Improve knowledge base

5 files changed, 344 insertions, 225 deletions

diff --git a/source/know/concept/cylindrical-parabolic-coordinates/index.md b/source/know/concept/cylindrical-parabolic-coordinates/index.md
index 766c9b6..58358dd 100644
--- a/source/know/concept/cylindrical-parabolic-coordinates/index.md
+++ b/source/know/concept/cylindrical-parabolic-coordinates/index.md
@@ -8,82 +8,97 @@ categories:
 layout: "concept"
 ---
 
-**Cylindrical parabolic coordinates** are a coordinate system
-that describes a point in space using three coordinates $$(\sigma, \tau, z)$$.
-The $$z$$-axis is unchanged from the Cartesian system,
-hence it is called a *cylindrical* system.
-In the $$z$$-isoplane, however, confocal parabolas are used.
-These coordinates can be converted to the Cartesian $$(x, y, z)$$ as follows:
+**Cylindrical parabolic coordinates** extend parabolic coordinates $$(\sigma, \tau)$$ to 3D,
+by describing a point in space using the variables $$(\sigma, \tau, z)$$.
+The $$z$$-axis is the same as in the Cartesian system, (hence the name *cylindrical*),
+while the coordinate lines of $$\sigma$$ and $$\tau$$ are confocal parabolas.
+
+[Cartesian coordinates](/know/concept/cartesian-coordinates/) $$(x, y, z)$$
+and this system $$(\sigma, \tau, z)$$ are related by:
 
 $$\begin{aligned}
     \boxed{
-        x = \frac{1}{2} (\tau^2 - \sigma^2 )
-        \qquad
-        y = \sigma \tau
-        \qquad
-        z = z
+        \begin{aligned}
+            x
+            &= \frac{1}{2} (\tau^2 - \sigma^2)
+            \\
+            y
+            &= \sigma \tau
+            \\
+            z
+            &= z
+        \end{aligned}
     }
 \end{aligned}$$
 
-Converting the other way is a bit trickier.
-It can be done by solving the following equations,
-and potentially involves some fiddling with signs:
+Conversely, a point given in $$(x, y, z)$$ can be converted
+to $$(\sigma, \tau, z)$$ using these formulae:
 
 $$\begin{aligned}
-    2 x
-    = \frac{y^2}{\sigma^2} - \sigma^2
-    \qquad \qquad
-    2 x
-    = - \frac{y^2}{\tau^2} + \tau^2
+    \boxed{
+        \begin{aligned}
+            \sigma
+            &= \sgn(x) \sqrt{\sqrt{x^2 + y^2} - x}
+            \\
+            \tau
+            &= \sqrt{\sqrt{x^2 + y^2} + x}
+            \\
+            z
+            &= z
+        \end{aligned}
+    }
 \end{aligned}$$
 
 Cylindrical parabolic coordinates form
 an [orthogonal curvilinear system](/know/concept/orthogonal-curvilinear-coordinates/),
-so we would like to find its scale factors $$h_\sigma$$, $$h_\tau$$ and $$h_z$$.
-The differentials of the Cartesian coordinates are as follows:
+whose **scale factors** $$h_\sigma$$, $$h_\tau$$ and $$h_z$$ we need.
+To get those, we calculate the unnormalized local basis:
 
 $$\begin{aligned}
-    \dd{x} = - \sigma \dd{\sigma} + \tau \dd{\tau}
-    \qquad
-    \dd{y} = \tau \dd{\sigma} + \sigma \dd{\tau}
-    \qquad
-    \dd{z} = \dd{z}
+    h_\sigma \vu{e}_\sigma
+    &= \vu{e}_x \pdv{x}{\sigma} + \vu{e}_y \pdv{y}{\sigma} + \vu{e}_z \pdv{z}{\sigma}
+    \\
+    &= - \vu{e}_x \sigma + \vu{e}_y \tau
+    \\
+    h_\tau \vu{e}_\tau
+    &= \vu{e}_x \pdv{x}{\tau} + \vu{e}_y \pdv{y}{\tau} + \vu{e}_z \pdv{z}{\tau}
+    \\
+    &= \vu{e}_x \tau + \vu{e}_y \sigma
+    \\
+    h_\sigma \vu{e}_\sigma
+    &= \vu{e}_x \pdv{x}{z} + \vu{e}_y \pdv{y}{z} + \vu{e}_z \pdv{z}{z}
+    \\
+    &= \vu{e}_z
 \end{aligned}$$
 
-We calculate the line segment $$\dd{\ell}^2$$,
-skipping many terms thanks to orthogonality:
-
-$$\begin{aligned}
-    \dd{\ell}^2
-    &= (\sigma^2 + \tau^2) \:\dd{\sigma}^2 + (\tau^2 + \sigma^2) \:\dd{\tau}^2 + \dd{z}^2
-\end{aligned}$$
-
-From this, we can directly read off the scale factors $$h_\sigma^2$$, $$h_\tau^2$$ and $$h_z^2$$,
-which turn out to be:
+By normalizing the **local basis vectors**
+$$\vu{e}_\sigma$$, $$\vu{e}_\tau$$ and $$\vu{e}_z$$,
+we arrive at these expressions,
+where we have defined the abbreviation $$\rho$$ for convenience:
 
