Improve knowledge base

5 files changed, 259 insertions, 51 deletions

diff --git a/source/know/concept/cartesian-coordinates/index.md b/source/know/concept/cartesian-coordinates/index.md
new file mode 100644
index 0000000..d198e84
--- /dev/null
+++ b/source/know/concept/cartesian-coordinates/index.md
@@ -0,0 +1,200 @@
+---
+title: "Cartesian coordinates"
+sort_title: "Cartesian coordinates"
+date: 2023-06-09
+categories:
+- Mathematics
+- Physics
+layout: "concept"
+---
+
+This article is a supplement to the ones on
+[orthogonal curvilinear systems](/know/concept/orthogonal-curvilinear-coordinates/),
+[spherical coordinates](/know/concept/spherical-coordinates/),
+[cylindrical polar coordinates](/know/concept/cylindrical-polar-coordinates/),
+and [cylindrical parabolic coordinates](/know/concept/cylindrical-parabolic-coordinates/).
+
+The well-known Cartesian coordinate system $$(x, y, z)$$
+has trivial **scale factors**:
+
+$$\begin{aligned}
+    \boxed{
+        h_x
+        = h_y
+        = h_z
+        = 1
+    }
+\end{aligned}$$
+
+With these, we can use the standard formulae for orthogonal curvilinear coordinates
+to write out various vector calculus operations.
+
+
+
+## Differential elements
+
+For line integrals,
+the tangent vector element $$\dd{\vb{\ell}}$$ for a curve is as follows:
+
+$$\begin{aligned}
+    \boxed{
+        \dd{\vb{\ell}}
+        = \vu{e}_x \dd{x}
+        + \: \vu{e}_y \dd{y}
+        + \: \vu{e}_z \dd{z}
+    }
+\end{aligned}$$
+
+For surface integrals,
+the normal vector element $$\dd{\vb{S}}$$ for a surface is given by:
+
+$$\begin{aligned}
+    \boxed{
+        \dd{\vb{S}}
+        = \vu{e}_x \dd{y} \dd{z}
+        + \: \vu{e}_y \dd{x} \dd{z}
+        + \: \vu{e}_z \dd{x} \dd{y}
+    }
+\end{aligned}$$
+
+And for volume integrals,
+the infinitesimal volume $$\dd{V}$$ takes the following form:
+
+$$\begin{aligned}
+    \boxed{
+        \dd{V}
+        = \dd{x} \dd{y} \dd{z}
+    }
+\end{aligned}$$
+
+
+
+## Common operations
+
+The basic vector operations (gradient, divergence, curl and Laplacian) are given by:
+
+$$\begin{aligned}
+    \boxed{
+        \nabla f
+        = \vu{e}_x \pdv{f}{x}
+        + \vu{e}_y \pdv{f}{y}
+        + \mathbf{e}_z \pdv{f}{z}
+    }
+\end{aligned}$$
+
+$$\begin{aligned}
+    \boxed{
+        \nabla \cdot \vb{V}
+        = \pdv{V_x}{x} + \pdv{V_y}{y} + \pdv{V_z}{z}
+    }
+\end{aligned}$$
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            \nabla \times \vb{V}
+            &= \quad \vu{e}_x \bigg( \pdv{V_z}{y} - \pdv{V_y}{z} \bigg)
+            \\
+            &\quad\: + \vu{e}_y \bigg( \pdv{V_x}{z} - \pdv{V_z}{x} \bigg)
+            \\
+            &\quad\: + \vu{e}_z \bigg( \pdv{V_y}{x} - \pdv{V_x}{y} \bigg)
