Minor fixes, rename "Ion Sound Wave" and "Ito Process"

9 files changed, 25 insertions, 22 deletions

diff --git a/config.toml b/config.toml
index d5a8d6d..3463bda 100644
--- a/config.toml
+++ b/config.toml
@@ -3,3 +3,6 @@ languageCode = "en-us"
 title = "Prefetch"
 
 disableKinds = ["taxonomy", "term"]
+
+[security.exec]
+allow = ['^dart-sass-embedded$', '^go$', '^npx$', '^postcss$', '^pandoc$']
diff --git a/content/know/concept/electric-dipole-approximation/index.pdc b/content/know/concept/electric-dipole-approximation/index.pdc
index 96b4fed..265babf 100644
--- a/content/know/concept/electric-dipole-approximation/index.pdc
+++ b/content/know/concept/electric-dipole-approximation/index.pdc
@@ -46,7 +46,7 @@ In an electromagnetic wave,
 $\vec{A}$ is oscillating sinusoidally in time and space as follows:
 
 $$\begin{aligned}
-    \vec{A}(\vec{r}, t) = \vec{A}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t)
+    \vec{A}(\vec{r}, t) = - i \vec{A}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t)
 \end{aligned}$$
 
 The corresponding perturbative
@@ -59,7 +59,7 @@ $$\begin{aligned}
     = \vec{E}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t)
 \end{aligned}$$
 
-Where $\vec{E}_0 = i \omega \vec{A}_0$.
+Where $\vec{E}_0 = \omega \vec{A}_0$.
 Let us restrict ourselves to visible light,
 whose wavelength $2 \pi / k \approx 10^{-6} \:\mathrm{m}$.
 Meanwhile, an atomic orbital is on the order of $10^{-10} \:\mathrm{m}$,
@@ -82,7 +82,7 @@ and the electron quantum-mechanically.
 Next, we want to convert $\hat{H}_1$
 to use the electric field $\vec{E}$ instead of the potential $\vec{A}$.
 To do so, we rewrite the momemtum $\vec{P} = m \: \dv*{\vec{r}}{t}$
-and evaluate this in the [Heisenberg picture](/know/concept/heisenberg-picture/):
+and evaluate this in the [interaction picture](/know/concept/interaction-picture/):
 
 $$\begin{aligned}
     \matrixel{2}{\dv*{\vec{r}}{t}}{1}
@@ -95,15 +95,15 @@ $$\begin{aligned}
 
 Therefore, $\vec{P} / m = i \omega_0 \vec{r}$,
 where $\omega_0 \equiv (E_2 - E_1) / \hbar$ is the resonance frequency of the transition,
-close to which we assume that $\vec{A}$ and $\vec{E}$ are oscillating.
+close to which we assume that $\vec{A}$ and $\vec{E}$ are oscillating, i.e. $\omega \approx \omega_0$.
 We thus get:
 
 $$\begin{aligned}
     \hat{H}_1(t)
     &= - \frac{q}{m} \vec{P} \cdot \vec{A}
-    = - i q \omega_0 \vec{r} \cdot \vec{A}_0 \exp\!(- i \omega t)
+    = - (- i i) q \omega_0 \vec{r} \cdot \vec{A}_0 \exp\!(- i \omega t)
     \\
-    &= - q \vec{r} \cdot \vec{E}_0 \exp\!(- i \omega t)
+    &\approx - q \vec{r} \cdot \vec{E}_0 \exp\!(- i \omega t)
     = - \vec{d} \cdot \vec{E}_0 \exp\!(- i \omega t)
 \end{aligned}$$
 
diff --git a/content/know/concept/guiding-center-theory/index.pdc b/content/know/concept/guiding-center-theory/index.pdc
index 12abac0..dee10ba 100644
--- a/content/know/concept/guiding-center-theory/index.pdc
+++ b/content/know/concept/guiding-center-theory/index.pdc
@@ -174,7 +174,7 @@ $$\begin{aligned}
     m \dv{\vb{u}_{gc}}{t}
     = q \big( \vb{E} + \vb{u}_{gc} \cross \vb{B}
     + \vb{u}_{gc} \cross (\vb{x}_L \cdot \nabla) \vb{B}
-    + + \vb{u}_L \cross (\vb{x}_L \cdot \nabla) \vb{B} \big)
+    + \vb{u}_L \cross (\vb{x}_L \cdot \nabla) \vb{B} \big)
 \end{aligned}$$
 
