deletions | additions       diff --git a/Theory.tex b/Theory.tex  index 2f1c034..f4c87d4 100644  --- a/Theory.tex  +++ b/Theory.tex 
...
\end{tiny}  One of the remarkable aspects of ducted whistler waves is that the phase velocity travels at a separate angle to the group velocity and direction of propogation of the wave. The group velocity is given by  $\vec{V}_{gr}=\frac{\mathrm\partial\omega}{\mathrm \partial\vec{k}}$ and the phase velocity is given by   $\V_{ph}=\frac{\omega}{k}$