Improve knowledge base

1 files changed, 184 insertions, 89 deletions

diff --git a/source/know/concept/cylindrical-polar-coordinates/index.md b/source/know/concept/cylindrical-polar-coordinates/index.md
index 43b4684..3c54ef8 100644
--- a/source/know/concept/cylindrical-polar-coordinates/index.md
+++ b/source/know/concept/cylindrical-polar-coordinates/index.md
@@ -8,11 +8,11 @@ categories:
 layout: "concept"
 ---
 
-**Cylindrical polar coordinates** are an extension of polar coordinates to 3D,
-which describes the location of a point in space
-using the coordinates $$(r, \varphi, z)$$.
-The $$z$$-axis is unchanged from Cartesian coordinates,
-hence it is called a *cylindrical* system.
+**Cylindrical polar coordinates** extend polar coordinates $$(r, \varphi)$$ to 3D,
+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*.
 
 Cartesian coordinates $$(x, y, z)$$
 and the cylindrical system $$(r, \varphi, z)$$ are related by:
@@ -20,78 +20,85 @@ and the cylindrical system $$(r, \varphi, z)$$ are related by:
 $$\begin{aligned}
     \boxed{
         \begin{aligned}
-            x &= r \cos\varphi \\
-            y &= r \sin\varphi \\
-            z &= z
+            x
+            &= r \cos{\varphi}
+            \\
+            y
+            &= r \sin{\varphi}
+            \\
+            z
+            &= z
         \end{aligned}
     }
 \end{aligned}$$
 
 Conversely, a point given in $$(x, y, z)$$
-can be converted to $$(r, \varphi, z)$$
-using these formulae:
+can be converted to $$(r, \varphi, z)$$ using these formulae,
+where $$\mathtt{atan2}$$ is the 2-argument arctangent,
+which is needed to handle the signs correctly:
 
 $$\begin{aligned}
     \boxed{
-        r = \sqrt{x^2 + y^2}
-        \qquad
-        \varphi = \mathtt{atan2}(y, x)
-        \qquad
-        z = z
+        \begin{aligned}
+            r
+            &= \sqrt{x^2 + y^2}
+            \\
+            \varphi
+            &= \mathtt{atan2}(y, x)
+            \\
+            z
+            &= z
+        \end{aligned}
     }
 \end{aligned}$$
 
-The cylindrical polar coordinates form
+Cylindrical polar coordinates form
 an [orthogonal curvilinear system](/know/concept/orthogonal-curvilinear-coordinates/),
-whose scale factors $$h_r$$, $$h_\varphi$$ and $$h_z$$ we want to find.
-To do so, we calculate the differentials of the Cartesian coordinates:
+whose **scale factors** $$h_r$$, $$h_\varphi$$ and $$h_z$$ we need.
+To get those, we calculate the unnormalized local basis:
 
 $$\begin{aligned}
-    \dd{x} = \dd{r} \cos\varphi - \dd{\varphi} r \sin\varphi
-    \qquad
-    \dd{y} = \dd{r} \sin\varphi + \dd{\varphi} r \cos\varphi
-    \qquad
-    \dd{z} = \dd{z}
-\end{aligned}$$
-
-And then we calculate the line element $$\dd{\ell}^2$$,
-skipping many terms thanks to orthogonality,
-
-$$\begin{aligned}
-    \dd{\ell}^2
-    &= \dd{r}^2 \big( \cos^2(\varphi) + \sin^2(\varphi) \big)
-    + \dd{\varphi}^2 \big( r^2 \sin^2(\varphi) + r^2 \cos^2(\varphi) \big)
-    + \dd{z}^2
+    h_r \vu{e}_r
+    &= \vu{e}_x \pdv{x}{r} + \vu{e}_y \pdv{y}{r} + \vu{e}_z \pdv{z}{r}
+    \\
+    &= \vu{e}_x \cos{\varphi} + \vu{e}_y \sin{\varphi}
+    \\
+    h_\varphi \vu{e}_\varphi
+    &= \vu{e}_x \pdv{x}{\varphi} + \vu{e}_y \pdv{y}{\varphi} + \vu{e}_z \pdv{z}{\varphi}
     \\
-    &= \dd{r}^2 + r^2 \: \dd{\varphi}^2 + \dd{z}^2
+    &= - \vu{e}_x \: r \sin{\varphi} + \vu{e}_y \: r \cos{\varphi}
+    \\
+    h_z \vu{e}_z
+    &= \vu{e}_x \pdv{x}{z} + \vu{e}_y \pdv{y}{z} + \vu{e}_z \pdv{z}{z}
+    \\
+    &= \vu{e}_z
 \end{aligned}$$
 
