Improve knowledge base

1 files changed, 209 insertions, 85 deletions

diff --git a/source/know/concept/spherical-coordinates/index.md b/source/know/concept/spherical-coordinates/index.md
index f037182..01c5a61 100644
--- a/source/know/concept/spherical-coordinates/index.md
+++ b/source/know/concept/spherical-coordinates/index.md
@@ -8,9 +8,9 @@ categories:
 layout: "concept"
 ---
 
-**Spherical coordinates** are an extension of polar coordinates to 3D.
+**Spherical coordinates** are an extension of polar coordinates $$(r, \varphi)$$ to 3D.
 The position of a given point in space is described by
-three coordinates $$(r, \theta, \varphi)$$, defined as:
+three variables $$(r, \theta, \varphi)$$, defined as:
 
 *   $$r$$: the **radius** or **radial distance**: distance to the origin.
 *   $$\theta$$: the **elevation**, **polar angle** or **colatitude**:
@@ -18,6 +18,10 @@ three coordinates $$(r, \theta, \varphi)$$, defined as:
 *   $$\varphi$$: the **azimuth**, **azimuthal angle** or **longitude**:
     angle from the positive $$x$$-axis, typically in the counter-clockwise sense.
 
+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:
 
@@ -32,104 +36,142 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Conversely, a point given in $$(x, y, z)$$
-can be converted to $$(r, \theta, \varphi)$$
-using these formulae:
+can be converted to $$(r, \theta, \varphi)$$ using these formulae,
+where $$\mathtt{atan2}$$ is the 2-argument arctangent,
+which is needed to handle the signs correctly:
 
 $$\begin{aligned}
     \boxed{
-        r = \sqrt{x^2 + y^2 + z^2}
-        \qquad
-        \theta = \arccos(z / r)
-        \qquad
-        \varphi = \mathtt{atan2}(y, x)
+        \begin{aligned}
+            r
+            &= \sqrt{x^2 + y^2 + z^2}
+            \\
+            \theta
+            &= \arccos(z / r)
+            \\
+            \varphi
+            &= \mathtt{atan2}(y, x)
+        \end{aligned}
     }
 \end{aligned}$$
 
-The spherical coordinate system is
+Spherical coordinates form
 an [orthogonal curvilinear system](/know/concept/orthogonal-curvilinear-coordinates/),
-whose scale factors $$h_r$$, $$h_\theta$$ and $$h_\varphi$$ we want to find.
-To do so, we calculate the differentials of the Cartesian coordinates:
+whose **scale factors** $$h_r$$, $$h_\theta$$ and $$h_\varphi$$ we need.
+To get those, we calculate the unnormalized local basis:
 