 $$\begin{aligned}
     \boxed{
-        h_\sigma = \sqrt{\sigma^2 + \tau^2}
-        \qquad
-        h_\tau = \sqrt{\sigma^2 + \tau^2}
-        \qquad
-        h_z = 1
+        \begin{aligned}
+            h_\sigma
+            &= \rho
+            \equiv \sqrt{\sigma^2 + \tau^2}
+            \\
+            h_\tau
+            &= \rho
+            \equiv \sqrt{\sigma^2 + \tau^2}
+            \\
+            h_z
+            &= 1
+        \end{aligned}
     }
-\end{aligned}$$
-
-With these scale factors, we can use
-the general formulae for orthogonal curvilinear coordinates
-to easily to convert things from the Cartesian system.
-The basis vectors are:
-
-$$\begin{aligned}
+    \qquad\qquad
     \boxed{
         \begin{aligned}
             \vu{e}_\sigma
-            &= \frac{- \sigma}{\sqrt{\sigma^2 + \tau^2}} \vu{e}_x + \frac{\tau}{\sqrt{\sigma^2 + \tau^2}} \vu{e}_y
+            &= - \vu{e}_x \frac{\sigma}{\rho} + \vu{e}_y \frac{\tau}{\rho}
             \\
             \vu{e}_\tau
-            &= \frac{\tau}{\sqrt{\sigma^2 + \tau^2}} \vu{e}_x + \frac{\sigma}{\sqrt{\sigma^2 + \tau^2}} \vu{e}_y
+            &= \vu{e}_x \frac{\tau}{\rho} + \vu{e}_y \frac{\sigma}{\rho}
             \\
             \vu{e}_z
             &= \vu{e}_z
@@ -91,13 +106,54 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-The basic vector operations (gradient, divergence, Laplacian and curl) are given by:
+Thanks to these scale factors, we can easily convert calculus from the Cartesian system
+using the standard formulae for orthogonal curvilinear coordinates.
+
+
+
+## Differential elements
+
+For line integrals,
+the tangent vector element $$\dd{\vb{\ell}}$$ for a curve is as follows:
+
+$$\begin{aligned}
+    \boxed{
+        \dd{\vb{\ell}}
+        = \vu{e}_\sigma \: \rho \dd{\sigma} + \: \vu{e}_\tau \: \rho \dd{\tau} + \: \vu{e}_z \dd{z}
+    }
+\end{aligned}$$
+
+For surface integrals,
+the normal vector element $$\dd{\vb{S}}$$ for a surface is given by:
+
+$$\begin{aligned}
+    \boxed{
+        \dd{\vb{S}}
+        = \vu{e}_\sigma \: \rho \dd{\tau} \dd{z} + \: \vu{e}_\tau \: \rho \dd{\sigma} \dd{z} + \: \vu{e}_z \: \rho^2 \dd{\sigma} \dd{\tau}
+    }
+\end{aligned}$$
+
+And for volume integrals,
+the infinitesimal volume $$\dd{V}$$ takes the following form:
+
+$$\begin{aligned}
+    \boxed{
+        \dd{V}
+        = \rho^2 \dd{\sigma} \dd{\tau} \dd{z}
+    }
+\end{aligned}$$
+
+
+
+## Common operations
+
+The basic vector operations (gradient, divergence, curl and Laplacian) are given by:
 
 $$\begin{aligned}
     \boxed{
         \nabla f
-        = \frac{\vu{e}_\sigma}{\sqrt{\sigma^2 + \tau^2}} \pdv{f}{\sigma}
-        + \frac{\vu{e}_\tau}{\sqrt{\sigma^2 + \tau^2}} \pdv{f}{\tau}
+        = \vu{e}_\sigma \frac{1}{\rho} \pdv{f}{\sigma}
+        + \vu{e}_\tau \frac{1}{\rho} \pdv{f}{\tau}
         + \vu{e}_z \pdv{f}{z}
     }
 \end{aligned}$$
@@ -105,78 +161,135 @@ $$\begin{aligned}
 $$\begin{aligned}
     \boxed{
         \nabla \cdot \vb{V}
-        = \frac{1}{\sigma^2 + \tau^2}
-        \Big( \pdv{(V_\sigma \sqrt{\sigma^2 + \tau^2})}{\sigma} + \pdv{(V_\tau \sqrt{\sigma^2 + \tau^2})}{\tau} \Big) + \pdv{V_z}{z}
+        = \frac{1}{\rho} \pdv{V_\sigma}{\sigma} + \frac{\sigma V_\sigma}{\rho^3}
+        + \frac{1}{\rho} \pdv{V_\tau}{\tau} + \frac{\tau V_\tau}{\rho^3}
+        + \pdv{V_z}{z}
+    }
+\end{aligned}$$
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            \nabla \times \vb{V}
+            &= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_z}{\tau} - \pdv{V_\tau}{z} \bigg)
+            \\
+            &\quad\: + \vu{e}_\tau \bigg( \pdv{V_\sigma}{z} - \frac{1}{\rho} \pdv{V_z}{\sigma} \bigg)
+            \\
+            &\quad\: + \vu{e}_z \bigg( \frac{1}{\rho} \pdv{V_\tau}{\sigma} + \frac{\sigma V_\tau}{\rho^3}
+            - \frac{1}{\rho} \pdv{V_\sigma}{\tau} - \frac{\tau V_\sigma}{\rho^3} \bigg)
+        \end{aligned}
     }
 \end{aligned}$$
 
 $$\begin{aligned}
     \boxed{
         \nabla^2 f
-        = \frac{1}{\sigma^2 + \tau^2} \Big( \pdvn{2}{f}{\sigma} + \pdvn{2}{f}{\tau} \Big) + \pdvn{2}{f}{z}
+        = \frac{1}{\rho^2} \pdvn{2}{f}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{f}{\tau} + \pdvn{2}{f}{z}
     }
 \end{aligned}$$
 
