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A natural operation on a [[Differentiable manifold|differentiable manifold]]  $  M $
that associates with a differentiable vector field  $  X $
and a differentiable geometric object  $  Q $
on  $  M $(
cf. [[Geometric objects, theory of|Geometric objects, theory of]]) a new geometric object  $  {\mathcal L} _ {X} Q $, 
which describes the rate of change of  $  Q $
with respect to the one-parameter (local) transformation group  $  \phi _ {t} $
of  $  M $
generated by  $  X $. 
The geometric object  $  {\mathcal L} _ {X} Q $
is called the Lie derivative of the geometric object  $  Q $
with respect to  $  X $(
cf. also [[Lie derivative|Lie derivative]]). Here it is assumed that transformations of  $  M $
induce transformations in the space of objects  $  Q $
in a natural way.

In the special case when  $  Q $
is a vector-valued function on  $  M $, 
its Lie derivative  $  {\mathcal L} _ {X} Q $
coincides with the derivative  $  \partial  _ {X} Q $
of the function  $  Q $
in the direction of the vector field  $  X $
and is given by the formula

$$ 
\left . ( {\mathcal L} _ {X} Q ) ( x)  = \ 

\frac{d}{dt}
 Q \circ \phi _ {t} ( x)
\right | _ {t=} 0 ,\  x \in M ,
$$

where  $  \phi _ {t} $
is the one-parameter local transformation group on  $  M $
generated by  $  X $, 
or, in the local coordinates  $  x  ^ {i} $, 
by the formula

$$ 
{\mathcal L} _ {X} Q ( x  ^ {i} )  = \ 
\sum _ { j } X  ^ {j}

\frac \partial {\partial  x  ^ {j} }
 Q ( x  ^ {i} ) ,
$$

where

$$ 
X  =  \sum _ { j } X  ^ {j} ( x)

\frac \partial {\partial  x  ^ {j} }
 .
$$

In the general case the definition of Lie differentiation consists in the following. Let  $  W $
be a  $   \mathop{\rm GL}  ^ {k} ( n) $-
space, that is, a manifold with a fixed action of the general differential group  $   \mathop{\rm GL}  ^ {k} ( n) $
of order  $  k $(
the group of  $  k $-
jets at the origin of diffeomorphisms  $  \phi :  \mathbf R  ^ {n} \rightarrow \mathbf R  ^ {n} $, 
$  \phi ( 0) = 0 $). 
Let  $  Q :  P  ^ {k} M \rightarrow W $
be a geometric object of order  $  k $
and type  $  W $
on an  $  n $-
dimensional manifold  $  M $, 
regarded as a  $   \mathop{\rm GL}  ^ {k} ( n) $-
equivariant mapping of the principal  $   \mathop{\rm GL}  ^ {k} ( n) $-
bundle of coframes  $  P  ^ {k} M $
of order  $  k $
on  $  M $
into  $  W $. 
The one-parameter local transformation group  $  \phi _ {t} $
on  $  M $
generated by a vector field  $  X $
on  $  M $
induces a one-parameter local transformation group  $  \phi _ {t}  ^ {(} k) $
on the manifold of coframes  $  P  ^ {k} M $. 
Its velocity field

$$ 
X  ^ {(} k)  =  \left . 
\frac{d}{dt}

\phi _ {t}  ^ {(} k) \right | _ {t=} 0
$$

is called the complete lift of  $  X $
to  $  P  ^ {k} M $. 
The Lie derivative of a geometric object  $  Q $
of type  $  W $
with respect to a vector field  $  X $
on  $  M $
is defined as the geometric object  $  {\mathcal L} _ {X} Q $
of type  $  TW $(
where  $  TW $
is the tangent bundle of  $  W $, 
regarded in a natural way as a  $   \mathop{\rm GL}  ^ {k} ( n) $-
space), given by the formula

$$ 
{\mathcal L} _ {X} Q  =  \left . 
\frac{d}{dt}

Q \circ \phi _ {t}  ^ {(} k) \right | _ {t=} 0 .
$$

The value of the Lie derivative  $  {\mathcal L} _ {X} Q $
at a point  $  p _ {k} \in P  ^ {k} M $
depends only on the  $  1 $-
jet of  $  Q $
at  $  p _ {k} $, 
and does so linearly, and on the value of  $  X  ^ {(} k) $
at this point (or, equivalently, on the  $  k $-
jet of  $  X $
at the corresponding point  $  x \in M $).

If the geometric object  $  Q $
is linear, that is, the corresponding  $   \mathop{\rm GL}  ^ {k} ( n) $-
space  $  W $
is a vector space with linear action of  $   \mathop{\rm GL}  ^ {k} ( n) $, 
then the tangent manifold  $  TW $
can in a natural way be identified with the direct product  $  W \times W $, 
and so the Lie derivative

$$ 
{\mathcal L} _ {X} Q :  P  ^ {k} M  \rightarrow  T W  =  W \times W
$$

can be regarded as a pair of geometric objects of type  $  W $. 
The first of these is  $  Q $
itself, and the second, which is usually identified with the Lie derivative of  $  Q $, 
is equal to the derivative  $  \partial  _ {X  ^ {(}  k) } Q $
of  $  Q $
in the direction of the vector field  $  X  ^ {(} k) $:

$$ 
{\mathcal L} _ {X} Q  =  ( Q , \partial  _ {X  ^ {(}  k) } Q ) .
$$

Thus, the Lie derivative of a linear geometric object can be regarded as a geometric object of the same type as  $  Q $.

