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

1 files changed, 13 insertions, 13 deletions

diff --git a/source/know/concept/young-dupre-relation/index.md b/source/know/concept/young-dupre-relation/index.md
index 686d2a8..ed41aee 100644
--- a/source/know/concept/young-dupre-relation/index.md
+++ b/source/know/concept/young-dupre-relation/index.md
@@ -12,7 +12,7 @@ layout: "concept"
 
 In fluid mechanics, the **Young-Dupré relation** relates the contact
 angle of a droplet at rest on a surface to the surface tensions of the interfaces.
-Let $\alpha_{gl}$, $\alpha_{sl}$ and $\alpha_{sg}$ respectively be
+Let $$\alpha_{gl}$$, $$\alpha_{sl}$$ and $$\alpha_{sg}$$ respectively be
 the energy costs of the liquid-gas, solid-liquid and solid-gas interfaces:
 
 $$\begin{aligned}
@@ -28,8 +28,8 @@ when you account for the surface tension force pulling along each interface.
 
 A more general derivation is possible by using the
 [calculus of variations](/know/concept/calculus-of-variations/).
-In 2D, the upper surface of the droplet is denoted by $y(x)$.
-Consider the following Lagrangian $\mathcal{L}$,
+In 2D, the upper surface of the droplet is denoted by $$y(x)$$.
+Consider the following Lagrangian $$\mathcal{L}$$,
 with the two first terms respectively being the energy costs
 of the top and bottom surfaces:
 
@@ -39,24 +39,24 @@ $$\begin{aligned}
 \end{aligned}$$
 
 And the last term comes from the constraint
-that the volume $V$ of the droplet must be constant:
+that the volume $$V$$ of the droplet must be constant:
 
 $$\begin{aligned}
     V = \int_0^L y \dd{x}
 \end{aligned}$$
 
 The total energy to be minimized is thus given by the following functional,
-where the endpoints of the droplet are $x = 0$ and $x = L$:
+where the endpoints of the droplet are $$x = 0$$ and $$x = L$$:
 
 $$\begin{aligned}
     E[y(x)]
     = \int_0^L \Big( \alpha_{gl} \sqrt{1 + (y')^2} + (\alpha_{sl} - \alpha_{sg}) + \lambda y \Big) \dd{x}
 \end{aligned}$$
 
-In this optimization problem, the endpoint $L$ is a free parameter,
-i.e. the $L$-value of the optimum is unknown and must be found.
-In such cases, the optimum $y(x)$ needs to satisfy the so-called *transversality condition*
-at the variable endpoint, in this case $x = L$:
+In this optimization problem, the endpoint $$L$$ is a free parameter,
+i.e. the $$L$$-value of the optimum is unknown and must be found.
+In such cases, the optimum $$y(x)$$ needs to satisfy the so-called *transversality condition*
+at the variable endpoint, in this case $$x = L$$:
 
 $$\begin{aligned}
     0
@@ -67,7 +67,7 @@ $$\begin{aligned}
     &= \bigg( \alpha_{gl} \frac{1}{\sqrt{1 + (y')^2}} + (\alpha_{sl} - \alpha_{sg}) + \lambda y \bigg)_{x = L}
 \end{aligned}$$
 
-Due to the droplet's shape, we have the boundary condition $y(L) = 0$,
+Due to the droplet's shape, we have the boundary condition $$y(L) = 0$$,
 so the last term vanishes.
 We are thus left with the following equation:
 
@@ -77,9 +77,9 @@ $$\begin{aligned}
 \end{aligned}$$
 
 At the edge of the droplet, imagine a small right-angled triangle
-with one side $\dd{x}$ on the $x$-axis,
-the hypotenuse on $y(x)$ having length $\dd{x} \sqrt{1 + (y')^2}$,
-and the corner between them being the contact point with angle $\theta$.
+with one side $$\dd{x}$$ on the $$x$$-axis,
+the hypotenuse on $$y(x)$$ having length $$\dd{x} \sqrt{1 + (y')^2}$$,
+and the corner between them being the contact point with angle $$\theta$$.
 Then, from the definition of the cosine:
 
 $$\begin{aligned}