+
+
+## 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 $$\nabla \cdot \overline{\overline{\vb{T}}}$$:
+
 $$\begin{aligned}
     \boxed{
         \begin{aligned}
-            \nabla \times \vb{V}
-            &= \vu{e}_\sigma \Big( \frac{\vu{e}_1}{\sqrt{\sigma^2 + \tau^2}} \pdv{V_z}{\tau} - \pdv{V_\tau}{z} \Big)
+            \nabla (\nabla \cdot \vb{V})
+            &= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\sigma} + \frac{1}{\rho^2} \mpdv{V_\tau}{\sigma}{\tau}
+            + \frac{1}{\rho} \mpdv{V_z}{\sigma}{z}
+            \\
+            &\qquad\qquad + \frac{\tau}{\rho^4} \pdv{V_\tau}{\sigma} - \frac{\sigma}{\rho^4} \pdv{V_\tau}{\tau}
+            + \frac{\rho^2 - 3 \sigma^2}{\rho^6} V_\sigma - \frac{3 \sigma \tau V_\tau}{\rho^6} \bigg)
             \\
-            &+ \vu{e}_\tau \Big( \pdv{V_\sigma}{z} - \frac{1}{\sqrt{\sigma^2 + \tau^2}} \pdv{V_z}{\sigma} \Big)
+            &\quad\: + \vu{e}_\tau \bigg( \frac{1}{\rho^2} \mpdv{V_\sigma}{\tau}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\tau}
+            + \frac{1}{\rho} \mpdv{V_z}{\tau}{z}
             \\
-            &+ \frac{\vu{e}_z}{\sigma^2 + \tau^2}
-            \Big( \pdv{(V_\tau \sqrt{\sigma^2 + \tau^2})}{\sigma} - \pdv{(V_\sigma \sqrt{\sigma^2 + \tau^2})}{\tau} \Big)
+            &\qquad\qquad - \frac{\tau}{\rho^4} \pdv{V_\sigma}{\sigma} + \frac{\sigma}{\rho^4} \pdv{V_\sigma}{\tau}
+            - \frac{3 \sigma \tau V_\sigma}{\rho^6} + \frac{\rho^2 - 3 \tau^2}{\rho^6} V_\tau \bigg)
+            \\
+            &\quad\: + \vu{e}_z \bigg( \frac{1}{\rho} \mpdv{V_\sigma}{z}{\sigma} + \frac{1}{\rho} \mpdv{V_\tau}{z}{\tau} + \pdvn{2}{V_z}{z}
+            + \frac{\sigma}{\rho^3} \pdv{V_\sigma}{z} + \frac{\tau}{\rho^3} \pdv{V_\tau}{z} \bigg)
         \end{aligned}
     }
 \end{aligned}$$
 
-The differential element of volume $$\dd{V}$$
-in cylindrical parabolic coordinates is given by:
-
 $$\begin{aligned}
     \boxed{
-        \dd{V} = (\sigma^2 + \tau^2) \dd{\sigma} \dd{\tau} \dd{z}
+        \begin{aligned}
+            \nabla \vb{V}
+            &= \quad \vu{e}_\sigma \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_\sigma}{\sigma} + \frac{\tau V_\tau}{\rho^3} \bigg)
+            + \vu{e}_\sigma \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{V_\tau}{\sigma} - \frac{\tau V_\sigma}{\rho^3} \bigg)
+            + \vu{e}_\sigma \vu{e}_z \frac{1}{\rho} \pdv{V_z}{\sigma}
+            \\
+            &\quad\: + \vu{e}_\tau \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_\sigma}{\tau} - \frac{\sigma V_\tau}{\rho^3} \bigg)
+            + \vu{e}_\tau \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{V_\tau}{\tau} + \frac{\sigma V_\sigma}{\rho^3} \bigg)
+            + \vu{e}_\tau \vu{e}_z \frac{1}{\rho} \pdv{V_z}{\tau}
+            \\
+            &\quad\: + \vu{e}_z \vu{e}_\sigma \pdv{V_\sigma}{z}
+            + \vu{e}_z \vu{e}_\tau \pdv{V_\tau}{z}
+            + \vu{e}_z \vu{e}_z \pdv{V_z}{z}
+        \end{aligned}
     }
 \end{aligned}$$
 
-The differential elements of the isosurfaces are as follows,
-where $$\dd{S_\sigma}$$ is the $$\sigma$$-isosurface, etc.:
-
 $$\begin{aligned}
     \boxed{
         \begin{aligned}
-            \dd{S_\sigma} &= \sqrt{\sigma^2 + \tau^2} \dd{\tau} \dd{z}
+            (\vb{U} \cdot \nabla) \vb{V}
+            &= \quad \vu{e}_\sigma \bigg( \frac{U_\sigma}{\rho} \pdv{V_\sigma}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_\sigma}{\tau} + U_z \pdv{V_\sigma}{z}
+            + \frac{\tau}{\rho^3} U_\sigma V_\tau - \frac{\sigma}{\rho^3} U_\tau V_\tau \bigg)
             \\
-            \dd{S_\tau} &= \sqrt{\sigma^2 + \tau^2} \dd{\sigma} \dd{z}
+            &\quad\: + \vu{e}_\tau \bigg( \frac{U_\sigma}{\rho} \pdv{V_\tau}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_\tau}{\tau} + U_z \pdv{V_\tau}{z}
+            + \frac{\sigma}{\rho^3} U_\tau V_\sigma - \frac{\tau}{\rho^3} U_\sigma V_\sigma \bigg)
             \\
-            \dd{S_z} &= (\sigma^2 + \tau^2) \dd{\sigma} \dd{\tau}
+            &\quad\: + \vu{e}_z \bigg( \frac{U_\sigma}{\rho} \pdv{V_z}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_z}{\tau} + U_z \pdv{V_z}{z} \bigg)
         \end{aligned}
     }
 \end{aligned}$$
 