+        \end{aligned}
+    }
+\end{aligned}$$
+
+$$\begin{aligned}
+    \boxed{
+        \nabla^2 f
+        = \pdvn{2}{f}{x} + \pdvn{2}{f}{y} + \pdvn{2}{f}{z}
+    }
+\end{aligned}$$
+
+
+
+## Uncommon operations
+
+Uncommon operations include:
+the gradient of a divergence $$\nabla (\nabla \cdot \vb{V})$$,
+the gradient of a vector $$\nabla \vb{V}$$,
+the advection of a vector $$(\vb{U} \cdot \nabla) \vb{V}$$ with respect to $$\vb{U}$$,
+the Laplacian of a vector $$\nabla^2 \vb{V}$$,
+and the divergence of a 2nd-order tensor $$\nabla \cdot \overline{\overline{\vb{T}}}$$:
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            \nabla (\nabla \cdot \vb{V})
+            &= \quad \vu{e}_x \bigg( \pdvn{2}{V_x}{x} + \mpdv{V_y}{x}{y} + \mpdv{V_z}{x}{z} \bigg)
+            \\
+            &\quad\: + \vu{e}_y \bigg( \mpdv{V_x}{y}{x} + \pdvn{2}{V_y}{y} + \mpdv{V_z}{y}{z} \bigg)
+            \\
+            &\quad\: + \vu{e}_z \bigg( \mpdv{V_x}{z}{x} + \mpdv{V_y}{z}{y} + \pdvn{2}{V_z}{z} \bigg)
+        \end{aligned}
+    }
+\end{aligned}$$
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            \nabla \vb{V}
+            &= \quad \vu{e}_x \vu{e}_x \pdv{V_x}{x}
+            + \vu{e}_x \vu{e}_y \pdv{V_y}{x}
+            + \vu{e}_x \vu{e}_z \pdv{V_z}{x}
+            \\
+            &\quad\: + \vu{e}_y \vu{e}_x \pdv{V_x}{y}
+            + \vu{e}_y \vu{e}_y \pdv{V_y}{y}
+            + \vu{e}_y \vu{e}_z \pdv{V_z}{y}
+            \\
+            &\quad\: + \vu{e}_z \vu{e}_x \pdv{V_x}{z}
+            + \vu{e}_z \vu{e}_y \pdv{V_y}{z}
+            + \vu{e}_z \vu{e}_z \pdv{V_z}{z}
+        \end{aligned}
+    }
+\end{aligned}$$
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            (\vb{U} \cdot \nabla) \vb{V}
+            &= \quad
+            \vu{e}_x \bigg( U_x \pdv{V_x}{x} + U_y \pdv{V_x}{y} + U_z \pdv{V_x}{z} \bigg)
+            \\
+            &\quad\: + \vu{e}_y \bigg( U_x \pdv{V_y}{x} + U_y \pdv{V_y}{y} + U_z \pdv{V_y}{z} \bigg)
+            \\
+            &\quad\: + \vu{e}_z \bigg( U_x \pdv{V_z}{x} + U_y \pdv{V_z}{y} + U_z \pdv{V_z}{z} \bigg)
+        \end{aligned}
+    }
+\end{aligned}$$
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            \nabla^2 \vb{V}
+            &= \quad \vu{e}_x \bigg( \pdvn{2}{V_x}{x} + \pdvn{2}{V_x}{y} + \pdvn{2}{V_x}{z} \bigg)
+            \\
+            &\quad\: + \vu{e}_y \bigg( \pdvn{2}{V_y}{x} + \pdvn{2}{V_y}{y} + \pdvn{2}{V_y}{z} \bigg)
+            \\
+            &\quad\: + \vu{e}_z \bigg( \pdvn{2}{V_z}{x} + \pdvn{2}{V_z}{y} + \pdvn{2}{V_z}{z} \bigg)
+        \end{aligned}
+    }
+\end{aligned}$$
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            \nabla \cdot \overline{\overline{\mathbf{T}}}
