Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

1 files changed, 32 insertions, 32 deletions

diff --git a/source/know/concept/screw-pinch/index.md b/source/know/concept/screw-pinch/index.md
index c7477b3..0a787dc 100644
--- a/source/know/concept/screw-pinch/index.md
+++ b/source/know/concept/screw-pinch/index.md
@@ -19,13 +19,13 @@ The general way of doing this is called a **screw pinch**.
 For simplicity, let the cylinder be infinitely long,
 so that it is natural to work in
 [cylindrical polar coordinates](/know/concept/cylindrical-polar-coordinates/)
-$(r, \theta, z)$.
+$$(r, \theta, z)$$.
 
 Using the framework of ideal [magnetohydrodynamics](/know/concept/magnetohydrodynamics/) (MHD),
 let us start by assuming that the fluid is stationary,
-and that the confining field $\vb{B}$ is fixed.
+and that the confining field $$\vb{B}$$ is fixed.
 From the (ideal) generalized Ohm's law, it then follows
-that the [electric field](/know/concept/electric-field/) $\vb{E} = 0$:
+that the [electric field](/know/concept/electric-field/) $$\vb{E} = 0$$:
 
 $$\begin{aligned}
     \vb{u}
@@ -41,10 +41,10 @@ $$\begin{aligned}
     = 0
 \end{aligned}$$
 
-To get the plasma's equilibrium state for a given $\vb{B}$,
+To get the plasma's equilibrium state for a given $$\vb{B}$$,
 we first solve [Ampère's law](/know/concept/maxwells-equations/)
-for the current density $\vb{J}$,
-and then the MHD momentum equation for the pressure $p$.
+for the current density $$\vb{J}$$,
+and then the MHD momentum equation for the pressure $$p$$.
 Symmetries should be used whenever possible to reduce these equations:
 
 $$\begin{aligned}
@@ -55,15 +55,15 @@ $$\begin{aligned}
     = \nabla p
 \end{aligned}$$
 
-Note that the latter implies that $\nabla p$ is always orthogonal to $\vb{J}$ and $\vb{B}$,
+Note that the latter implies that $$\nabla p$$ is always orthogonal to $$\vb{J}$$ and $$\vb{B}$$,
 meaning that the current density and magnetic field must follow
 surfaces of constant pressure.
 
 
 ## ϴ-pinch
 
-In a so-called **ϴ-pinch**, the confining field $\vb{B}$
-is parallel to the $z$-axis, and its magntiude $B_z$ may only depend on $r$.
+In a so-called **ϴ-pinch**, the confining field $$\vb{B}$$
+is parallel to the $$z$$-axis, and its magntiude $$B_z$$ may only depend on $$r$$.
 Concretely, we have:
 
 $$\begin{aligned}
@@ -71,13 +71,13 @@ $$\begin{aligned}
     = B_z(r) \: \vu{e}_z
 \end{aligned}$$
 
-Where $\vu{e}_z$ is the basis vector of the $z$-axis.
-This $\vb{B}$ confines the plasma thanks to
+Where $$\vu{e}_z$$ is the basis vector of the $$z$$-axis.
+This $$\vb{B}$$ confines the plasma thanks to
 the [Lorentz force](/know/concept/lorentz-force/),
 which makes charged particles gyrate around magnetic field lines.
 
-Using Ampère's law, we find that the resulting current density $\vb{J}$,
-expressed in $(r, \theta, z)$:
+Using Ampère's law, we find that the resulting current density $$\vb{J}$$,
+expressed in $$(r, \theta, z)$$:
 
 $$\begin{aligned}
     \vb{J}
@@ -91,15 +91,15 @@ $$\begin{aligned}
     = -\frac{1}{\mu_0} \pdv{B_z}{r} \: \vu{e}_\theta
 \end{aligned}$$
 
-Where we have used that only $B_z$ is nonzero,
-and that it only depends on $r$.
-This yields a circular current parallel to $\vu{e}_\theta$,
+Where we have used that only $$B_z$$ is nonzero,
+and that it only depends on $$r$$.
+This yields a circular current parallel to $$\vu{e}_\theta$$,
 hence the name *ϴ-pinch*.
 