-The normal element $$\dd{\vu{S}}$$ of a surface and
-the tangent element $$\dd{\vu{\ell}}$$ of a curve are respectively:
-
 $$\begin{aligned}
     \boxed{
-        \dd{\vu{S}}
-        = \vu{e}_\sigma \sqrt{\sigma^2 + \tau^2} \dd{\tau} \dd{z}
-        + \vu{e}_\tau \sqrt{\sigma^2 + \tau^2} \dd{\sigma} \dd{z}
-        + \vu{e}_z (\sigma^2 + \tau^2) \dd{\sigma} \dd{\tau}
+        \begin{aligned}
+            \nabla^2 \vb{V}
+            &= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\tau} + \pdvn{2}{V_\sigma}{z}
+            + \frac{2 \tau}{\rho^4} \pdv{V_\tau}{\sigma} - \frac{2 \sigma}{\rho^4} \pdv{V_\tau}{\tau} - \frac{V_\sigma}{\rho^4} \bigg)
+            \\
+            &\quad\: + \vu{e}_\tau \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\tau} + \pdvn{2}{V_\tau}{z}
+            - \frac{2 \tau}{\rho^4} \pdv{V_\sigma}{\sigma} + \frac{2 \sigma}{\rho^4} \pdv{V_\sigma}{\tau} - \frac{V_\tau}{\rho^4} \bigg)
+            \\
+            &\quad\: + \vu{e}_z \bigg( \frac{1}{\rho^2} \pdvn{2}{V_z}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_z}{\tau} + \pdvn{2}{V_z}{z} \bigg)
+        \end{aligned}
     }
 \end{aligned}$$
 
 $$\begin{aligned}
     \boxed{
-        \dd{\vu{\ell}}
-        = \vu{e}_\sigma \sqrt{\sigma^2 + \tau^2} \dd{\sigma}
-        + \vu{e}_\tau \sqrt{\sigma^2 + \tau^2} \dd{\tau}
-        + \vu{e}_z \dd{z}
+        \begin{aligned}
+            \nabla \cdot \overline{\overline{\mathbf{T}}}
+            &= \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{T_{\sigma \sigma}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau \sigma}}{\tau} + \pdv{T_{z \sigma}}{z}
+            + \frac{\sigma T_{\sigma \sigma}}{\rho^3} + \frac{\tau T_{\sigma \tau}}{\rho^3}
+            + \frac{\tau T_{\tau \sigma}}{\rho^3} - \frac{\sigma T_{\tau \tau}}{\rho^3} \bigg)
+            \\
+            &+ \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{T_{\sigma \tau}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau \tau}}{\tau} + \pdv{T_{k \tau}}{z}
+            - \frac{\tau T_{\sigma \sigma}}{\rho^3} + \frac{\sigma T_{\sigma \tau}}{\rho^3}
+            + \frac{\sigma T_{\tau \sigma}}{\rho^3} + \frac{\tau T_{\tau \tau}}{\rho^3} \bigg)
+            \\
+            &+ \vu{e}_z \bigg( \frac{1}{\rho} \pdv{T_{\sigma z}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau z}}{\tau} + \pdv{T_{zz}}{z}
+            + \frac{\sigma T_{\sigma z}}{\rho^3} + \frac{\tau T_{\tau z}}{\rho^3} \bigg)
+        \end{aligned}
     }
 \end{aligned}$$
 
 
+
 ## References
 1.  M.L. Boas,
     *Mathematical methods in the physical sciences*, 2nd edition,
diff --git a/source/know/concept/cylindrical-polar-coordinates/index.md b/source/know/concept/cylindrical-polar-coordinates/index.md
index fe7d7c1..cf227a6 100644
--- a/source/know/concept/cylindrical-polar-coordinates/index.md
+++ b/source/know/concept/cylindrical-polar-coordinates/index.md
@@ -12,9 +12,9 @@ layout: "concept"
 by describing the location of a point in space
 using the variables $$(r, \varphi, z)$$.
 The $$z$$-axis is unchanged from the Cartesian system,
-hence it is called *cylindrical*.
+hence the name *cylindrical*.
 
-Cartesian coordinates $$(x, y, z)$$
+[Cartesian coordinates](/know/concept/cartesian-coordinates/) $$(x, y, z)$$
 and the cylindrical system $$(r, \varphi, z)$$ are related by:
 
 $$\begin{aligned}
diff --git a/source/know/concept/drude-model/index.md b/source/know/concept/drude-model/index.md
index b175e64..c4faf81 100644
--- a/source/know/concept/drude-model/index.md
+++ b/source/know/concept/drude-model/index.md
@@ -9,124 +9,117 @@ categories:
 layout: "concept"
 ---
 
-The **Drude model** classically predicts
-the dielectric function and electric conductivity of a gas of free charge carriers,
+The **Drude model**, also known as
+the **Drude-Lorentz model** due to its analogy
+to the *Lorentz oscillator model*
+classically predicts the [dielectric function](/know/concept/dielectric-function/)
+and electric conductivity of a gas of free charges,
 as found in metals and doped semiconductors.
 