-Finally, we can simply read off
-the squares of the desired scale factors
-$$h_r^2$$, $$h_\varphi^2$$ and $$h_z^2$$:
+By normalizing the **local basis vectors**
+$$\vu{e}_r$$, $$\vu{e}_\varphi$$ and $$\vu{e}_z$$,
+we arrive at these expressions:
 
 $$\begin{aligned}
     \boxed{
-        h_r = 1
-        \qquad
-        h_\varphi = r
-        \qquad
-        h_z = 1
+        \begin{aligned}
+            h_r
+            &= 1
+            \\
+            h_\varphi
+            &= r
+            \\
+            h_z
+            &= 1
+        \end{aligned}
     }
-\end{aligned}$$
-
-With these factors, we can easily convert things from the Cartesian system
-using the standard formulae for orthogonal curvilinear coordinates.
-The basis vectors are:
-
-$$\begin{aligned}
+    \qquad\qquad
     \boxed{
         \begin{aligned}
             \vu{e}_r
-            &= \cos\varphi \:\vu{e}_x + \sin\varphi \:\vu{e}_y
+            &= \vu{e}_x \cos{\varphi} + \vu{e}_y \sin{\varphi}
             \\
             \vu{e}_\varphi
-            &= - \sin\varphi \:\vu{e}_x + \cos\varphi \:\vu{e}_y
+            &= - \vu{e}_x \sin{\varphi} + \vu{e}_y \cos{\varphi}
             \\
             \vu{e}_z
             &= \vu{e}_z
@@ -99,7 +106,52 @@ $$\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:
+
+$$\begin{aligned}
+    \boxed{
+        \dd{\vb{\ell}}
+        = \vu{e}_r \dd{r}
+        + \: \vu{e}_\varphi \: r \dd{\varphi}
+        + \: \vu{e}_z \dd{z}
+    }
+\end{aligned}$$
+
+For surface integrals,
+the normal vector element $$\dd{\vb{S}}$$ for a surface is given by:
+
+$$\begin{aligned}
+    \boxed{
+        \dd{\vb{S}}
+        = \vu{e}_r \: r \dd{\varphi} \dd{z}
+        + \: \vu{e}_\varphi \dd{r} \dd{z}
+        + \: \vu{e}_z \: r \dd{r} \dd{\varphi}
+    }
+\end{aligned}$$
+
+And for volume integrals,
+the infinitesimal volume $$\dd{V}$$ takes the following form:
+
+$$\begin{aligned}
+    \boxed{
+        \dd{V}
+        = r \dd{r} \dd{\varphi} \dd{z}
+    }
+\end{aligned}$$
+
+
+
+## Common operations
+
+The basic vector operations (gradient, divergence, curl and Laplacian) are given by:
 
 $$\begin{aligned}
     \boxed{
@@ -113,7 +165,7 @@ $$\begin{aligned}
 $$\begin{aligned}
     \boxed{
         \nabla \cdot \vb{V}
-        = \frac{1}{r} \pdv{(r V_r)}{r}
+        = \pdv{V_r}{r} + \frac{V_r}{r}
         + \frac{1}{r} \pdv{V_\varphi}{\varphi}
         + \pdv{V_z}{z}
     }
@@ -121,81 +173,124 @@ $$\begin{aligned}
 
 $$\begin{aligned}
     \boxed{
+        \begin{aligned}
+            \nabla \times \vb{V}
+            &= \quad \vu{e}_r \bigg( \frac{1}{r} \pdv{V_z}{\varphi} - \pdv{V_\varphi}{z} \bigg)
+            \\
+            &\quad\: + \vu{e}_\varphi \bigg( \pdv{V_r}{z} - \pdv{V_z}{r} \bigg)
+            \\
+            &\quad\: + \vu{e}_z \bigg( \pdv{V_\varphi}{r} + \frac{V_\varphi}{r} - \frac{1}{r} \pdv{V_r}{\varphi} \bigg)
+        \end{aligned}
+    }
+\end{aligned}$$
+
+$$\begin{aligned}
+    \boxed{
         \nabla^2 f
-        = \frac{1}{r} \pdv{}{r}\Big( r \pdv{f}{r} \Big)
+        = \pdvn{2}{f}{r} + \frac{1}{r} \pdv{f}{r}
         + \frac{1}{r^2} \pdvn{2}{f}{\varphi}
         + \pdvn{2}{f}{z}
     }
 \end{aligned}$$
 
