View source for Differentiation along the flow of a dynamical system

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An operator defined as follows. Let

$$ \tag{* }
\dot{x}  =  f ( x)
$$

be an [[Autonomous system|autonomous system]], let  $  x \in \mathbf R  ^ {n} $, 
let  $  f = ( f _ {1} \dots f _ {n} ) $, 
and let  $  f _ {j} :  G \rightarrow \mathbf R $
be smooth mappings, where  $  G $
is a domain in  $  \mathbf R  ^ {n} $. 
Let a smooth mapping  $  \phi :  G \rightarrow \mathbf R $
be given. The derivative  $  \theta _ {f} \phi $
along the flow of the system (*) of  $  \phi $
at a point  $  x  ^ {0} \in G $
is defined by

$$ 
\left . ( \theta _ {f} \phi ) x  ^ {0}  =  \sum _ {j = 1 } ^ { n } 

\frac{\partial  \phi ( x  ^ {0} ) }{\partial  x _ {j} }
 f _ {j} ( x  ^ {0} )
 =  
\frac{d}{dt}
 ( \phi ( x ( t , x  ^ {0} ) ) ) \right | _ {t = t  ^ {0}  } ,
$$

where  $  x ( t , x  ^ {0} ) $
is a solution of the system (*) such that  $  x ( t  ^ {0} , x  ^ {0} ) = x  ^ {0} $. 
The operator  $  \theta _ {f} $
displays the following properties: 1) linearity in  $  \phi $; 
and 2)  $  \theta _ {f} ( \phi _ {1} \phi _ {2} ) = \phi _ {1} \theta _ {f} \phi _ {2} + \phi _ {2} \theta _ {f} \phi _ {1} $. 
The function  $  ( \theta _ {f} \phi ) ( x) $
coincides with the derivative of  $  \phi $
with respect to the vector field  $  f $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,   "Ordinary differential equations" , Addison-Wesley  (1962)  (Translated from Russian)</TD></TR></table>

====Comments====
In terms of the canonical basis  $  ( \partial  / {\partial  x _ {1} } \dots \partial  / {\partial  x _ {n} } ) $
of the tangent space  $  T _ {x} \mathbf R  ^ {n} $
at a point  $  x $, 
the vector field  $  f $
is written

$$ 
\sum f _ {j} ( x) 
\frac \partial {\partial  x _ {j} }
 .
$$

This first-order differential operator defines a derivation of the ring (cf. [[Derivation in a ring|Derivation in a ring]]) of smooth functions  $  C  ^  \infty  ( \mathbf R  ^ {n} ) $
into itself. Moreover, this sets up a bijective correspondence between vector fields on  $  \mathbf R  ^ {n} $
and derivations on  $  C  ^  \infty  ( \mathbf R  ^ {n} ) $. 
Using local coordinates this extends to the case of smooth differentiable manifolds. And indeed it is quite customary to define vector fields on a manifold  $  M $
as derivations of  $  C  ^  \infty  ( M) $
and to observe subsequently that the notion corresponds to a section of the tangent bundle. In this setting a tangent vector at  $  x \in M $
can be defined as a derivation on the local algebra of germs of smooth functions at  $  x $
on  $  M $. 
Thus, differentiation along the flow of a dynamical system given by the vector field  $  f $
simply means applying the derivation on  $  C  ^  \infty  ( M) $
given by  $  f $.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Y. Choquet-Bruhat,   C. DeWitt-Morette,   M. Dillard-Bleick,   "Analysis, manifolds and physics" , North-Holland  (1977)  pp. Sect. III.B  (Translated from French)</TD></TR></table>