 $$\begin{aligned}
-    \dd{x} &= \dd{r} \sin\theta \cos\varphi + \dd{\theta} r \cos\theta \cos\varphi - \dd{\varphi} r \sin\theta \sin\varphi
+    h_r \vu{e}_r
+    &= \vu{e}_x \pdv{x}{r} + \vu{e}_y \pdv{y}{r} + \vu{e}_z \pdv{z}{r}
     \\
-    \dd{y} &= \dd{r} \sin\theta \sin\varphi + \dd{\theta} r \cos\theta \sin\varphi + \dd{\varphi} r \sin\theta \cos\varphi
+    &= \vu{e}_x \sin{\theta} \cos{\varphi} + \vu{e}_y \sin{\theta} \sin{\varphi} + \vu{e}_z \cos{\theta}
     \\
-    \dd{z} &= \dd{r} \cos\theta - \dd{\theta} r \sin\theta
-\end{aligned}$$
-
-And then we calculate the line element $$\dd{\ell}^2$$,
-skipping many terms thanks to orthogonality:
-
-$$\begin{aligned}
-    \dd{\ell}^2
-    &= \:\:\:\: \dd{r}^2 \big( \sin^2(\theta) \cos^2(\varphi) + \sin^2(\theta) \sin^2(\varphi) + \cos^2(\theta) \big)
+    h_\theta \vu{e}_\theta
+    &= \vu{e}_x \pdv{x}{\theta} + \vu{e}_y \pdv{y}{\theta} + \vu{e}_z \pdv{z}{\theta}
     \\
-    &\quad + \dd{\theta}^2 \big( r^2 \cos^2(\theta) \cos^2(\varphi) + r^2 \cos^2(\theta) \sin^2(\varphi) + r^2 \sin^2(\theta) \big)
+    &= \vu{e}_x \: r \cos{\theta} \cos{\varphi} + \vu{e}_y \: r \cos{\theta} \sin{\varphi} - \vu{e}_z \: r \sin{\theta}
     \\
-    &\quad + \dd{\varphi}^2 \big( r^2 \sin^2(\theta) \sin^2(\varphi) + r^2 \sin^2(\theta) \cos^2(\varphi) \big)
+    h_\varphi \vu{e}_\varphi
+    &= \vu{e}_x \pdv{x}{\varphi} + \vu{e}_y \pdv{y}{\varphi} + \vu{e}_z \pdv{z}{\varphi}
     \\
-    &= \dd{r}^2 + r^2 \: \dd{\theta}^2 + r^2 \sin^2(\theta) \: \dd{\varphi}^2
+    &= - \vu{e}_x \: r \sin{\theta} \sin{\varphi} + \vu{e}_y \: r \sin{\theta} \cos{\varphi}
 \end{aligned}$$
 
-Finally, we can simply read off
-the squares of the desired scale factors
-$$h_r^2$$, $$h_\theta^2$$ and $$h_\varphi^2$$:
+By normalizing the **local basis vectors**
+$$\vu{e}_r$$, $$\vu{e}_\theta$$ and $$\vu{e}_\varphi$$,
+we arrive at these expressions:
 
 $$\begin{aligned}
     \boxed{
-        h_r = 1
-        \qquad
-        h_\theta = r
-        \qquad
-        h_\varphi = r \sin\theta
+        \begin{aligned}
+            h_r
+            &= 1
+            \\
+            h_\theta
+            &= r
+            \\
+            h_\varphi
+            &= r \sin{\theta}
+        \end{aligned}
     }
-\end{aligned}$$
-
-With these factors, we can easily convert things from the Cartesian system
-using the standard formulae for orthogonal curvilinear coordinates.
-The basis vectors are:
-
-$$\begin{aligned}
+    \qquad\qquad
     \boxed{
         \begin{aligned}
             \vu{e}_r
-            &= \sin\theta \cos\varphi \:\vu{e}_x + \sin\theta \sin\varphi \:\vu{e}_y + \cos\theta \:\vu{e}_z
+            &= \vu{e}_x \sin{\theta} \cos{\varphi} + \vu{e}_y \sin{\theta} \sin{\varphi} + \vu{e}_z \cos{\theta}
             \\
             \vu{e}_\theta
-            &= \cos\theta \cos\varphi \:\vu{e}_x + \cos\theta \sin\varphi \:\vu{e}_y - \sin\theta \:\vu{e}_z
+            &= \vu{e}_x \cos{\theta} \cos{\varphi} + \vu{e}_y \cos{\theta} \sin{\varphi} - \vu{e}_z \sin{\theta}
             \\
             \vu{e}_\varphi
-            &= - \sin\varphi \:\vu{e}_x + \cos\varphi \:\vu{e}_y
+            &= - \vu{e}_x \sin{\varphi} + \vu{e}_y \cos{\varphi}
         \end{aligned}
     }
 \end{aligned}$$
 