 
+
 ## Metals
 
-An [electromagnetic wave](/know/concept/electromagnetic-wave-equation/)
-has an oscillating [electric field](/know/concept/electric-field/)
-$$E(t) = E_0 \exp(- i \omega t)$$
-that exerts a force on the charge carriers,
-which have mass $$m$$ and charge $$q$$.
-They thus obey the following equation of motion,
-where $$\gamma$$ is a frictional damping coefficient:
+In a metal, the conduction electrons can roam freely.
+When an [electromagnetic wave](/know/concept/electromagnetic-wave-equation/)
+passes by, its oscillating [electric field](/know/concept/electric-field/)
+$$\vb{E}(t) = \vb{E}_0 e^{- i \omega t}$$ exerts a force on those electrons,
+so the displacement $$\vb{x}(t)$$ of a particle from its initial position
+obeys this equation of motion:
 
 $$\begin{aligned}
-    m \dvn{2}{x}{t} + m \gamma \dv{x}{t}
-    = q E_0 \exp(- i \omega t)
+    m \dvn{2}{\vb{x}}{t}
+    = q \vb{E} - \gamma m \dv{\vb{x}}{t}
 \end{aligned}$$
 
-Inserting the ansatz $$x(t) = x_0 \exp(- i \omega t)$$
-and isolating for the displacement $$x_0$$ yields:
+Where $$m$$ and $$q < 0$$ are the mass and charge of the electron.
+The first term is Newton's third law,
+and the last term represents a damping force
+slowing down the electrons at rate $$\gamma$$.
 
-$$\begin{aligned}
-    - x_0 m \omega^2 - i x_0 m \gamma \omega
-    = q E_0
-    \quad \implies \quad
-    x_0
-    = - \frac{q E_0}{m (\omega^2 + i \gamma \omega)}
-\end{aligned}$$
-
-The polarization density $$P(t)$$ is therefore as shown below.
-Note that the dipole moment $$p$$ goes from negative to positive,
-and the electric field $$E$$ from positive to negative.
-Let $$N$$ be the density of carriers in the gas, then:
+Inserting the ansatz $$\vb{x}(t) = \vb{x}_0 e^{- i \omega t}$$
+and isolating for the displacement $$\vb{x}$$, we find:
 
 $$\begin{aligned}
-    P(t)
-    = N p(t)
-    = N q x(t)
-    = - \frac{N q^2}{m (\omega^2 + i \gamma \omega)} E(t)
+    \vb{x}(t)
+    = \vb{x}_0 e^{- i \omega t}
+    = - \frac{q \vb{E}}{m (\omega^2 + i \gamma \omega)}
 \end{aligned}$$
 
-The electric displacement field $$D$$ is thus as follows,
-where $$\varepsilon_r$$ is the unknown relative permittivity of the gas,
-which we will find shortly:
+The polarization density $$\vb{P}(t)$$ is therefore as shown below.
+Note that the dipole moment vector $$\vb{p}$$ is defined
+as pointing from negative to positive,
+whereas the electric field $$\vb{E}$$ goes from positive to negative.
+Let $$N$$ be the metal's electron density, then:
 
 $$\begin{aligned}
-    D
-    = \varepsilon_0 \varepsilon_r E
-    = \varepsilon_0 E + P
-    = \varepsilon_0 \bigg( 1 - \frac{N q^2}{\varepsilon_0 m} \frac{1}{\omega^2 + i \gamma \omega} \bigg) E
+    \vb{P}(t)
+    = N \vb{p}(t)
+    = N q \vb{x}(t)
+    = - \frac{N q^2}{m (\omega^2 + i \gamma \omega)} \vb{E}(t)
 \end{aligned}$$
 
-The parenthesized expression is the desired dielectric function $$\varepsilon_r$$,
-which depends on $$\omega$$:
+The electric displacement field $$\vb{D}$$ is then as follows,
+where the parenthesized expression is the dielectric function
+$$\varepsilon_r$$ of the material:
 
 $$\begin{aligned}
-    \boxed{
-        \varepsilon_r(\omega)
-        = 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega}
-    }
+    \vb{D}
+    = \varepsilon_0 \vb{E} + \vb{P}
+    = \varepsilon_0 \bigg( 1 - \frac{N q^2}{\varepsilon_0 m} \frac{1}{\omega^2 + i \gamma \omega} \bigg) \vb{E}
+    = \varepsilon_0 \varepsilon_r \vb{E}
 \end{aligned}$$
 
-Where we have defined the important so-called **plasma frequency** like so:
+From this, we define the **plasma frequency** $$\omega_p$$
+at which the conductor "resonates",
+leading to so-called **plasma oscillations** of the electron density
+(see also [Langmuir waves](/know/concept/langmuir-waves/)):
 
 $$\begin{aligned}
+    \varepsilon_r(\omega)
+    = 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega}
+    \qquad\qquad
     \boxed{
         \omega_p
         \equiv \sqrt{\frac{N q^2}{\varepsilon_0 m}}
     }
 \end{aligned}$$
 
-If $$\gamma = 0$$, then $$\varepsilon_r$$ is
-negative $$\omega < \omega_p$$,
-positive for $$\omega > \omega_p$$,
-and zero for $$\omega = \omega_p$$.
-Respectively, this leads to
-an imaginary index $$\sqrt{\varepsilon_r}$$ (high absorption),
-a real index tending to $$1$$ (transparency),
-and the possibility of self-sustained plasma oscillations.
-For metals, $$\omega_p$$ lies in the UV.
-
-We can refine this result for $$\varepsilon_r$$,
-by recognizing the (mean) velocity $$v = \idv{x}{t}$$,
-and rewriting the equation of motion accordingly:
-
-$$\begin{aligned}
-    m \dv{v}{t} + m \gamma v = q E(t)
-\end{aligned}$$
+Suppose that $$\gamma = 0$$,
+then we can identify three distinct scenarios for $$\varepsilon_r$$ here:
 
