Definition talk:Regular Curve

A note on terminology
In cases like this, which hinge upon specific details of a particular case, it is unwise and potentially inflammatory to stamp an exposition as "wrong". --prime mover (talk) 12:57, 20 October 2022 (UTC)


 * With curve endpoints included we may need to consider a smooth manifold with a boundary. In the literature sometimes this information is mentioned in the parenthesis, i.e. "smooth manifold (with or without a boundary)", but sometimes it has to be understood from the context. I will tell more when I come back from work.--Julius (talk) 13:34, 20 October 2022 (UTC)

Definition of $\gamma'$
Is the definition of velocity $\map {\gamma '} t$ for curves on a manifold still missing in ?

This Definition:Velocity of Smooth Curve is only for curves in $\R^n$. --Usagiop (talk) 15:42, 10 July 2023 (UTC)


 * For personal reasons I copied mostly from Lee's book on Riemannian manifolds. In the same book, the topics for smooth manifold are covered not to full depth. I copied only the Euclidean definition. If instead we open Lee's book on smooth manifolds, then for velocity of a curve we have the following:


 * Let $\gamma : I \to M$.


 * Let $t_0 \in I$


 * Then the velocity of $\gamma$ is the map $\gamma' : I \to T_\gamma M$ such that:


 * $\ds \map {\gamma'} {t_0} = \map {\rd \gamma} {\valueat{\dfrac d {dt} }{t \mathop = t_0} }$


 * Furthermore, suppose $\tuple {U, \phi}$ is a smooth chart with coordinate functions $\tuple {x^i}$.


 * Also suppose $\map \gamma {t_0} \in U$.


 * Then we can write the coordinate representation of $\gamma$ as $\map \gamma t = \tuple {\map {\gamma^1} t, \map {\gamma^2} t, \ldots \map {\gamma^n} t}$


 * And the coordinate formula for velocity reads $\map {\gamma'} {t_0} = \map {\dfrac {d \gamma^i}{d t} } {t_0} \valueat {\dfrac {\partial}{\partial x^i} } {\map \gamma {t_0} }$ with Einstein summation implied.


 * At the moment I cannot make a proper page out from it. I would need to nit pick a few more things before I have a nice article. You can start building something. Maybe later in the evening I will put together something with a few red links.--Julius (talk) 13:27, 11 July 2023 (UTC)