 We approximate this by taking the average over a single gyration,
diff --git a/content/know/concept/hellmann-feynman-theorem/index.pdc b/content/know/concept/hellmann-feynman-theorem/index.pdc
index 3b88cd8..1d2fe82 100644
--- a/content/know/concept/hellmann-feynman-theorem/index.pdc
+++ b/content/know/concept/hellmann-feynman-theorem/index.pdc
@@ -74,7 +74,7 @@ to minimize energies with respect to $\lambda$:
 $$\begin{aligned}
     \boxed{
         \nabla_\lambda E_n
-        = \matrixel{\psi_m}{\nabla_\lambda \hat{H}}{\psi_n}
+        = \matrixel{\psi_n}{\nabla_\lambda \hat{H}}{\psi_n}
     }
 \end{aligned}$$
 
diff --git a/content/know/concept/ion-sound-waves/index.pdc b/content/know/concept/ion-sound-wave/index.pdc
index c56188b..5cba1d0 100644
--- a/content/know/concept/ion-sound-waves/index.pdc
+++ b/content/know/concept/ion-sound-wave/index.pdc
@@ -1,5 +1,5 @@
 ---
-title: "Ion sound waves"
+title: "Ion sound wave"
 firstLetter: "I"
 publishDate: 2021-10-31
 categories:
@@ -11,7 +11,7 @@ draft: false
 markup: pandoc
 ---
 
-# Ion sound waves
+# Ion sound wave
 
 In a plasma, electromagnetic interactions allow
 compressional longitudinal waves to propagate
diff --git a/content/know/concept/ito-calculus/index.pdc b/content/know/concept/ito-process/index.pdc
index 3d4dd67..d27a2fb 100644
--- a/content/know/concept/ito-calculus/index.pdc
+++ b/content/know/concept/ito-process/index.pdc
@@ -1,5 +1,5 @@
 ---
-title: "Itō calculus"
+title: "Itō process"
 firstLetter: "I"
 publishDate: 2021-11-06
 categories:
@@ -11,7 +11,7 @@ draft: false
 markup: pandoc
 ---
 
-# Itō calculus
+# Itō process
 
 Given two [stochastic processes](/know/concept/stochastic-process/)
 $F_t$ and $G_t$, consider the following random variable $X_t$,
@@ -197,7 +197,7 @@ This is done by defining $Y_t \equiv h(X_t)$, with $h(x)$ given by:
 $$\begin{aligned}
     \boxed{
         h(x)
-        = \int_{x_0}^x \exp\!\bigg( \!-\!\! \int_{x_1}^x \frac{2 f(y)}{g^2(y)} \dd{y} \bigg)
+        = \int_{x_0}^x \exp\!\bigg( \!-\!\! \int_{x_1}^y \frac{2 f(z)}{g^2(z)} \dd{z} \bigg) \dd{y}
     }
 \end{aligned}$$
 
diff --git a/content/know/concept/langmuir-waves/index.pdc b/content/know/concept/langmuir-waves/index.pdc
index cd9b449..3c3f1d2 100644
--- a/content/know/concept/langmuir-waves/index.pdc
+++ b/content/know/concept/langmuir-waves/index.pdc
@@ -34,7 +34,7 @@ We also need [Gauss' law](/know/concept/maxwells-equations/):
 
 $$\begin{aligned}
     \varepsilon_0 \nabla \cdot \vb{E}
-    = q_e (n_i - n_e)
+    = q_e (n_e - n_i)
 \end{aligned}$$
 
 We split $n_e$, $\vb{u}_e$ and $\vb{E}$ into a base component
@@ -85,10 +85,10 @@ and use the plasma's quasi-neutrality $n_i = n_{e0}$ to get:
 
 $$\begin{aligned}
     \varepsilon_0 \nabla \cdot \big( \vb{E}_0 \!+\! \vb{E}_1 \big)
-    = q_e (n_i - n_{e0} \!-\! n_{e1} )
+    = q_e (n_{e0} + n_{e1} - n_i)
     \quad \implies \quad
     \varepsilon_0 \nabla \cdot \vb{E}_1
-    = - q_e n_{e1}
+    = q_e n_{e1}
 \end{aligned}$$
 