Local coordinates  $  x  ^ {i} $
in the manifold  $  M $
determine local coordinates  $  x  ^ {i} , y _ {j}  ^ {i} $
in the manifold  $  P  ^ {1} M $
of coframes of order 1: for  $  \theta \in P  ^ {1} M $
one has

$$ 
\theta  =  \sum _ { j } y _ {j}  ^ {i}  d x  ^ {j} .
$$

In these coordinates the Lie derivative of any geometric object  $  Q = Q ( x  ^ {i} , y _ {j}  ^ {i)} $
of order 1 (for example, a tensor field) in the direction of the vector field

$$ 
X  =  \sum _ { j } X  ^ {j} 
\frac \partial {\partial  x  ^ {j} }

$$

is given by the formula

$$ 
( {\mathcal L} _ {X} Q ) ( x  ^ {i} , y _ {j}  ^ {i} )  = \ 
\sum _ { j } 
\frac \partial {\partial  x  ^ {i} }

Q - \sum _ { i,j,l }
y _ {l}  ^ {i} X _ {j}  ^ {l} 
\frac \partial {\partial  y _ {j}  ^ {i} }
 Q ,
$$

where

$$ 
X _ {j}  ^ {l}  =  
\frac \partial {\partial  x  ^ {j} }
 X  ^ {l} .
$$

A similar formula holds for the Lie derivative of a geometric object of arbitrary order.

The Lie derivative  $  {\mathcal L} _ {X} $
in the space of differential forms on a manifold  $  M $
can be expressed in terms of the operator of exterior differentiation  $  d $
and the operator of interior multiplication  $  i _ {X} $(
defined as the contraction of a vector field with a differential form) by means of the following homotopy formula:

$$ 
{\mathcal L} _ {X}  =  d \circ i _ {X} + i _ {X} \circ d .
$$

Conversely, the operator of exterior differentiation  $  d $, 
acting on a  $  p $-
form  $  \omega $, 
can be expressed in terms of the Lie derivative by the formula

$$ 
d \omega ( X _ {1} \dots X _ {p+} 1 ) =
$$

$$ 
= \ 
\sum _ { i= } 1 ^ { p+ }  1 (- 1)  ^ {i+} 1 {\mathcal L} _ {X _ {i}  } \omega
( X _ {1} \dots \widehat{X}  _ {i} \dots X _ {p+} 1 ) +
$$

$$ 
+ 
\sum _ {i < j } (- 1)  ^ {i+} j \omega ( {\mathcal L} _ {X _ {i}  } X _ {j} , X _ {1} \dots \widehat{X}  _ {i} \dots \widehat{X}  _ {j} \dots X _ {p+} 1 ) ,
$$

where  $   \widehat{ {}}   $
means that the corresponding symbol must be omitted, and the  $  X _ {1} \dots X _ {p+} 1 $
are vector fields.

In contrast to [[Covariant differentiation|covariant differentiation]], which requires the introduction of a connection, the operation of Lie differentiation is determined by the structure of the differentiable manifold, and the Lie derivative of a geometric object  $  Q $
in the direction of a vector field  $  X $
is a concomitant of the geometric objects  $  X $
and  $  Q $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Slebodziński,   "Sur les équations canonique de Hamilton"  ''Bull. Cl. Sci. Acad. Roy. Belgique'' , '''17'''  (1931)  pp. 864–870</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.L. Laptev,   "Lie differentiation"  ''Progress in Math.'' , '''6'''  (1970)  pp. 229–269  ''Itogi. Nauk. Algebra Topol. Geom. 1965''  (1967)  pp. 429–465</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. Yano,   "The theory of Lie derivatives and its applications" , North-Holland  (1957)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Sternberg,   "Lectures on differential geometry" , Prentice-Hall  (1964)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.V. Vagner,   "Theory of geometric objects and theory of finite and infinite continuous transformation groups"  ''Dokl. Akad. Nauk SSSR'' , '''46'''  (1945)  pp. 347–349  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B.L. Laptev,   "Lie derivative in a space of supporting elements"  ''Trudy Sem. Vektor. Tenzor. Anal.'' , '''10'''  (1956)  pp. 227–248  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  L.E. Evtushik,   "The Lie derivative and differential field equations of a geometric object"  ''Soviet Math. Dokl.'' , '''1'''  (1960)  pp. 687–690  ''Dokl. Akad. Nauk SSSR'' , '''132'''  (1960)  pp. 998–1001</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  R.S. Palais,   "A definition of the exterior derivative in terms of Lie derivatives"  ''Proc. Amer. Math. Soc.'' , '''5'''  (1954)  pp. 902–908</TD></TR></table>