In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by For other meanings of mathematics or math, see mathematics (disambiguation). ... Marius Sophus Lie (December 17, 1842 - February 18, 1899) was a Norwegian-born mathematician who largely created the theory of continuous symmetry, and applied it to the study of geometric structures and differential equations. ... In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A → A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

The Lie derivatives are represented by vector fields, as infinitesimal generators of flows (active diffeomorphisms) on M. Looking at it the other way round, the group of diffeomorphisms of M has the associated Lie algebra structure, of Lie derivatives, in a way directly analogous to the Lie group theory. Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... An active transformation is one which actually changes the physical state of a system and makes sense even in the absence of a coordinate system whereas a passive transformation is merely a change in the coordinate system of no physical significance. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ...

Definition

The Lie derivative may be defined in several equivalent ways. In this section, to keep things simple, we begin by defining the Lie derivative acting on scalar functions and vector fields. The Lie derivative can also be defined to act on general tensors, as developed later in the article.

The Lie derivative of a function

One might start by defining the Lie derivative in terms of the differential of a function. Thus, given a function and a vector field X defined on M, one defines the Lie derivative of f at point as In mathematics, the word differential has various meanings: In calculus, a differential is an infinitesimal change in the value of a function. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

the usual derivative of f along the vector field X.

In fancier terms, this can be restated using the dual pairing between the tangent bundle and cotangent bundle of M as follows: In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ...

where df is the differential of f. That is, is the 1-form given by In mathematics, the word differential has various meanings: In calculus, a differential is an infinitesimal change in the value of a function. ... (Redirected from 1-form) A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

Here, the dx^a are the basis vectors for the cotangent bundle T ^* M. (The Einstein summation convention is implied in the formula.) Thus, the notation means that the inner product of the differential of f (at point p in M) is being taken with the vector field X (at point p). Writing X in the x^a coordinates, In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ... For other topics related to Einstein see Einstein (disambig) In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate equations or formulas. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ...

we have

which recovers the original definition of the Lie derivative of a function.

Alternately, one might start by showing that a smooth vector field X on M defines a family of curves on M. That is, one shows that for any point p in M there exists a curve γ(t) on M such that In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...

with p = γ(0). The existence of solutions to this first-order ordinary differential equation is given by the Picard-Lindelöf theorem (more generally, one says the existence of such curves is given by the Frobenius theorem). One then defines the Lie derivative as In mathematics, and particularly in analysis, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ... In mathematics, the Picard-Lindelöf theorem on existence and uniqueness of solutions of differential equations (Picard 1890, Lindelöf 1894) states that an initial value problem has exactly one solution if f is Lipschitz continuous in , continuous in as long as stays bounded. ... In mathematics, Frobenius theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an underdetermined system of first-order homogeneous linear partial differential equations. ...

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The Lie derivative of a vector field

The Lie derivative of a function has now been defined in several ways. In each case, the Lie derivative of a function agrees with the usual idea of differentiation along a vector field from multivariable calculus. The Lie derivative can be defined for vector fields by first defining the Lie bracket of a pair of vector fields X and Y, denoted [X,Y]. There are several approaches to defining the Lie bracket, all of which are equivalent. Regardless of the chosen definition, one then defines the Lie derivative of the vector field Y to be equal to the Lie bracket of X and Y, that is, Multivariable calculus is the extension of calculus in one variable to calculus in several variables: the functions which are differentiated and integrated involve several variables rather than one variable. ... A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

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The first definition of the Lie bracket uses the local coordinate expressions of the vector fields X and Y. Let x^a be coordinates on M. One starts by noting that the basis vectors for the tangent bundle can be written as , and so a vector field, expressed in terms of this selected set of basis vectors is written as See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space...

One defines the Lie bracket [X,Y] of a pair of vector fields as A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

The second definition is intrinsic in that it does not rely on the use of coordinates. Since a vector field can be identified with a first-order differential operator on functions, the Lie bracket of two vector fields can be defined as follows. If X and Y are two vector fields, then the Lie bracket of X and Y is also a vector field, denoted by [X,Y], defined by the equation:

[X,Y](f) = X(Y(f)) − Y(X(f)).

Using a local coordinate expression for X and Y, one can prove that this is equivalent to the previous definition of the Lie bracket.

Other equivalent definitions are:

The Lie derivative of differential forms

The Lie derivative can also be defined on differential forms. In this context, it 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 antiderivation or equivalently an interior product, after which the relationships fall out as a set of identities. A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... In mathematics, the exterior algebra (also known as the Grassmann algebra, after Hermann Grassmann) of a given vector space V over a field K is a certain unital associative algebra which contains V as a subspace. ...

Let M be a manifold and X a vector field on M. Let be a k+1-form. The interior product of X and ω is

Note that

and that i_X is a -antiderivation. That is, i_X is R-linear, and In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A → A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ...

for and η another differential form. Also, for a function , that is a real or complex-valued function on M, one has

i_fXω = fi_Xω

The relationship between exterior derivatives and Lie derivatives can then be summarized as follows. For an ordinary function f, the Lie derivative is just the contraction of the exterior derivative with the vector field X: In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

For a general differential form, the Lie derivative is likewise a contraction, taking into account the variation in X:

.