-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:
 
 $$\begin{aligned}
     \boxed{
-        \nabla f
-        = \vu{e}_r \pdv{f}{r}
-        + \vu{e}_\theta \frac{1}{r} \pdv{f}{\theta} + \mathbf{e}_\varphi \frac{1}{r \sin\theta} \pdv{f}{\varphi}
+        \dd{\vb{\ell}}
+        = \vu{e}_r \dd{r}
+        + \: \vu{e}_\theta \: r \dd{\theta}
+        + \: \vu{e}_\varphi \: r \sin{\theta} \dd{\varphi}
     }
 \end{aligned}$$
 
+For surface integrals,
+the normal vector element $$\dd{\vb{S}}$$ for a surface is given by:
+
 $$\begin{aligned}
     \boxed{
-        \nabla \cdot \vb{V}
-        = \frac{1}{r^2} \pdv{(r^2 V_r)}{r}
-        + \frac{1}{r \sin\theta} \pdv{(\sin\theta V_\theta)}{\theta}
-        + \frac{1}{r \sin\theta} \pdv{V_\varphi}{\varphi}
+        \dd{\vb{S}}
+        = \vu{e}_r \: r^2 \sin{\theta} \dd{\theta} \dd{\varphi}
+        + \: \vu{e}_\theta \: r \sin{\theta} \dd{r} \dd{\varphi}
+        + \: \vu{e}_\varphi \: r \dd{r} \dd{\theta}
     }
 \end{aligned}$$
 
+And for volume integrals,
+the infinitesimal volume $$\dd{V}$$ takes the following form:
+
 $$\begin{aligned}
     \boxed{
-        \nabla^2 f
-        = \frac{1}{r^2} \pdv{}{r}\Big( r^2 \pdv{f}{r} \Big)
-        + \frac{1}{r^2 \sin\theta} \pdv{}{\theta}\Big( \sin\theta \pdv{f}{\theta} \Big)
-        + \frac{1}{r^2 \sin^2(\theta)} \pdvn{2}{f}{\varphi}
+        \dd{V}
+        = r^2 \sin{\theta} \dd{r} \dd{\theta} \dd{\varphi}
+    }
+\end{aligned}$$
+
+
+
+## Common operations
+
+The basic vector operations (gradient, divergence, curl and Laplacian) are given by:
+
+$$\begin{aligned}
+    \boxed{
+        \nabla f
+        = \vu{e}_r \pdv{f}{r}
+        + \vu{e}_\theta \frac{1}{r} \pdv{f}{\theta} + \mathbf{e}_\varphi \frac{1}{r \sin{\theta}} \pdv{f}{\varphi}
+    }
+\end{aligned}$$
+
+$$\begin{aligned}
+    \boxed{
+        \nabla \cdot \vb{V}
+        = \pdv{V_r}{r} + \frac{2}{r} V_r
+        + \frac{1}{r} \pdv{V_\theta}{\theta} + \frac{\cot{\theta}}{r} V_\theta
+        + \frac{1}{r \sin\theta} \pdv{V_\varphi}{\varphi}
     }
 \end{aligned}$$
 
@@ -137,70 +179,152 @@ $$\begin{aligned}
     \boxed{
         \begin{aligned}
             \nabla \times \vb{V}
-            &= \frac{\vu{e}_r}{r \sin\theta} \Big( \pdv{(\sin\theta V_\varphi)}{\theta} - \pdv{V_\theta}{\varphi} \Big)
+            &= \quad \vu{e}_r \bigg( \frac{1}{r} \pdv{V_\varphi}{\theta} + \frac{\cot{\theta}}{r} V_\varphi
+            - \frac{1}{r \sin{\theta}} \pdv{V_\theta}{\varphi} \bigg)
             \\
-            &+ \frac{\vu{e}_\theta}{r} \Big( \frac{1}{\sin\theta} \pdv{V_r}{\varphi} - \pdv{(r V_\varphi)}{r} \Big)
+            &\quad\: + \vu{e}_\theta \bigg( \frac{1}{r \sin{\theta}} \pdv{V_r}{\varphi}
+            - \pdv{V_\varphi}{r} - \frac{V_\varphi}{r} \bigg)
             \\
-            &+ \frac{\vu{e}_\varphi}{r} \Big( \pdv{(r V_\theta)}{r} - \pdv{V_r}{\theta} \Big)
+            &\quad\: + \vu{e}_\varphi \bigg( \pdv{V_\theta}{r} + \frac{V_\theta}{r}
+            - \frac{1}{r} \pdv{V_r}{\theta} \bigg)
         \end{aligned}
     }
 \end{aligned}$$
 
