In Warner's book on page 36 a curve $\gamma:(a,b)\rightarrow M$ is defined to be an integral curve iff
$d\gamma(\frac{d}{dr}|_t)=X(\gamma(t))$
Could anyone explain to me the left side in detail and break it into coordinates? Here $X$ is a vector field on $M$.
A curve $\sigma$ is an integral in $M$ if $\dot{\sigma(t)}=X(\sigma(t)) \forall t \in \text{domain of $\sigma$ }$.