The derivative of products is distributed:

Properties

The Lie derivative has a number of properties. Let be the algebra of functions defined on the manifold M. Then Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...

is a derivation on the algebra . That is, is R-linear and In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A → A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ...

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Similarly, it is a derivation on where is the set of vector fields on M:

which is may also be written in the equivalent notation

where the tensor product symbol is used to emphasize the fact that the product of a function times a vector field is being taken over the entire manifold. In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...

Additional properties are consistent with that of the Lie bracket. Thus, for example, considered as a derivation on a vector field, A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

one finds the above to be just the Jacobi identity. Thus, one has the important result that the space of vector fields over M, equipped with the Lie bracket, forms a Lie algebra. In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...

The Lie derivative also has important properties when acting on differential forms. Let α and β be two differential forms on M, and let X and Y be two vector fields. Then

• 
• 
• where i denotes interior multiplication between vector fields and differential forms.

Lie derivative of tensor fields

More generally, if we have a differentiable tensor field T of rank (p,q) and a differentiable vector field Y (i.e. a differentiable section of the tangent bundle TM), then we can define the Lie derivative of T along Y. Let φ:M×R→M be the one-parameter semigroup of local diffeomorphisms of M induced by the vector flow of Y and denote φ_t(p) := φ(p, t). For each sufficiently small t, φ_t is a diffeomorphism from an neighborhood in M to another neighborhood in M, and φ₀ is the identity diffeomorphism. The Lie derivative of T is defined at a point p by In mathematics, the derivative of a function is one of the two central concepts of calculus. ... Note: This is a fairly abstract mathematical approach to tensors. ... Note: This is a fairly abstract mathematical approach to tensors. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space... In mathematics, the vector flow refers to a set of closely related concepts of the flow determined by a vector field. ... In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ...

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where (φ_t)_* is the pushforward along the diffeomorphism. In other words, if you have a tensor field T and an infinitesimal generator of a diffeomorphism given by a vector field Y, then is nothing other than the infinitesimal change in T under the infinitesimal diffeomorphism. This article needs to be cleaned up to conform to a higher standard of quality. ...

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 the directional derivative of the function. So if f is a real valued function on M, then

Axiom 2. The Lie derivative of a vector field is the Lie bracket. So if X is a vector field, one has

Axiom 3. The Lie derivative of a differential form is the anticommutator of the interior product with the exterior derivative. So if α is a differential form,

Axiom 4. The Lie derivative obeys the Leibniz rule. For any tensor fields S and T, we have

Explicitly, let T be a tensor field of type (p,q). Consider T to be a differentiable multilinear map of smooth sections α¹, α², …, α^q of the cotangent bundle T*M and of sections X₁, X₂, … X_p of the tangent bundle TM, written T(α¹, α², …, X₁, X₂, …) into R. Define the Lie derivative of T along Y by the formula In linear algebra, a multilinear map is a mathematical function of several vector variables that is linear in each variable. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space...

The analytic and algebraic definitions can be proven to be equivalent using the properties of the pushforward and the Leibniz rule for differentiation. At least two results in calculus are called Leibnizs rule or the Leibniz rule, in honor of Gottfried Leibniz. ...

Coordinate expressions

Let x^a be a system of coordinates. For a type (r,s) tensor field T, the Lie derivative along X is See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ...

here, the notation ∇ means taking the gradient in the x coordinate system. In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. ...

Alternatively, if we are using a torsion-free connection, then ∇ could also mean the covariant derivative. For a torsion-free connection, both definitions are equivalent. In differential geometry, the torsion tensor is one of the tensorial invariants of a connection on the tangent bundle. ... In differential geometry, a connection (also connexion) or covariant derivative is a way of specifying a derivative of a vector field along another vector field on a manifold. ... In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ...

Generalizations

Various generalizations of the Lie derivative play an important role in differential geometry.

Nijenhuis-Lie derivative

This article has defined the usual Lie derivative of a differential form along a vector field. One generalization, due to Albert Nijenhuis, allows one to define the Lie derivative of a differential form along any contravariant tensor field. In detail, if K is a contravariant tensor and α is a differential p-form, then it is possible define the interior product i_Kα of K and α. The Nijenhuis-Lie derivative is then the anticommutator of the interior product and the exterior derivative: Contravariant is a mathematical term with a precise definition in tensor analysis. ...

The Nijenhuis-Lie derivative enjoys many algebraic properties similar to those of the Lie derivative, with one notable exception: it is not a derivation in the usual sense.

See also

• Killing field
• Lie group
• Geodesic
• Covariant derivative
• Connection (mathematics)
• Frölicher-Nijenhuis bracket

In mathematics, a Killing vector field is a vector field on a Riemannian manifold that preserves the metric. ... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. Definition of geodesic depends on the type of curved space. If the space carries a natural metric then geodesics are defined to be (locally) the shortest path between points on the space. ... In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ... In differential geometry, a connection (also connexion) or covariant derivative is a way of specifying a derivative of a vector field along another vector field on a manifold. ... In mathematics, the Frölicher-Nijenhuis bracket is a bracket on vector-valued differential forms. ...

References

• Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin ISBN 3-540-4267-2 See section 1.6.
• Ralph Abraham and Jarrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See section 2.2.
• David Bleecker, Gauge Theory and Variational Principles, (1981), Addison-Wesley Publishing, ISBN 0-201-10096-7. See Chapter 0.