-Note that $$m v$$ is simply the momentum $$p$$.
-We define the **momentum scattering time** $$\tau \equiv 1 / \gamma$$,
-which represents the average time between collisions,
-where each collision resets the involved particles' momentums to zero.
-Or, more formally:
+*   $$\omega < \omega_p$$, so $$\varepsilon_r < 0$$,
+    so the refractive index $$\sqrt{\varepsilon_r}$$ is imaginary,
+    meaning high absorption and high reflectivity
+    (due to the large complex index difference between media).
+*   $$\omega = \omega_p$$, so $$\varepsilon = 0$$,
+    allowing for self-sustained plasma oscillations.
+*   $$\omega > \omega_p$$, so $$\varepsilon_r > 0$$,
+    so the index $$\sqrt{\varepsilon}$$ is real and asymptotically goes to $$1$$,
+    leading to high transparency and low reflectivity from air.
 
-$$\begin{aligned}
-    \dv{p}{t}
-    = - \frac{p}{\tau} + q E
-\end{aligned}$$
+For most metals $$\omega_p$$ is ultraviolet,
+which explains why they typically appear shiny to us.
+In reality $$\gamma > 0$$, reducing the reflectivity somewhat when $$\omega < \omega_p$$.
 
-Returning to the equation for the mean velocity $$v$$,
-we insert the ansatz $$v(t) = v_0 \exp(- i \omega t)$$,
-for the same electric field $$E(t) = E_0 \exp(-i \omega t)$$ as before:
+The Drude model also lets us calculate the metal's conductivity.
+We already have an expression for $$\vb{x}(t)$$,
+which we differentiate to get the velocity $$\vb{v}(t)$$:
 
 $$\begin{aligned}
-    - i m \omega v_0 + \frac{m}{\tau} v_0 = q E_0
-    \quad \implies \quad
-    v_0 = \frac{q \tau}{m (1 - i \omega \tau)} E_0
+    \vb{v}(t)
+    = \dv{\vb{x}}{t}
+    = - i \omega \vb{x}
+    = \frac{i \omega q \vb{E}}{m (\omega^2 + i \gamma \omega)}
+    = \frac{q \vb{E}}{m (\gamma - i \omega)}
 \end{aligned}$$
 
-From $$v(t)$$, we find the resulting average current density $$J(t)$$ to be as follows:
+Consequently the average current density $$\vb{J}(t)$$ is found to be:
 
 $$\begin{aligned}
-    J(t)
-    = - N q v(t)
-    = \sigma E(t)
+    \vb{J}(t)
+    = N q \vb{v}(t)
+    = \sigma \vb{E}(t)
 \end{aligned}$$
 
 Where $$\sigma(\omega)$$ is the **AC conductivity**,
@@ -134,57 +127,76 @@ which depends on the **DC conductivity** $$\sigma_0$$:
 
 $$\begin{aligned}
     \boxed{
-        \sigma
-        = \frac{\sigma_0}{1 - i \omega \tau}
+        \sigma(\omega)
+        = \frac{\gamma \sigma_0}{\gamma - i \omega}
     }
-    \qquad \quad
+    \qquad\qquad
     \boxed{
         \sigma_0
-        = \frac{N q^2 \tau}{m}
+        \equiv \frac{N q^2}{\gamma m}
     }
 \end{aligned}$$
 
-We can use these quantities to rewrite
-the dielectric function $$\varepsilon_r$$ from earlier:
+Recall that $$\gamma$$ measures friction.
+Specifically, Drude assumed that the electrons often collide with obstacles,
+each time resetting their momentum to zero;
+in that case $$\vb{v}$$ should be interpreted as the average "drift"
+of many electrons in an ensemble.
+The mean time between those collisions is
+the **momentum scattering time** $$\tau \equiv 1 / \gamma$$, so:
+
+$$\begin{aligned}
+    \sigma(\omega)
+    = \frac{\sigma_0}{1 - i \omega \tau}
+    \qquad\qquad
+    \sigma_0
+    = \frac{N q^2 \tau}{m}
+\end{aligned}$$
+
+After defining all those quantities,
+the dielectric function $$\varepsilon_r(\omega)$$ can be written as:
 
 $$\begin{aligned}
     \boxed{
-        \varepsilon_r(\omega)
-        = 1 + \frac{i \sigma(\omega)}{\varepsilon_0 \omega}
+        \begin{aligned}
+            \varepsilon_r(\omega)
+            &= 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega}
+            \\
+            &= 1 + \frac{i \sigma(\omega)}{\varepsilon_0 \omega}
+        \end{aligned}
     }
 \end{aligned}$$
 