+            &= \quad \vu{e}_x \bigg( \pdv{T_{xx}}{x} + \pdv{T_{yx}}{y} + \pdv{T_{zx}}{z} \bigg)
+            \\
+            &\quad\: + \vu{e}_y \bigg( \pdv{T_{xy}}{x} + \pdv{T_{yy}}{y} + \pdv{T_{zy}}{z} \bigg)
+            \\
+            &\quad\: + \vu{e}_z \bigg( \pdv{T_{xz}}{x} + \pdv{T_{yz}}{y} + \pdv{T_{zz}}{z} \bigg)
+        \end{aligned}
+    }
+\end{aligned}$$
+
+
+
+## References
+1.  M.L. Boas,
+    *Mathematical methods in the physical sciences*, 2nd edition,
+    Wiley.
diff --git a/source/know/concept/cylindrical-polar-coordinates/index.md b/source/know/concept/cylindrical-polar-coordinates/index.md
index 3c54ef8..fe7d7c1 100644
--- a/source/know/concept/cylindrical-polar-coordinates/index.md
+++ b/source/know/concept/cylindrical-polar-coordinates/index.md
@@ -257,14 +257,14 @@ $$\begin{aligned}
     \boxed{
         \begin{aligned}
             \nabla^2 \vb{V}
-            &= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r} \pdv{V_r}{r} + \frac{1}{r^2} \pdvn{2}{V_r}{\varphi}
-            + \pdvn{2}{V_r}{z} - \frac{2}{r^2} \pdv{V_\varphi}{\varphi} - \frac{V_r}{r^2} \bigg)
+            &= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r^2} \pdvn{2}{V_r}{\varphi} + \pdvn{2}{V_r}{z}
+            + \frac{1}{r} \pdv{V_r}{r} - \frac{2}{r^2} \pdv{V_\varphi}{\varphi} - \frac{V_r}{r^2} \bigg)
             \\
-            &\quad\: + \vu{e}_\varphi \bigg( \pdvn{2}{V_\varphi}{r} + \frac{1}{r} \pdv{V_\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\varphi}
-            + \pdvn{2}{V_\varphi}{z} + \frac{2}{r^2} \pdv{V_r}{\varphi} - \frac{V_\varphi}{r^2} \bigg)
+            &\quad\: + \vu{e}_\varphi \bigg( \pdvn{2}{V_\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\varphi} + \pdvn{2}{V_\varphi}{z}
+            + \frac{2}{r^2} \pdv{V_r}{\varphi} + \frac{1}{r} \pdv{V_\varphi}{r} - \frac{V_\varphi}{r^2} \bigg)
             \\
-            &\quad\: + \vu{e}_z \bigg( \pdvn{2}{V_z}{r} + \frac{1}{r} \pdv{V_z}{r}
-            + \frac{1}{r^2} \pdvn{2}{V_z}{\varphi} + \pdvn{2}{V_z}{z} \bigg)
+            &\quad\: + \vu{e}_z \bigg( \pdvn{2}{V_z}{r} + \frac{1}{r^2} \pdvn{2}{V_z}{\varphi}
+            + \pdvn{2}{V_z}{z} + \frac{1}{r} \pdv{V_z}{r} \bigg)
         \end{aligned}
     }
 \end{aligned}$$
diff --git a/source/know/concept/material-derivative/index.md b/source/know/concept/material-derivative/index.md
index d11287d..6bb83c5 100644
--- a/source/know/concept/material-derivative/index.md
+++ b/source/know/concept/material-derivative/index.md
@@ -88,6 +88,9 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
+To evaluate this in various coordinate systems,
+see [orthogonal curvilinear coordinates](/know/concept/orthogonal-curvilinear-coordinates/).
+
 