+
+
+## 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 $$\nabla \cdot \overline{\overline{\vb{T}}}$$:
+
 $$\begin{aligned}
     \boxed{
         \begin{aligned}
-            \nabla \times \vb{V}
-            &= \vu{e}_r \Big( \frac{1}{r} \pdv{V_z}{\varphi} - \pdv{V_\varphi}{z} \Big)
+            \nabla (\nabla \cdot \vb{V})
+            &= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r} \mpdv{V_\varphi}{r}{\varphi} + \mpdv{V_z}{r}{z}
+            + \frac{1}{r} \pdv{V_r}{r} - \frac{1}{r^2} \pdv{V_\varphi}{\varphi} - \frac{V_r}{r^2} \bigg)
             \\
-            &+ \vu{e}_\varphi \Big( \pdv{V_r}{z} - \pdv{V_z}{r} \Big)
+            &\quad\: + \vu{e}_\varphi \bigg( \frac{1}{r} \mpdv{V_r}{\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\varphi}
+            + \frac{1}{r} \mpdv{V_z}{\varphi}{z} + \frac{1}{r^2} \pdv{V_r}{\varphi} \bigg)
             \\
-            &+ \frac{\vu{e}_z}{r} \Big( \pdv{(r V_\varphi)}{r} - \pdv{V_r}{\varphi} \Big)
+            &\quad\: + \vu{e}_z \bigg( \mpdv{V_r}{z}{r} + \frac{1}{r} \mpdv{V_\varphi}{z}{\varphi} + \pdvn{2}{V_z}{z} + \frac{1}{r} \pdv{V_r}{z} \bigg)
         \end{aligned}
     }
 \end{aligned}$$
 
-The differential element of volume $$\dd{V}$$
-takes the following form:
-
 $$\begin{aligned}
     \boxed{
-        \dd{V}
-        = r \dd{r} \dd{\varphi} \dd{z}
+        \begin{aligned}
+            \nabla \vb{V}
+            &= \quad \vu{e}_r \vu{e}_r \pdv{V_r}{r}
+            + \vu{e}_r \vu{e}_\varphi \pdv{V_\varphi}{r}
+            + \vu{e}_r \vu{e}_z \pdv{V_z}{r}
+            \\
+            &\quad\: + \vu{e}_\varphi \vu{e}_r \bigg( \frac{1}{r} \pdv{V_r}{\varphi} - \frac{V_\varphi}{r} \bigg)
+            + \vu{e}_\varphi \vu{e}_\varphi \bigg( \frac{1}{r} \pdv{V_\varphi}{\varphi} + \frac{V_r}{r} \bigg)
+            + \vu{e}_\varphi \vu{e}_z \frac{1}{r} \pdv{V_z}{\varphi}
+            \\
+            &\quad\: + \vu{e}_z \vu{e}_r \pdv{V_r}{z}
+            + \vu{e}_z \vu{e}_\varphi \pdv{V_\varphi}{z}
+            + \vu{e}_z \vu{e}_z \pdv{V_z}{z}
+        \end{aligned}
     }
 \end{aligned}$$
 