 Since we are looking for linear waves,
@@ -110,7 +110,7 @@ Inserting this into the continuity equation and Gauss' law yields, respectively:
 $$\begin{aligned}
     - i \omega n_{e1} = - i n_{e0} \vb{k} \cdot \vb{u}_{e1}
     \qquad \quad
-    i \varepsilon_0 \vb{k} \cdot \vb{E}_1 = q_e n_{e1}
+    -\! i \varepsilon_0 \vb{k} \cdot \vb{E}_1 = q_e n_{e1}
 \end{aligned}$$
 
 However, there are three unknowns $n_{e1}$, $\vb{u}_{e1}$ and $\vb{E}_1$,
diff --git a/content/know/concept/magnetohydrodynamics/index.pdc b/content/know/concept/magnetohydrodynamics/index.pdc
index 6672a74..89d23db 100644
--- a/content/know/concept/magnetohydrodynamics/index.pdc
+++ b/content/know/concept/magnetohydrodynamics/index.pdc
@@ -333,7 +333,7 @@ This term can be dropped in any of the following cases:
 
 $$\begin{gathered}
     1
-    \gg \frac{\big| \vb{J} \cross \vb{B} / q_e n_e \big|}{\big| \vb{U} \cross \vb{B} \big|}
+    \gg \frac{\big| \vb{J} \cross \vb{B} / q_e n_e \big|}{\big| \vb{u} \cross \vb{B} \big|}
     \sim \frac{\rho v_\mathrm{char} / \tau_\mathrm{char}}{v_\mathrm{char} |\vb{B}| q_i n_i}
     \approx \frac{m_i n_i}{|\vb{B}| q_i n_i \tau_\mathrm{char}}
     = \frac{1}{\omega_{ci} \tau_\mathrm{char}}
@@ -366,7 +366,7 @@ where we have used Ampère's law to find $\vb{J} \sim \vb{B} / \mu_0 \lambda_\ma
 
 $$\begin{aligned}
     1
-    \ll \frac{\big| \vb{U} \cross \vb{B} \big|}{\big| \eta \vb{J} \big|}
+    \ll \frac{\big| \vb{u} \cross \vb{B} \big|}{\big| \eta \vb{J} \big|}
     \sim \frac{v_\mathrm{char} |\vb{B}|}{\eta \vb{J}}
     \sim \frac{v_\mathrm{char} |\vb{B}|}{\eta |\vb{B}| / \mu_0 \lambda_\mathrm{char}}
     = \mathrm{R_m}
diff --git a/content/know/concept/time-dependent-perturbation-theory/index.pdc b/content/know/concept/time-dependent-perturbation-theory/index.pdc
index 1fbd9ce..76ab684 100644
--- a/content/know/concept/time-dependent-perturbation-theory/index.pdc
+++ b/content/know/concept/time-dependent-perturbation-theory/index.pdc
@@ -59,13 +59,13 @@ We then take the inner product with an arbitrary stationary basis state $\ket{m}
 
 $$\begin{aligned}
     0
-    &= \sum_{n} \Big( \lambda c_n \matrixel{m}{\hat{H}_1}{n} - i \hbar \frac{d c_n}{dt} \braket{m}{n} \Big) \exp\!(- i E_n t / \hbar)
+    &= \sum_{n} \Big( \lambda c_n \matrixel{m}{\hat{H}_1}{n} - i \hbar \dv{c_n}{t} \braket{m}{n} \Big) \exp\!(- i E_n t / \hbar)
 \end{aligned}$$
 
 Thanks to orthonormality, this removes the latter term from the summation:
 
 $$\begin{aligned}
-    i \hbar \frac{d c_m}{dt} \exp\!(- i E_m t / \hbar)
+    i \hbar \dv{c_m}{t} \exp\!(- i E_m t / \hbar)
     &= \lambda \sum_{n} c_n \matrixel{m}{\hat{H}_1}{n} \exp\!(- i E_n t / \hbar)
 \end{aligned}$$
 
@@ -74,7 +74,7 @@ $\omega_{mn} \equiv (E_m - E_n) / \hbar$ to get:
 
 $$\begin{aligned}
     \boxed{
-        i \hbar \frac{d c_m}{dt}
+        i \hbar \dv{c_m}{t}
         = \lambda \sum_{n} c_n(t) \matrixel{m}{\hat{H}_1(t)}{n} \exp\!(i \omega_{mn} t)
     }
 \end{aligned}$$