 
 ## References
diff --git a/source/know/concept/spherical-coordinates/index.md b/source/know/concept/spherical-coordinates/index.md
index 01c5a61..7f6d111 100644
--- a/source/know/concept/spherical-coordinates/index.md
+++ b/source/know/concept/spherical-coordinates/index.md
@@ -170,7 +170,7 @@ $$\begin{aligned}
     \boxed{
         \nabla \cdot \vb{V}
         = \pdv{V_r}{r} + \frac{2}{r} V_r
-        + \frac{1}{r} \pdv{V_\theta}{\theta} + \frac{\cot{\theta}}{r} V_\theta
+        + \frac{1}{r} \pdv{V_\theta}{\theta} + \frac{V_\theta}{r \tan{\theta}}
         + \frac{1}{r \sin\theta} \pdv{V_\varphi}{\varphi}
     }
 \end{aligned}$$
@@ -179,7 +179,7 @@ $$\begin{aligned}
     \boxed{
         \begin{aligned}
             \nabla \times \vb{V}
-            &= \quad \vu{e}_r \bigg( \frac{1}{r} \pdv{V_\varphi}{\theta} + \frac{\cot{\theta}}{r} V_\varphi
+            &= \quad \vu{e}_r \bigg( \frac{1}{r} \pdv{V_\varphi}{\theta} + \frac{V_\varphi}{r \tan{\theta}}
             - \frac{1}{r \sin{\theta}} \pdv{V_\theta}{\varphi} \bigg)
             \\
             &\quad\: + \vu{e}_\theta \bigg( \frac{1}{r \sin{\theta}} \pdv{V_r}{\varphi}
@@ -195,7 +195,7 @@ $$\begin{aligned}
     \boxed{
         \nabla^2 f
         = \pdvn{2}{f}{r} + \frac{2}{r} \pdv{f}{r}
-        + \frac{1}{r^2} \pdvn{2}{f}{\theta} + \frac{\cot{\theta}}{r^2} \pdv{f}{\theta}
+        + \frac{1}{r^2} \pdvn{2}{f}{\theta} + \frac{1}{r^2 \tan{\theta}} \pdv{f}{\theta}
         + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{f}{\varphi}
     }
 \end{aligned}$$
@@ -219,12 +219,12 @@ $$\begin{aligned}
             + \frac{2}{r} \pdv{V_r}{r} - \frac{1}{r^2} \pdv{V_\theta}{\theta}
             \\
             &\qquad\qquad - \frac{1}{r^2 \sin{\theta}} \pdv{V_\varphi}{\varphi}
-            + \frac{\cot{\theta}}{r} \pdv{V_\theta}{r} - \frac{2}{r^2} V_r - \frac{\cot{\theta}}{r^2} V_\theta \bigg)
+            + \frac{1}{r \tan{\theta}} \pdv{V_\theta}{r} - \frac{2 V_r}{r^2} - \frac{V_\theta}{r^2 \tan{\theta}} \bigg)
             \\
             &\quad\: + \vu{e}_\theta \bigg( \frac{1}{r} \mpdv{V_r}{\theta}{r} + \frac{1}{r^2} \pdvn{2}{V_\theta}{\theta}
             + \frac{1}{r^2 \sin{\theta}} \mpdv{V_\varphi}{\theta}{\varphi} + \frac{2}{r^2} \pdv{V_r}{\theta}
             \\
-            &\qquad\qquad + \frac{\cot{\theta}}{r^2} \pdv{V_\theta}{\theta}
+            &\qquad\qquad + \frac{1}{r^2 \tan{\theta}} \pdv{V_\theta}{\theta}
             - \frac{\cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\varphi}{\varphi} - \frac{V_\theta}{r^2 \sin^2{\theta}} \bigg)
             \\
             &\quad\: + \vu{e}_\varphi \bigg( \frac{1}{r \sin{\theta}} \mpdv{V_r}{\varphi}{r} + \frac{1}{r^2 \sin{\theta}} \mpdv{V_\theta}{\varphi}{\theta}
@@ -246,10 +246,10 @@ $$\begin{aligned}
             + \vu{e}_\theta \vu{e}_\varphi \frac{1}{r} \pdv{V_\varphi}{\theta}
             \\
             &\quad\: + \vu{e}_\varphi \vu{e}_r \bigg( \frac{1}{r \sin{\theta}} \pdv{V_r}{\varphi} - \frac{V_\varphi}{r} \bigg)
