# What is Lie derivative: Definition and 60 Discussions

In differential geometry, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold.
Functions, tensor fields and forms can be differentiated with respect to a vector field. If T is a tensor field and X is a vector field, then the Lie derivative of T with respect to X is denoted

L

X

(
T
)

{\displaystyle {\mathcal {L}}_{X}(T)}
. The differential operator

T

L

X

(
T
)

{\displaystyle T\mapsto {\mathcal {L}}_{X}(T)}
is a derivation of the algebra of tensor fields of the underlying manifold.
The Lie derivative commutes with contraction and the exterior derivative on differential forms.
Although there are many concepts of taking a derivative in differential geometry, they all agree when the expression being differentiated is a function or scalar field. Thus in this case the word "Lie" is dropped, and one simply speaks of the derivative of a function.
The Lie derivative of a vector field Y with respect to another vector field X is known as the "Lie bracket" of X and Y, and is often denoted [X,Y] instead of

L

X

(
Y
)

{\displaystyle {\mathcal {L}}_{X}(Y)}
. The space of vector fields forms a Lie algebra with respect to this Lie bracket. The Lie derivative constitutes an infinite-dimensional Lie algebra representation of this Lie algebra, due to the identity

L

[
X
,
Y
]

T
=

L

X

L

Y

T

L

Y

L

X

T
,

{\displaystyle {\mathcal {L}}_{[X,Y]}T={\mathcal {L}}_{X}{\mathcal {L}}_{Y}T-{\mathcal {L}}_{Y}{\mathcal {L}}_{X}T,}
valid for any vector fields X and Y and any tensor field T.
Considering vector fields as infinitesimal generators of flows (i.e. one-dimensional groups of diffeomorphisms) on M, the Lie derivative is the differential of the representation of the diffeomorphism group on tensor fields, analogous to Lie algebra representations as infinitesimal representations associated to group representation in Lie group theory.
Generalisations exist for spinor fields, fibre bundles with connection and vector-valued differential forms.

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17. ### Component of Lie Derivative expression vector field

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25. ### A Definition of the Lie derivative

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26. ### I Lie derivative of a differential form

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27. ### Lie derivative of tensor field with respect to Lie bracket

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28. ### Lie Derivative of one-form: an identity

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32. ### Showing that the Lie derivative of a function is the directional deriv

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33. ### Lie derivative of covariant vector

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34. ### Understanding the Lie Derivative of Tensors: A Step-by-Step Approach

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35. ### Lie derivative of two left invariant vector fields

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46. ### Coordinate independence of Lie derivative

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47. ### Lie derivative versus covariant derivative

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48. ### Adjoint of the scalar Lie derivative?

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49. ### Can a Lie Derivative be Taken in the Direction of a Scalar Function?

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