-The differential element of volume $$\dd{V}$$
-takes the following form:
-
 $$\begin{aligned}
     \boxed{
-        \dd{V}
-        = r^2 \sin\theta \dd{r} \dd{\theta} \dd{\varphi}
+        \nabla^2 f
+        = \pdvn{2}{f}{r} + \frac{2}{r} \pdv{f}{r}
+        + \frac{1}{r^2} \pdvn{2}{f}{\theta} + \frac{\cot{\theta}}{r^2} \pdv{f}{\theta}
+        + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{f}{\varphi}
     }
 \end{aligned}$$
 
-So, for example, an integral over all of space is converted like so:
 
-$$\begin{aligned}
-    \iiint_{-\infty}^\infty f(x, y, z) \dd{V}
-    = \int_0^{2\pi} \int_0^\pi \int_0^\infty f(r, \theta, \varphi) \: r^2 \sin\theta \dd{r} \dd{\theta} \dd{\varphi}
-\end{aligned}$$
 
-The isosurface elements are as follows, where $$S_r$$ is a surface at constant $$r$$, etc.:
+## 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 $$\nabla \cdot \overline{\overline{\vb{T}}}$$:
 
 $$\begin{aligned}
     \boxed{
         \begin{aligned}
-            \dd{S}_r = r^2 \sin\theta \dd{\theta} \dd{\varphi}
-            \qquad
-            \dd{S}_\theta = r \sin\theta \dd{r} \dd{\varphi}
-            \qquad
-            \dd{S}_\varphi = r \dd{r} \dd{\theta}
+            \nabla (\nabla \cdot \vb{V})
+            &= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r} \mpdv{V_\theta}{r}{\theta} + \frac{1}{r \sin{\theta}} \mpdv{V_\varphi}{\varphi}{r}
+            + \frac{2}{r} \pdv{V_r}{r} - \frac{1}{r^2} \pdv{V_\theta}{\theta}
+            \\
+            &\qquad\qquad - \frac{1}{r^2 \sin{\theta}} \pdv{V_\varphi}{\varphi}
+            + \frac{\cot{\theta}}{r} \pdv{V_\theta}{r} - \frac{2}{r^2} V_r - \frac{\cot{\theta}}{r^2} V_\theta \bigg)
+            \\
+            &\quad\: + \vu{e}_\theta \bigg( \frac{1}{r} \mpdv{V_r}{\theta}{r} + \frac{1}{r^2} \pdvn{2}{V_\theta}{\theta}
+            + \frac{1}{r^2 \sin{\theta}} \mpdv{V_\varphi}{\theta}{\varphi} + \frac{2}{r^2} \pdv{V_r}{\theta}
+            \\
+            &\qquad\qquad + \frac{\cot{\theta}}{r^2} \pdv{V_\theta}{\theta}
+            - \frac{\cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\varphi}{\varphi} - \frac{V_\theta}{r^2 \sin^2{\theta}} \bigg)
+            \\
+            &\quad\: + \vu{e}_\varphi \bigg( \frac{1}{r \sin{\theta}} \mpdv{V_r}{\varphi}{r} + \frac{1}{r^2 \sin{\theta}} \mpdv{V_\theta}{\varphi}{\theta}
+            + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\varphi}{\varphi}
+            \\
+            &\qquad\qquad + \frac{2}{r^2 \sin{\theta}} \pdv{V_r}{\varphi} + \frac{\cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\theta}{\varphi} \bigg)
         \end{aligned}
     }
 \end{aligned}$$
 