-So, for example, an integral over all of space is converted like so:
-
-$$\begin{aligned}
-    \iiint_{-\infty}^\infty f(x, y, z) \dd{V}
-    = \int_{-\infty}^{\infty} \int_0^{2\pi} \int_0^\infty f(r, \varphi, z) \: r \dd{r} \dd{\varphi} \dd{z}
-\end{aligned}$$
-
-The isosurface elements are as follows, where $$S_r$$ is a surface at constant $$r$$, etc.:
-
 $$\begin{aligned}
     \boxed{
         \begin{aligned}
-            \dd{S}_r = r \dd{\varphi} \dd{z}
-            \qquad
-            \dd{S}_\varphi = \dd{r} \dd{z}
-            \qquad
-            \dd{S}_z = r \dd{r} \dd{\varphi}
+            (\vb{U} \cdot \nabla) \vb{V}
+            &= \quad \vu{e}_r \bigg( U_r \pdv{V_r}{r} + \frac{U_\varphi}{r} \pdv{V_r}{\varphi} + U_z \pdv{V_r}{z}
+            - \frac{U_\varphi V_\varphi}{r} \bigg)
+            \\
+            &\quad\: + \vu{e}_\varphi \bigg( U_r \pdv{V_\varphi}{r} + \frac{U_\varphi}{r} \pdv{V_\varphi}{\varphi} + U_z \pdv{V_\varphi}{z}
+            + \frac{U_\varphi V_r}{r} \bigg)
+            \\
+            &\quad\: + \vu{e}_z \bigg( U_r \pdv{V_z}{r} + \frac{U_\varphi}{r} \pdv{V_z}{\varphi} + U_z \pdv{V_z}{z} \bigg)
         \end{aligned}
     }
 \end{aligned}$$
 
-Similarly, the normal vector element $$\dd{\vu{S}}$$ for an arbitrary surface is given by:
-
 $$\begin{aligned}
     \boxed{
-        \dd{\vu{S}}
-        = \vu{e}_r \: r \dd{\varphi} \dd{z}
-        + \vu{e}_\varphi \dd{r} \dd{z}
-        + \vu{e}_z \: r \dd{r} \dd{\varphi}
+        \begin{aligned}
+            \nabla^2 \vb{V}
+            &= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r} \pdv{V_r}{r} + \frac{1}{r^2} \pdvn{2}{V_r}{\varphi}
+            + \pdvn{2}{V_r}{z} - \frac{2}{r^2} \pdv{V_\varphi}{\varphi} - \frac{V_r}{r^2} \bigg)
+            \\
+            &\quad\: + \vu{e}_\varphi \bigg( \pdvn{2}{V_\varphi}{r} + \frac{1}{r} \pdv{V_\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\varphi}
+            + \pdvn{2}{V_\varphi}{z} + \frac{2}{r^2} \pdv{V_r}{\varphi} - \frac{V_\varphi}{r^2} \bigg)
+            \\
+            &\quad\: + \vu{e}_z \bigg( \pdvn{2}{V_z}{r} + \frac{1}{r} \pdv{V_z}{r}
+            + \frac{1}{r^2} \pdvn{2}{V_z}{\varphi} + \pdvn{2}{V_z}{z} \bigg)
+        \end{aligned}
     }
 \end{aligned}$$
 
-And finally, the tangent vector element $$\dd{\vu{\ell}}$$ of a given curve is as follows:
-
 $$\begin{aligned}
     \boxed{
-        \dd{\vu{\ell}}
-        = \vu{e}_r \dd{r}
-        + \vu{e}_\varphi \: r \dd{\varphi}
-        + \vu{e}_z \dd{z}
+        \begin{aligned}
+            \nabla \cdot \overline{\overline{\mathbf{T}}}
+            &= \quad \vu{e}_r \bigg( \pdv{T_{rr}}{r} + \frac{1}{r} \pdv{T_{\varphi r}}{\varphi} + \pdv{T_{zr}}{z}
+            + \frac{T_{rr}}{r} - \frac{T_{\varphi \varphi}}{r} \bigg)
+            \\
+            &\quad\: + \vu{e}_\varphi \bigg( \pdv{T_{r \varphi}}{r} + \frac{1}{r} \pdv{T_{\varphi \varphi}}{\varphi} + \pdv{T_{z \varphi}}{z}
+            + \frac{T_{r \varphi}}{r} + \frac{T_{\varphi r}}{r} \bigg)
+            \\
+            &\quad\: + \vu{e}_z \bigg( \pdv{T_{rz}}{r} + \frac{1}{r} \pdv{T_{\varphi z}}{\varphi} + \pdv{T_{zz}}{z}
+            + \frac{T_{rz}}{r} \bigg)
+        \end{aligned}
     }
 \end{aligned}$$
 
 
+
 ## References
 1.  M.L. Boas,
     *Mathematical methods in the physical sciences*, 2nd edition,
     Wiley.
+2.  B. Lautrup,
+    *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition,
+    CRC Press.