-            + \vu{e}_\varphi \vu{e}_\theta \bigg( \frac{1}{r \sin{\theta}} \pdv{V_\theta}{\varphi} - \frac{\cot{\theta}}{r} V_\varphi \bigg)
+            + \vu{e}_\varphi \vu{e}_\theta \bigg( \frac{1}{r \sin{\theta}} \pdv{V_\theta}{\varphi} - \frac{V_\varphi}{r \tan{\theta}} \bigg)
             \\
             &\quad\: + \vu{e}_\varphi \vu{e}_\varphi
-            \bigg( \frac{1}{r \sin{\theta}} \pdv{V_\varphi}{\varphi} + \frac{V_r}{r} + \frac{\cot{\theta}}{r} V_\theta \bigg)
+            \bigg( \frac{1}{r \sin{\theta}} \pdv{V_\varphi}{\varphi} + \frac{V_r}{r} + \frac{V_\theta}{r \tan{\theta}}  \bigg)
         \end{aligned}
         }
 \end{aligned}$$
@@ -262,10 +262,10 @@ $$\begin{aligned}
             + \frac{U_\varphi}{r \sin{\theta}} \pdv{V_r}{\varphi} - \frac{U_\theta V_\theta}{r} - \frac{U_\varphi V_\varphi}{r} \bigg)
             \\
             &\quad\: + \vu{e}_\theta \bigg( U_r \pdv{V_\theta}{r} + \frac{U_\theta}{r} \pdv{V_\theta}{\theta}
-            + \frac{U_\varphi}{r \sin{\theta}} \pdv{V_\theta}{\varphi} + \frac{U_\theta V_r}{r} - \frac{\cot{\theta}}{r} U_\varphi V_\varphi \bigg)
+            + \frac{U_\varphi}{r \sin{\theta}} \pdv{V_\theta}{\varphi} + \frac{U_\theta V_r}{r} - \frac{U_\varphi V_\varphi}{r \tan{\theta}} \bigg)
             \\
             &\quad\: + \vu{e}_\varphi \bigg( U_r \pdv{V_\varphi}{r} + \frac{U_\theta}{r} \pdv{V_\varphi}{\theta}
-            + \frac{U_\varphi}{r \sin{\theta}} \pdv{V_\varphi}{\varphi} + \frac{U_\varphi V_r}{r} + \frac{\cot{\theta}}{r} U_\varphi V_\theta \bigg)
+            + \frac{U_\varphi}{r \sin{\theta}} \pdv{V_\varphi}{\varphi} + \frac{U_\varphi V_r}{r} + \frac{U_\varphi V_\theta}{r \tan{\theta}} \bigg)
         \end{aligned}
     }
 \end{aligned}$$
@@ -275,22 +275,22 @@ $$\begin{aligned}
         \begin{aligned}
             \nabla^2 \vb{V}
             &= \quad\: \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r^2} \pdvn{2}{V_r}{\theta} + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_r}{\varphi}
-            + \frac{2}{r} \pdv{V_r}{r} + \frac{\cot{\theta}}{r^2} \pdv{V_r}{\theta}
+            + \frac{2}{r} \pdv{V_r}{r} + \frac{1}{r^2 \tan{\theta}} \pdv{V_r}{\theta}
             \\
             &\qquad\qquad - \frac{2}{r^2} \pdv{V_\theta}{\theta} - \frac{2}{r^2 \sin{\theta}} \pdv{V_\varphi}{\varphi}
-            - \frac{2}{r^2} V_r - \frac{2 \cot{\theta}}{r^2} V_\theta \bigg)
+            - \frac{2 V_r}{r^2} - \frac{2 V_\theta}{r^2 \tan{\theta}} \bigg)
             \\
             &\quad\: + \vu{e}_\theta \bigg( \pdvn{2}{V_\theta}{r} + \frac{1}{r^2} \pdvn{2}{V_\theta}{\theta}
             + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\theta}{\varphi} + \frac{2}{r^2} \pdv{V_r}{\theta} + \frac{2}{r} \pdv{V_\theta}{r}
             \\
-            &\qquad\qquad + \frac{\cot{\theta}}{r^2} \pdv{V_\theta}{\theta}
+            &\qquad\qquad + \frac{1}{r^2 \tan{\theta}} \pdv{V_\theta}{\theta}