 
+
 ## Doped semiconductors
 
 Doping a semiconductor introduces
-free electrons (n-type)
-or free holes (p-type),
-which can be treated as free particles
-moving in the bands of the material.
-
-The Drude model can also be used in this case,
-by replacing the actual carrier mass $$m$$
-by the effective mass $$m^*$$.
+free electrons (n-type doping) or free holes (p-type doping),
+which can be treated as free charge carriers moving through the material,
+so the Drude model is also relevant in this case.
+
+We must replace the carriers' true mass $$m$$ with their *effective mass* $$m^*$$
+found from the material's electronic band structure.
 Furthermore, semiconductors already have
-a high intrinsic permittivity $$\varepsilon_{\mathrm{int}}$$
-before the dopant is added,
-so the diplacement field $$D$$ is:
+a high intrinsic dielectric function $$\varepsilon_{\mathrm{int}}$$
+before being doped, so the displacement field $$\vb{D}$$ becomes:
 
 $$\begin{aligned}
-    D
-    = \varepsilon_0 E + P_{\mathrm{int}} + P_{\mathrm{free}}
-    = \varepsilon_{\mathrm{int}} \varepsilon_0 E - \frac{N q^2}{m^* (\omega^2 + i \gamma \omega)} E
+    \vb{D}
+    = \varepsilon_0 \vb{E} + \vb{P}_{\mathrm{int}} + \vb{P}_{\mathrm{free}}
+    = \varepsilon_0 \varepsilon_{\mathrm{int}} \vb{E} - \frac{N q^2}{m^* (\omega^2 + i \gamma \omega)} \vb{E}
+    = \varepsilon_0 \varepsilon_r \vb{E}
 \end{aligned}$$
 
-Where $$P_{\mathrm{int}}$$ is the intrinsic undoped polarization,
-and $$P_{\mathrm{free}}$$ is the contribution of the free carriers.
+Where $$\vb{P}_{\mathrm{int}}$$ is the intrinsic polarization before doping,
+and $$\vb{P}_{\mathrm{free}}$$ is the expression we calculated above for metals.
 The dielectric function $$\varepsilon_r(\omega)$$ is therefore given by:
 
 $$\begin{aligned}
     \boxed{
         \varepsilon_r(\omega)
-        = \varepsilon_{\mathrm{int}} \Big( 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega} \Big)
+        = \varepsilon_{\mathrm{int}} \bigg( 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega} \bigg)
     }
 \end{aligned}$$
 
@@ -194,29 +206,28 @@ to include $$\varepsilon_\mathrm{int}$$:
 $$\begin{aligned}
     \boxed{
         \omega_p
-        = \sqrt{\frac{N q^2}{\varepsilon_{\mathrm{int}} \varepsilon_0 m^*}}
+        \equiv \sqrt{\frac{N q^2}{\varepsilon_0 \varepsilon_{\mathrm{int}} m^*}}
     }
 \end{aligned}$$
 
 The meaning of $$\omega_p$$ is the same as for metals,
-with high absorption for $$\omega < \omega_p$$.
-However, due to the lower carrier density $$N$$ in a semiconductor,
-$$\omega_p$$ lies in the IR rather than UV.
+but the free carrier density $$N$$ is typically lower in this case,
+so $$\omega_p$$ is usually infrared rather than ultraviolet.
 
-However, instead of asymptotically going to $$1$$ for $$\omega > \omega_p$$ like a metal,
-$$\varepsilon_r$$ tends to $$\varepsilon_\mathrm{int}$$ instead,
-and crosses $$1$$ along the way,
-at which point the reflectivity is zero.
-This occurs at:
+Furthermore, instead of $$\varepsilon_r \to 1$$
+for $$\omega \to \infty$$ like a metal,
+now $$\varepsilon_r \to \varepsilon_\mathrm{int}$$.
+Along the way, there is a point where $$\varepsilon_r = 1$$
+and the reflectivity becomes zero. This occurs at:
 
 $$\begin{aligned}
     \omega^2
     = \frac{\varepsilon_{\mathrm{int}}}{\varepsilon_{\mathrm{int}} - 1} \omega_p^2
 \end{aligned}$$
 
-This is used to experimentally determine the effective mass $$m^*$$
-of the doped semiconductor,
-by finding which value of $$m^*$$ gives the measured $$\omega$$.
+If $$N$$ and $$\varepsilon_\mathrm{int}$$ are known,
+this can be used to experimentally determine $$m^*$$
+by finding which value of $$\omega_p$$ would lead to the measured zero-reflectivity point.
 
 
 
diff --git a/source/know/concept/orthogonal-curvilinear-coordinates/index.md b/source/know/concept/orthogonal-curvilinear-coordinates/index.md
index c7299ee..669358c 100644
--- a/source/know/concept/orthogonal-curvilinear-coordinates/index.md
+++ b/source/know/concept/orthogonal-curvilinear-coordinates/index.md
@@ -21,7 +21,8 @@ where the coordinate surfaces are always perpendicular.
 Examples of such orthogonal curvilinear systems include
 [spherical coordinates](/know/concept/spherical-coordinates/),
 [cylindrical polar coordinates](/know/concept/cylindrical-polar-coordinates/),
-and [cylindrical parabolic coordinates](/know/concept/cylindrical-parabolic-coordinates/).
+[cylindrical parabolic coordinates](/know/concept/cylindrical-parabolic-coordinates/),
+and (trivially) [Cartesian coordinates](/know/concept/cartesian-coordinates/).
 
