deletions | additions       diff --git a/paragraph_2_Mimic_the_derivation__.tex b/paragraph_2_Mimic_the_derivation__.tex  index 77337a6..e59fbdf 100644  --- a/paragraph_2_Mimic_the_derivation__.tex  +++ b/paragraph_2_Mimic_the_derivation__.tex 
...
U - c,U,U,U + c where $U = \varepsilon_{1}u + \varepsilon_{2}u$ and c is the speed of sound $c = \sqrt{\frac{\gamma p}{\rho} }$. It can be verified that there are 4 independent eigenvectors.  \paragraph{3. Show that if u satisfy }  f(u)=\left\{ \[  \left\{  \begin{array}{ll}  u_{t} + f(u)_{x} = 0\\  u(x,0) = u_{0}(x)\\ 
...
\end{array}  \right   \]