             - \frac{2 \cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\varphi}{\varphi} - \frac{V_\theta}{r^2 \sin^2{\theta}} \bigg)
             \\
             &\quad\: + \vu{e}_\varphi \bigg( \pdvn{2}{V_\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\theta}
             + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\varphi}{\varphi} + \frac{2}{r^2 \sin{\theta}} \pdv{V_r}{\varphi}
             \\
             &\qquad\qquad + \frac{2 \cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\theta}{\varphi}
-            + \frac{2}{r} \pdv{V_\varphi}{r} + \frac{\cot{\theta}}{r^2} \pdv{V_\varphi}{\theta}
+            + \frac{2}{r} \pdv{V_\varphi}{r} + \frac{1}{r^2 \tan{\theta}} \pdv{V_\varphi}{\theta}
             - \frac{V_\varphi}{r^2 \sin^2{\theta}} \bigg)
         \end{aligned}
     }
@@ -302,19 +302,20 @@ $$\begin{aligned}
             \nabla \cdot \overline{\overline{\mathbf{T}}}
             &= \quad \vu{e}_r \bigg( \pdv{T_{rr}}{r} + \frac{1}{r} \pdv{T_{\theta r}}{\theta} + \frac{1}{r \sin{\theta}} \pdv{T_{\varphi r}}{\varphi}
             \\
-            &\qquad\qquad + \frac{2}{r} T_{rr} + \frac{\cot{\theta}}{r} T_{\theta r} - \frac{T_{\theta \theta}}{r} - \frac{T_{\varphi \varphi}}{r} \bigg)
+            &\qquad\qquad + \frac{2 T_{rr}}{r} + \frac{T_{\theta r}}{r \tan{\theta}}
+            - \frac{T_{\theta \theta}}{r} - \frac{T_{\varphi \varphi}}{r} \bigg)
             \\
             &\quad\: + \vu{e}_\theta \bigg(\pdv{T_{r \theta}}{r} + \frac{1}{r} \pdv{T_{\theta \theta}}{\theta}
             + \frac{1}{r \sin{\theta}} \pdv{T_{\varphi \theta}}{\varphi}
             \\
-            &\qquad\qquad + \frac{2}{r} T_{r \theta} + \frac{T_{\theta r}}{r}
-            + \frac{\cot{\theta}}{r} T_{\theta \theta} - \frac{\cot{\theta}}{r} T_{\varphi \varphi} \bigg)
+            &\qquad\qquad + \frac{2 T_{r \theta}}{r} + \frac{T_{\theta r}}{r}
+            + \frac{T_{\theta \theta}}{r \tan{\theta}} - \frac{T_{\varphi \varphi}}{r \tan{\theta}} \bigg)
             \\
             &\quad\: + \vu{e}_\varphi \bigg( \pdv{T_{r \varphi}}{r} + \frac{1}{r} \pdv{T_{\theta \varphi}}{\theta}
             + \frac{1}{r \sin{\theta}} \pdv{T_{\varphi \varphi}}{\varphi}
             \\
-            &\qquad\qquad + \frac{2}{r} T_{r \varphi} + \frac{\cot{\theta}}{r} T_{\theta \varphi}
-            + \frac{T_{\varphi r}}{r} + \frac{\cot{\theta}}{r} T_{\varphi \theta} \bigg)
+            &\qquad\qquad + \frac{2 T_{r \varphi}}{r} + \frac{T_{\theta \varphi}}{r \tan{\theta}}
+            + \frac{T_{\varphi r}}{r} + \frac{T_{\varphi \theta}}{r \tan{\theta}} \bigg)
         \end{aligned}
     }
 \end{aligned}$$
diff --git a/source/know/concept/sturm-liouville-theory/index.md b/source/know/concept/sturm-liouville-theory/index.md
index 0ac7476..bff57af 100644
--- a/source/know/concept/sturm-liouville-theory/index.md
+++ b/source/know/concept/sturm-liouville-theory/index.md
@@ -18,6 +18,7 @@ and that the corresponding eigenvalue problem, known as a
 of eigenfunctions.
 
