt}\biggr|_{t=0} \Phi_t = X \circ \Phi_0 , one has\mathcal{L}_X T = \bigl(\Phi_0^{-1}\bigr)^* {d \over dt}\biggr|_{t=0} \Phi_t^* T = - {d \over dt}\biggr|_{t=0} \bigl(\Phi_t^{-1}\bigr)^* \Phi_0^* T \, .
Algebraic definition We now give an algebraic definition. The algebraic definition for the Lie derivative of a tensor field follows from the following four axioms: :
Axiom 1. The Lie derivative of a function is equal to the directional derivative of the function. This fact is often expressed by the formula ::\mathcal{L}_Yf=Y(f) :
Axiom 2. The Lie derivative obeys the following version of Leibniz's rule: For any tensor fields
S and
T, we have ::\mathcal{L}_Y(S\otimes T)=(\mathcal{L}_YS)\otimes T+S\otimes (\mathcal{L}_YT). :
Axiom 3. The Lie derivative obeys the Leibniz rule with respect to
contraction: :: \mathcal{L}_X (T(Y_1, \ldots, Y_n)) = (\mathcal{L}_X T)(Y_1,\ldots, Y_n) + T((\mathcal{L}_X Y_1), \ldots, Y_n) + \cdots + T(Y_1, \ldots, (\mathcal{L}_X Y_n)) :
Axiom 4. The Lie derivative commutes with exterior derivative on functions: :: [\mathcal{L}_X, d] = 0 Using the first and third axioms, applying the Lie derivative \mathcal{L}_X to Y(f) shows that ::\mathcal{L}_X Y (f) = X(Y(f)) - Y(X(f)), which is one of the standard definitions for the
Lie bracket. The Lie derivative acting on a differential form is the
anticommutator of the
interior product with the exterior derivative. So if α is a differential form, ::\mathcal{L}_Y\alpha=i_Yd\alpha+di_Y\alpha. This follows easily by checking that the expression commutes with exterior derivative, is a derivation (being an anticommutator of graded derivations) and does the right thing on functions. This is '''Cartan's magic formula'''. See
interior product for details. Explicitly, let
T be a tensor field of type . Consider
T to be a differentiable
multilinear map of
smooth sections α1,
α2, ...,
αp of the cotangent bundle
T∗
M and of sections
X1,
X2, ...,
Xq of the
tangent bundle TM, written
T(
α1,
α2, ...,
X1,
X2, ...) into
R. Define the Lie derivative of
T along
Y by the formula :(\mathcal{L}_Y T)(\alpha_1, \alpha_2, \ldots, X_1, X_2, \ldots) =Y(T(\alpha_1,\alpha_2,\ldots,X_1,X_2,\ldots)) ::- T(\mathcal{L}_Y\alpha_1, \alpha_2, \ldots, X_1, X_2, \ldots) - T(\alpha_1, \mathcal{L}_Y\alpha_2, \ldots, X_1, X_2, \ldots) -\ldots ::- T(\alpha_1, \alpha_2, \ldots, \mathcal{L}_YX_1, X_2, \ldots) - T(\alpha_1, \alpha_2, \ldots, X_1, \mathcal{L}_YX_2, \ldots) - \ldots The analytic and algebraic definitions can be proven to be equivalent using the properties of the pushforward and the
Leibniz rule for differentiation. The Lie derivative commutes with the contraction.
The Lie derivative of a differential form A particularly important class of tensor fields is the class of
differential forms. The restriction of the Lie derivative to the space of differential forms is closely related to the
exterior derivative. Both the Lie derivative and the exterior derivative attempt to capture the idea of a derivative in different ways. These differences can be bridged by introducing the idea of an
interior product, after which the relationships falls out as an identity known as '''Cartan's formula'''. Cartan's formula can also be used as a definition of the Lie derivative on the space of differential forms. Let M be a manifold and X a vector field on M. Let \omega \in \Lambda^k(M) be a k-
form, i.e., for each p \in M, \omega(p) is an
alternating multilinear map from (T_p M)^k to the real numbers. The
interior product of X and \omega is the (k-1)-form i_X\omega defined as (i_X\omega) (X_1, \ldots, X_{k-1}) = \omega (X,X_1, \ldots, X_{k-1})\, The differential form i_X\omega is also called the
contraction of \omega with X, and i_X:\Lambda^k(M) \rightarrow \Lambda^{k-1}(M) is a
\wedge-
antiderivation where
\wedge is the
wedge product on differential forms. That is, i_X is \mathbb{R}-linear, and i_X (\omega \wedge \eta) = (i_X \omega) \wedge \eta + (-1)^k \omega \wedge (i_X \eta) for \omega \in \Lambda^k(M) and \eta another differential form. Also, for a function f \in \Lambda^0(M), that is, a real- or complex-valued function on M, one has i_{fX} \omega = f\,i_X\omega, where f X denotes the product of f and X. The relationship between
exterior derivatives and Lie derivatives can then be summarized as follows. First, since the Lie derivative of a function
f with respect to a vector field
X is the same as the directional derivative
X(
f), it is also the same as the
contraction of the exterior derivative of
f with
X: :\mathcal{L}_Xf = i_X \, df For a general differential form, the Lie derivative is likewise a contraction, taking into account the variation in
X: :\mathcal{L}_X\omega = i_Xd\omega + d(i_X \omega). This identity is known variously as
Cartan formula,
Cartan homotopy formula or '''Cartan's magic formula'''. See
interior product for details. The Cartan formula can be used as a definition of the Lie derivative of a differential form. Cartan's formula shows in particular that :d\mathcal{L}_X\omega = \mathcal{L}_X(d\omega). The Lie derivative also satisfies the relation :\mathcal{L}_{fX}\omega = f\mathcal{L}_X\omega + df \wedge i_X \omega . ==Coordinate expressions==