Geodesic equation proof confusing me

[Superposed_Cat]

Hi all, I was looking through this proof and have no idea where the "u" comes from., any help apreciated.

http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++&bg=eedbbd&fg=000000&s=0

http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++&bg=eedbbd&fg=000000&s=0

http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++&bg=eedbbd&fg=000000&s=0

http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++&bg=eedbbd&fg=000000&s=0

http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++&bg=eedbbd&fg=000000&s=0

where

 

[Matterwave]

It looks like they have defined $u^m\equiv \frac{dx^m}{d\tau}$