 
+
 ## General operator
 
 Consider the most general form of a second-order linear
@@ -65,7 +66,7 @@ $$[p_0(f^* g' - (f^*)' g)]_a^b = 0$$, leaving:
 
 $$\begin{aligned}
     \inprod{f}{\hat{L} g}
-    &= \big[ p_0 \big( f^* g' - (f^*)' g \big) \big]_a^b + \Inprod{\hat{L}^\dagger f}{g}
+    &= \big[ p_0 \big( f^* g' - (f^*)' g \big) \big]_a^b + \inprod{\hat{L}^\dagger f}{g}
     = \inprod{\hat{L}^\dagger f}{g}
 \end{aligned}$$
 
@@ -115,18 +116,19 @@ $$\begin{aligned}
 
 The latter is a differential equation for $$p(x)$$, which we solve by integration:
 
-$$\begin{gathered}
-    \frac{p_1(x)}{p_0(x)} = \frac{1}{p(x)} \dv{p}{x}
-    \quad \implies \quad
-    \frac{p_1(x)}{p_0(x)} \dd{x} = \frac{1}{p(x)} \dd{p}
+$$\begin{aligned}
+    \frac{p_1(x)}{p_0(x)} \dd{x}
+    &= \frac{1}{p(x)} \dd{p}
     \\
     \implies \quad
-    \int_a^x \frac{p_1(\xi)}{p_0(\xi)} \dd{\xi} = \int_{p(a)}^{p(x)} \frac{1}{f} \dd{f}
-    = \ln\!\Big( \frac{p(x)}{p(a)} \Big)
+    \int \frac{p_1(x)}{p_0(x)} \dd{x}
+    &= \int \frac{1}{p} \dd{p}
+    = \ln\!\big( p(x) \big)
     \\
-    \implies \quad
-    p(x) = p(a) \exp\!\Big( \int_a^x \frac{p_1(\xi)}{p_0(\xi)} \dd{\xi} \Big)
-\end{gathered}$$
+    \implies \qquad\qquad
+    p(x)
+    &= \exp\!\bigg( \int \frac{p_1(x)}{p_0(x)} \dd{x} \bigg)
+\end{aligned}$$
 
 Now that we have $$p(x)$$ and $$q(x)$$, we can define a new operator $$\hat{L}_p$$ as follows:
 
@@ -153,6 +155,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 
+
 ## Eigenvalue problem
 
 A **Sturm-Liouville problem** (SLP) is analogous to a matrix eigenvalue problem,
@@ -166,7 +169,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Necessarily, $$w(x) > 0$$ except in isolated points, where $$w(x) = 0$$ is allowed;
-the point is that any inner product $$\Inprod{f}{w g}$$ may never be zero due to $$w$$'s fault.
+the point is that any inner product $$\inprod{f}{w g}$$ may never be zero due to $$w$$'s fault.
 Furthermore, the convention is that $$u(x)$$ cannot be trivially zero.
 
 In our derivation of $$\hat{L}_{SL}$$,
@@ -212,7 +215,7 @@ $$\begin{aligned}
     &= (\lambda_m^* - \lambda_n) \int_a^b u_n u_m^* w \:dx
     \\
     \inprod{u_m}{\hat{L}_{SL} u_n} - \inprod{\hat{L}_{SL} u_m}{u_n}
-    &= (\lambda_m^* - \lambda_n) \Inprod{u_m}{w u_n}
+    &= (\lambda_m^* - \lambda_n) \inprod{u_m}{w u_n}
 \end{aligned}$$
 
 The operator $$\hat{L}_{SL}$$ is self-adjoint by definition,
@@ -220,17 +223,17 @@ so the left-hand side vanishes, leaving us with:
 
 $$\begin{aligned}
     0
-    &= (\lambda_m^* - \lambda_n) \Inprod{u_m}{w u_n}
+    &= (\lambda_m^* - \lambda_n) \inprod{u_m}{w u_n}
 \end{aligned}$$
 
-When $$m = n$$, the inner product $$\Inprod{u_n}{w u_n}$$ is real and positive
+When $$m = n$$, the inner product $$\inprod{u_n}{w u_n}$$ is real and positive
 (assuming $$u_n$$ is not trivially zero, in which case it would be disqualified anyway).
 In this case we thus know that $$\lambda_n^* = \lambda_n$$,
 i.e. the eigenvalue $$\lambda_n$$ is real for any $$n$$.
 
 When $$m \neq n$$, then $$\lambda_m^* - \lambda_n$$ may or may not be zero,
 depending on the degeneracy. If there is no degeneracy, we
-see that $$\Inprod{u_m}{w u_n} = 0$$, i.e. the eigenfunctions are orthogonal.
+see that $$\inprod{u_m}{w u_n} = 0$$, i.e. the eigenfunctions are orthogonal.
 
