More improvements to knowledge base

1 files changed, 7 insertions, 5 deletions

diff --git a/source/know/concept/langmuir-waves/index.md b/source/know/concept/langmuir-waves/index.md
index be47567..2dbce8f 100644
--- a/source/know/concept/langmuir-waves/index.md
+++ b/source/know/concept/langmuir-waves/index.md
@@ -22,7 +22,7 @@ tell us that:
 $$\begin{aligned}
     m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t}
     = q_e n_e \vb{E} - \nabla p_e
-    \qquad \quad
+    \qquad \qquad
     \pdv{n_e}{t} + \nabla \cdot (n_e \vb{u}_e) = 0
 \end{aligned}$$
 
@@ -50,7 +50,7 @@ $$\begin{aligned}
 
 Where the perturbations $$n_{e1}$$, $$\vb{u}_{e1}$$ and $$\vb{E}_1$$ are very small,
 and the equilibrium components $$n_{e0}$$, $$\vb{u}_{e0}$$ and $$\vb{E}_0$$
-by definition satisfy:
+are assumed to satisfy:
 
 $$\begin{aligned}
     \pdv{n_{e0}}{t} = 0
@@ -64,7 +64,7 @@ $$\begin{aligned}
     \vb{E}_0 = 0
 \end{aligned}$$
 
-We insert this decomposistion into the electron continuity equation,
+We insert this decomposition into the electron continuity equation,
 arguing that $$n_{e1} \vb{u}_{e1}$$ is small enough to neglect, leading to:
 
 $$\begin{aligned}
@@ -114,6 +114,7 @@ However, there are three unknowns $$n_{e1}$$, $$\vb{u}_{e1}$$ and $$\vb{E}_1$$,
 so one more equation is needed.
 
 
+
 ## Cold Langmuir waves
 
 We therefore turn to the electron momentum equation.
@@ -172,7 +173,8 @@ $$\begin{aligned}
 Note that this is a dispersion relation $$\omega(k) = \omega_p$$,
 but that $$\omega_p$$ does not contain $$k$$.
 This means that cold Langmuir waves do not propagate:
-the oscillation is "stationary".
+the oscillation is stationary.
+
 
 
 ## Warm Langmuir waves
@@ -181,7 +183,7 @@ Next, we generalize this result to nonzero $$T_e$$,
 in which case the pressure $$p_e$$ is involved:
 
 $$\begin{aligned}
-    m_e n_{e0} \pdv{}{\vb{u}{e1}}{t}
+    m_e n_{e0} \pdv{\vb{u}_{e1}}{t}
     = q_e n_{e0} \vb{E}_1 - \nabla p_e
 \end{aligned}$$