# A Lie derivative of vector field defined through integral curv

#### Emil_M

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!

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#### Matterwave

Gold Member
I think this is a pure math question and belongs in the differential geometry forum. :)

#### Emil_M

I think this is a pure math question and belongs in the differential geometry forum. :)

Edit: since crossposting is banned, how do I delete this post?

#### Ibix

Edit: since crossposting is banned, how do I delete this post?
Don't. Just report your original post (menu at the middle of the bottom of the post) and ask for it to be moved to the differential geometry forum.

#### Emil_M

Don't. Just report your original post and ask for it to be moved to the differential geometry forum.
Thanks!

"Lie derivative of vector field defined through integral curv"

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