 In case of degeneracy, manual orthogonalization is needed, but as it turns out,
 this is guaranteed to be doable, using e.g. the [Gram-Schmidt method](/know/concept/gram-schmidt-method/).
@@ -240,8 +243,8 @@ and all the corresponding eigenfunctions $$u(x)$$ are mutually orthogonal**:
 
 $$\begin{aligned}
     \boxed{
-        \Inprod{u_m(x)}{w(x) u_n(x)}
-        = \Inprod{u_n}{w u_n} \delta_{nm}
+        \inprod{u_m(x)}{w(x) u_n(x)}
+        = \inprod{u_n}{w u_n} \delta_{nm}
         = A_n \delta_{nm}
     }
 \end{aligned}$$
@@ -257,6 +260,7 @@ in other words, there always exists a *lowest* eigenvalue $$\lambda_0 > -\infty$
 known as the **ground state**.
 
 
+
 ## Completeness
 
 Not only are the eigenfunctions $$u_n(x)$$ of an SLP orthogonal, they
@@ -287,18 +291,18 @@ By integrating we get inner products on both the left and the right:
 
 $$\begin{aligned}
     \int_a^b f(x) w(x) u_m^*(x) \dd{x}
-    &= \int_a^b \Big(\sum_{n = 0}^\infty a_n u_n(x) w(x) u_m^*(x)\Big) \dd{x}
+    &= \int_a^b \bigg( \sum_{n = 0}^\infty a_n u_n(x) w(x) u_m^*(x) \bigg) \dd{x}
     \\
-    \Inprod{u_m}{w f}
-    &= \sum_{n = 0}^\infty a_n \Inprod{u_m}{w u_n}
+    \inprod{u_m}{w f}
+    &= \sum_{n = 0}^\infty a_n \inprod{u_m}{w u_n}
 \end{aligned}$$
 
 Because the eigenfunctions of an SLP are mutually orthogonal,
 the summation disappears:
 
 $$\begin{aligned}
-    \Inprod{u_m}{w f}
-    &= \sum_{n = 0}^\infty a_n \Inprod{u_m}{w u_n}
+    \inprod{u_m}{w f}
+    &= \sum_{n = 0}^\infty a_n \inprod{u_m}{w u_n}
     = \sum_{n = 0}^\infty a_n A_n \delta_{nm}
     = a_m A_m
 \end{aligned}$$
@@ -310,8 +314,8 @@ function $$f(x)$$ onto the normalized eigenfunctions $$u_n(x) / A_n$$:
 $$\begin{aligned}
     \boxed{
         a_n
-        = \frac{\Inprod{u_n}{w f}}{A_n}
-        = \frac{\Inprod{u_n}{w f}}{\Inprod{u_n}{w u_n}}
+        = \frac{\inprod{u_n}{w f}}{A_n}
+        = \frac{\inprod{u_n}{w f}}{\inprod{u_n}{w u_n}}
     }
 \end{aligned}$$
 
@@ -321,10 +325,10 @@ after inserting the expression for $$a_n$$:
 
 $$\begin{aligned}
     f(x)
-    &= \sum_{n = 0}^\infty \frac{1}{A_n} \Inprod{u_n}{w f} u_n(x)
-    = \int_a^b \Big(\sum_{n = 0}^\infty \frac{1}{A_n} u_n^*(\xi) w(\xi) f(\xi) u_n(x) \Big) \dd{\xi}
+    &= \sum_{n = 0}^\infty \frac{1}{A_n} \inprod{u_n}{w f} u_n(x)
+    = \int_a^b \bigg(\sum_{n = 0}^\infty \frac{1}{A_n} u_n^*(\xi) w(\xi) f(\xi) u_n(x) \bigg) \dd{\xi}
     \\
-    &= \int_a^b f(\xi) \Big(\sum_{n = 0}^\infty \frac{1}{A_n} u_n^*(\xi) w(\xi) u_n(x) \Big) \dd{\xi}
+    &= \int_a^b f(\xi) \bigg(\sum_{n = 0}^\infty \frac{1}{A_n} u_n^*(\xi) w(\xi) u_n(x) \bigg) \dd{\xi}
 \end{aligned}$$
 
 Upon closer inspection, the parenthesized summation