-Next, we use the MHD momentum equation to find the pressure gradient $\nabla p$.
+Next, we use the MHD momentum equation to find the pressure gradient $$\nabla p$$.
 The cross product is easy to evaluate,
-since $\vb{B}$ is parallel to $\vu{e}_z$,
-and $\vb{J}$ is parallel to $\vu{e}_\theta$:
+since $$\vb{B}$$ is parallel to $$\vu{e}_z$$,
+and $$\vb{J}$$ is parallel to $$\vu{e}_\theta$$:
 
 $$\begin{aligned}
     \nabla p
@@ -109,9 +109,9 @@ $$\begin{aligned}
     = - \frac{1}{\mu_0} \pdv{B_z}{r} B_z \: \vu{e}_r
 \end{aligned}$$
 
-Consequently, $\nabla p$ is parallel to $\vu{e}_r$,
-and only depends on $r$ through $B_z$.
-Along the $r$-direction, the above equation can be rewritten
+Consequently, $$\nabla p$$ is parallel to $$\vu{e}_r$$,
+and only depends on $$r$$ through $$B_z$$.
+Along the $$r$$-direction, the above equation can be rewritten
 into the following equilibrium condition:
 
 $$\begin{aligned}
@@ -121,22 +121,22 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-In other words, the parenthesized expression does not depend on $r$.
+In other words, the parenthesized expression does not depend on $$r$$.
 
 
 ## Z-pinch
 
 Meanwhile, in a so-called **Z-pinch**,
-we create an $r$-dependent current $\vb{J}$ parallel to the $z$-axis:
+we create an $$r$$-dependent current $$\vb{J}$$ parallel to the $$z$$-axis:
 
 $$\begin{aligned}
     \vb{J}
     = J_z(r) \: \vu{e}_z
 \end{aligned}$$
 
-We can then deduce $\vb{B}$ from Ampère's law,
-using that only $J_z$ is nonzero,
-and that $\ipdv{B_r}{\theta} = 0$ due to circular symmetry:
+We can then deduce $$\vb{B}$$ from Ampère's law,
+using that only $$J_z$$ is nonzero,
+and that $$\ipdv{B_r}{\theta} = 0$$ due to circular symmetry:
 
 $$\begin{aligned}
     \vb{J}
@@ -150,11 +150,11 @@ $$\begin{aligned}
     = \frac{1}{\mu_0 r} \pdv{(r B_\theta)}{r} \: \vu{e}_z
 \end{aligned}$$
 
-Therefore, $\vb{J}$ induces a circular $\vb{B} = B_\theta(r) \: \vu{e}_\theta$,
+Therefore, $$\vb{J}$$ induces a circular $$\vb{B} = B_\theta(r) \: \vu{e}_\theta$$,
 which confines the plasma for the same reason as in the ϴ-pinch:
 the Lorentz force makes particles gyrate around magnetic field lines.
 
-Next, the resulting pressure gradient $\nabla p$ is found from the MHD momentum equation:
+Next, the resulting pressure gradient $$\nabla p$$ is found from the MHD momentum equation:
 
 $$\begin{aligned}
     \nabla p
@@ -164,8 +164,8 @@ $$\begin{aligned}
     = - \frac{1}{\mu_0 r} \pdv{(r B_\theta)}{r} B_\theta \: \vu{e}_r
 \end{aligned}$$
 
-Once again, $\nabla p$ is parallel to $\vu{e}_r$ and only depends on $r$.
-After rearranging, we thus arrive at the following equilibrium condition in the $r$-direction:
+Once again, $$\nabla p$$ is parallel to $$\vu{e}_r$$ and only depends on $$r$$.
+After rearranging, we thus arrive at the following equilibrium condition in the $$r$$-direction:
 
 $$\begin{aligned}
     \boxed{
@@ -179,7 +179,7 @@ $$\begin{aligned}
 
 Thanks to the linearity of electromagnetism,
 a ϴ-pinch and Z-pinch can be combined to create a **screw pinch**,
-where $\vb{J}$ and $\vb{B}$ both have nonzero $\theta$ and $z$-components.
+where $$\vb{J}$$ and $$\vb{B}$$ both have nonzero $$\theta$$ and $$z$$-components.
 By performing the above procedure again,
 the following equilibrium condition is obtained: