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Consider ##X## and ##Y## two vector fields on ##M ##. Fix ##x## a point in ##M## , and consider the integral

curve of ##X## passing through ##x## . This integral curve is given by the local flow of ##X## , denoted

##\phi _ { t } ( p ) .##

Now consider $$t \mapsto a _ { t } \left( \phi _ { t } ( x ) \right) \equiv b \left( t , \phi _ { t } ( x ) \right)$$

where ##a _ { t } \left( \phi _ { t } ( x ) \right) = \frac { \partial \phi _ { - t } ^ { i } } { \partial x ^ { j } } \left( \phi _ { t } ( x ) \right)##.

We will denote ##\left(\partial \phi _ { t } ^ { i }\right)^{-1}=\partial \phi _{ - t } ^ { i }##.

The time derivative of ##a _ { t } \left( \phi _ { t } ( x ) \right)##is thus calculated by applying the chain rule. The following is the solution: $$\frac { d } { d t } \left( a\ _ { t } \left( \phi _ { t } ( x ) \right) \right) = \dot { a } _ { t } \left( \phi _ { t } ( x ) \right) + \left( \partial _ { k } a _ { t } \right) \left( \phi _ { t } ( x ) \right) \dot { \phi } _ { t } ^ { k } ( x )$$

I don't understand how to get there, though, so I would greatly appreciate help!

curve of ##X## passing through ##x## . This integral curve is given by the local flow of ##X## , denoted

##\phi _ { t } ( p ) .##

Now consider $$t \mapsto a _ { t } \left( \phi _ { t } ( x ) \right) \equiv b \left( t , \phi _ { t } ( x ) \right)$$

where ##a _ { t } \left( \phi _ { t } ( x ) \right) = \frac { \partial \phi _ { - t } ^ { i } } { \partial x ^ { j } } \left( \phi _ { t } ( x ) \right)##.

We will denote ##\left(\partial \phi _ { t } ^ { i }\right)^{-1}=\partial \phi _{ - t } ^ { i }##.

The time derivative of ##a _ { t } \left( \phi _ { t } ( x ) \right)##is thus calculated by applying the chain rule. The following is the solution: $$\frac { d } { d t } \left( a\ _ { t } \left( \phi _ { t } ( x ) \right) \right) = \dot { a } _ { t } \left( \phi _ { t } ( x ) \right) + \left( \partial _ { k } a _ { t } \right) \left( \phi _ { t } ( x ) \right) \dot { \phi } _ { t } ^ { k } ( x )$$

I don't understand how to get there, though, so I would greatly appreciate help!

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