-Similarly, the normal vector element $$\dd{\vu{S}}$$ for an arbitrary surface is given by:
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            \nabla \vb{V}
+            &= \quad \vu{e}_r \vu{e}_r \pdv{V_r}{r} + \vu{e}_r \vu{e}_\theta \pdv{V_\theta}{r} + \vu{e}_r \vu{e}_\varphi \pdv{V_\varphi}{r}
+            \\
+            &\quad\: + \vu{e}_\theta \vu{e}_r \bigg( \frac{1}{r} \pdv{V_r}{\theta} - \frac{V_\theta}{r} \bigg)
+            + \vu{e}_\theta \vu{e}_\theta \bigg( \frac{1}{r} \pdv{V_\theta}{\theta} + \frac{V_r}{r} \bigg)
+            + \vu{e}_\theta \vu{e}_\varphi \frac{1}{r} \pdv{V_\varphi}{\theta}
+            \\
+            &\quad\: + \vu{e}_\varphi \vu{e}_r \bigg( \frac{1}{r \sin{\theta}} \pdv{V_r}{\varphi} - \frac{V_\varphi}{r} \bigg)
+            + \vu{e}_\varphi \vu{e}_\theta \bigg( \frac{1}{r \sin{\theta}} \pdv{V_\theta}{\varphi} - \frac{\cot{\theta}}{r} V_\varphi \bigg)
+            \\
+            &\quad\: + \vu{e}_\varphi \vu{e}_\varphi
+            \bigg( \frac{1}{r \sin{\theta}} \pdv{V_\varphi}{\varphi} + \frac{V_r}{r} + \frac{\cot{\theta}}{r} V_\theta \bigg)
+        \end{aligned}
+        }
+\end{aligned}$$
 
 $$\begin{aligned}
     \boxed{
-        \dd{\vu{S}}
-        = \vu{e}_r \: r^2 \sin\theta \dd{\theta} \dd{\varphi}
-        + \vu{e}_\theta \: r \sin\theta \dd{r} \dd{\varphi}
-        + \vu{e}_\varphi \: r \dd{r} \dd{\theta}
+        \begin{aligned}
+            (\vb{U} \cdot \nabla) \vb{V}
+            &= \quad \vu{e}_r \bigg( U_r \pdv{V_r}{r} + \frac{U_\theta}{r} \pdv{V_r}{\theta}
+            + \frac{U_\varphi}{r \sin{\theta}} \pdv{V_r}{\varphi} - \frac{U_\theta V_\theta}{r} - \frac{U_\varphi V_\varphi}{r} \bigg)
+            \\
+            &\quad\: + \vu{e}_\theta \bigg( U_r \pdv{V_\theta}{r} + \frac{U_\theta}{r} \pdv{V_\theta}{\theta}
+            + \frac{U_\varphi}{r \sin{\theta}} \pdv{V_\theta}{\varphi} + \frac{U_\theta V_r}{r} - \frac{\cot{\theta}}{r} U_\varphi V_\varphi \bigg)
+            \\
+            &\quad\: + \vu{e}_\varphi \bigg( U_r \pdv{V_\varphi}{r} + \frac{U_\theta}{r} \pdv{V_\varphi}{\theta}
+            + \frac{U_\varphi}{r \sin{\theta}} \pdv{V_\varphi}{\varphi} + \frac{U_\varphi V_r}{r} + \frac{\cot{\theta}}{r} U_\varphi V_\theta \bigg)
+        \end{aligned}
     }
 \end{aligned}$$
 