 
 
@@ -690,12 +691,9 @@ When this index notation is written out in full, it becomes:
 
 $$\begin{aligned}
     \nabla^2 f
-    = \frac{1}{h_1 h_2 h_3}
-    \bigg(
-        \pdv{}{c_1}\Big(\! \frac{h_2 h_3}{h_1} \pdv{f}{c_1} \!\Big)
-        + \pdv{}{c_2}\Big(\! \frac{h_1 h_3}{h_2} \pdv{f}{c_2} \!\Big)
-        + \pdv{}{c_3}\Big(\! \frac{h_1 h_2}{h_3} \pdv{f}{c_3} \!\Big)
-    \bigg)
+    = \frac{1}{h_1 h_2 h_3} \bigg( \pdv{}{c_1} \Big( \frac{h_2 h_3}{h_1} \pdv{f}{c_1} \Big)
+    + \pdv{}{c_2} \Big( \frac{h_1 h_3}{h_2} \pdv{f}{c_2} \Big)
+    + \pdv{}{c_3} \Big( \frac{h_1 h_2}{h_3} \pdv{f}{c_3} \Big) \bigg)
 \end{aligned}$$
 
 This is trivial to prove: $$\nabla^2 f = \nabla \cdot (\nabla f)$$,
diff --git a/source/know/concept/spherical-coordinates/index.md b/source/know/concept/spherical-coordinates/index.md
index 7f6d111..1607b61 100644
--- a/source/know/concept/spherical-coordinates/index.md
+++ b/source/know/concept/spherical-coordinates/index.md
@@ -22,8 +22,8 @@ Note that this is the standard notation among physicists,
 but mathematicians often switch the definitions of $$\theta$$ and $$\varphi$$,
 while still writing $$(r, \theta, \varphi)$$.
 
-Cartesian coordinates $$(x, y, z)$$ and the spherical system
-$$(r, \theta, \varphi)$$ are related by:
+[Cartesian coordinates](/know/concept/cartesian-coordinates/) $$(x, y, z)$$
+and the spherical system $$(r, \theta, \varphi)$$ are related by:
 
 $$\begin{aligned}
     \boxed{
@@ -114,8 +114,6 @@ using the standard formulae for orthogonal curvilinear coordinates.
 
 
 
-
-
 ## Differential elements
 
 For line integrals,
@@ -169,7 +167,7 @@ $$\begin{aligned}
 $$\begin{aligned}
     \boxed{
         \nabla \cdot \vb{V}
-        = \pdv{V_r}{r} + \frac{2}{r} V_r
+        = \pdv{V_r}{r} + \frac{2 V_r}{r}
         + \frac{1}{r} \pdv{V_\theta}{\theta} + \frac{V_\theta}{r \tan{\theta}}
         + \frac{1}{r \sin\theta} \pdv{V_\varphi}{\varphi}
     }
@@ -216,15 +214,15 @@ $$\begin{aligned}
         \begin{aligned}
             \nabla (\nabla \cdot \vb{V})
             &= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r} \mpdv{V_\theta}{r}{\theta} + \frac{1}{r \sin{\theta}} \mpdv{V_\varphi}{\varphi}{r}
-            + \frac{2}{r} \pdv{V_r}{r} - \frac{1}{r^2} \pdv{V_\theta}{\theta}
             \\
-            &\qquad\qquad - \frac{1}{r^2 \sin{\theta}} \pdv{V_\varphi}{\varphi}
+            &\qquad\qquad + \frac{2}{r} \pdv{V_r}{r} - \frac{1}{r^2} \pdv{V_\theta}{\theta}
+            - \frac{1}{r^2 \sin{\theta}} \pdv{V_\varphi}{\varphi}
             + \frac{1}{r \tan{\theta}} \pdv{V_\theta}{r} - \frac{2 V_r}{r^2} - \frac{V_\theta}{r^2 \tan{\theta}} \bigg)
             \\
             &\quad\: + \vu{e}_\theta \bigg( \frac{1}{r} \mpdv{V_r}{\theta}{r} + \frac{1}{r^2} \pdvn{2}{V_\theta}{\theta}
-            + \frac{1}{r^2 \sin{\theta}} \mpdv{V_\varphi}{\theta}{\varphi} + \frac{2}{r^2} \pdv{V_r}{\theta}
+            + \frac{1}{r^2 \sin{\theta}} \mpdv{V_\varphi}{\theta}{\varphi}
             \\
-            &\qquad\qquad + \frac{1}{r^2 \tan{\theta}} \pdv{V_\theta}{\theta}
+            &\qquad\qquad + \frac{2}{r^2} \pdv{V_r}{\theta} + \frac{1}{r^2 \tan{\theta}} \pdv{V_\theta}{\theta}
             - \frac{\cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\varphi}{\varphi} - \frac{V_\theta}{r^2 \sin^2{\theta}} \bigg)
             \\
             &\quad\: + \vu{e}_\varphi \bigg( \frac{1}{r \sin{\theta}} \mpdv{V_r}{\varphi}{r} + \frac{1}{r^2 \sin{\theta}} \mpdv{V_\theta}{\varphi}{\theta}
@@ -275,23 +273,22 @@ $$\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{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}
+            &\qquad\qquad + \frac{2}{r} \pdv{V_r}{r} + \frac{1}{r^2 \tan{\theta}} \pdv{V_r}{\theta}
+            - \frac{2}{r^2} \pdv{V_\theta}{\theta} - \frac{2}{r^2 \sin{\theta}} \pdv{V_\varphi}{\varphi}
             - \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}
+            + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\theta}{\varphi}
             \\
-            &\qquad\qquad + \frac{1}{r^2 \tan{\theta}} \pdv{V_\theta}{\theta}
+            &\qquad\qquad + \frac{2}{r^2} \pdv{V_r}{\theta} + \frac{2}{r} \pdv{V_\theta}{r} + \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}
+            + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\varphi}{\varphi}
             \\
-            &\qquad\qquad + \frac{2 \cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\theta}{\varphi}
-            + \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)
+            &\qquad\qquad + \frac{2}{r^2 \sin{\theta}} \pdv{V_r}{\varphi} + \frac{2 \cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\theta}{\varphi}
+            + \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}
     }
 \end{aligned}$$