-And finally, the tangent vector element $$\dd{\vu{\ell}}$$ of a given curve is as follows:
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            \nabla^2 \vb{V}
+            &= \quad\: \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r^2} \pdvn{2}{V_r}{\theta} + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_r}{\varphi}
+            + \frac{2}{r} \pdv{V_r}{r} + \frac{\cot{\theta}}{r^2} \pdv{V_r}{\theta}
+            \\
+            &\qquad\qquad - \frac{2}{r^2} \pdv{V_\theta}{\theta} - \frac{2}{r^2 \sin{\theta}} \pdv{V_\varphi}{\varphi}
+            - \frac{2}{r^2} V_r - \frac{2 \cot{\theta}}{r^2} V_\theta \bigg)
+            \\
+            &\quad\: + \vu{e}_\theta \bigg( \pdvn{2}{V_\theta}{r} + \frac{1}{r^2} \pdvn{2}{V_\theta}{\theta}
+            + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\theta}{\varphi} + \frac{2}{r^2} \pdv{V_r}{\theta} + \frac{2}{r} \pdv{V_\theta}{r}
+            \\
+            &\qquad\qquad + \frac{\cot{\theta}}{r^2} \pdv{V_\theta}{\theta}
+            - \frac{2 \cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\varphi}{\varphi} - \frac{V_\theta}{r^2 \sin^2{\theta}} \bigg)
+            \\
+            &\quad\: + \vu{e}_\varphi \bigg( \pdvn{2}{V_\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\theta}
+            + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\varphi}{\varphi} + \frac{2}{r^2 \sin{\theta}} \pdv{V_r}{\varphi}
+            \\
+            &\qquad\qquad + \frac{2 \cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\theta}{\varphi}
+            + \frac{2}{r} \pdv{V_\varphi}{r} + \frac{\cot{\theta}}{r^2} \pdv{V_\varphi}{\theta}
+            - \frac{V_\varphi}{r^2 \sin^2{\theta}} \bigg)
+        \end{aligned}
+    }
+\end{aligned}$$
 
 $$\begin{aligned}
     \boxed{
-        \dd{\vu{\ell}}
-        = \vu{e}_r \: \dd{r}
-        + \vu{e}_\theta \: r \dd{\theta}
-        + \vu{e}_\varphi \: r \sin\theta \dd{\varphi}
+        \begin{aligned}
+            \nabla \cdot \overline{\overline{\mathbf{T}}}
+            &= \quad \vu{e}_r \bigg( \pdv{T_{rr}}{r} + \frac{1}{r} \pdv{T_{\theta r}}{\theta} + \frac{1}{r \sin{\theta}} \pdv{T_{\varphi r}}{\varphi}
+            \\
+            &\qquad\qquad + \frac{2}{r} T_{rr} + \frac{\cot{\theta}}{r} T_{\theta r} - \frac{T_{\theta \theta}}{r} - \frac{T_{\varphi \varphi}}{r} \bigg)
+            \\
+            &\quad\: + \vu{e}_\theta \bigg(\pdv{T_{r \theta}}{r} + \frac{1}{r} \pdv{T_{\theta \theta}}{\theta}
+            + \frac{1}{r \sin{\theta}} \pdv{T_{\varphi \theta}}{\varphi}
+            \\
+            &\qquad\qquad + \frac{2}{r} T_{r \theta} + \frac{T_{\theta r}}{r}
+            + \frac{\cot{\theta}}{r} T_{\theta \theta} - \frac{\cot{\theta}}{r} T_{\varphi \varphi} \bigg)
+            \\
+            &\quad\: + \vu{e}_\varphi \bigg( \pdv{T_{r \varphi}}{r} + \frac{1}{r} \pdv{T_{\theta \varphi}}{\theta}
+            + \frac{1}{r \sin{\theta}} \pdv{T_{\varphi \varphi}}{\varphi}
+            \\
+            &\qquad\qquad + \frac{2}{r} T_{r \varphi} + \frac{\cot{\theta}}{r} T_{\theta \varphi}
+            + \frac{T_{\varphi r}}{r} + \frac{\cot{\theta}}{r} T_{\varphi \theta} \bigg)
+        \end{aligned}
     }
 \end{aligned}$$
 
 
+
 ## 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.