From 3138ead6bfd6e88e8cdbf9e4c32df64e18bc4595 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Fri, 9 Jun 2023 19:52:54 +0200
Subject: Improve knowledge base
---
source/know/concept/capillary-action/index.md | 2 +-
.../concept/cylindrical-polar-coordinates/index.md | 273 ++++++++++++-------
source/know/concept/hilbert-space/index.md | 55 ++--
source/know/concept/material-derivative/index.md | 20 --
.../orthogonal-curvilinear-coordinates/index.md | 250 +++++++++---------
source/know/concept/spherical-coordinates/index.md | 294 +++++++++++++++------
6 files changed, 549 insertions(+), 345 deletions(-)
(limited to 'source')
diff --git a/source/know/concept/capillary-action/index.md b/source/know/concept/capillary-action/index.md
index 4b9e76c..fea6ef8 100644
--- a/source/know/concept/capillary-action/index.md
+++ b/source/know/concept/capillary-action/index.md
@@ -58,7 +58,7 @@ $$\begin{aligned}
}
\end{aligned}$$
-The right-most side gives an alternative way of understanding $$\mathrm{Bo}$$:
+The rightmost side gives an alternative way of understanding $$\mathrm{Bo}$$:
$$m$$ is the mass of a cube with side $$L$$, such that the numerator is the weight force,
and the denominator is the tension force of the surface.
In any case, capillary action can be observed when $$\mathrm{Bo \ll 1}$$.
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{
@@ -113,89 +165,132 @@ $$\begin{aligned}
$$\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}
}
\end{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}$$
+
## References
1. M.L. Boas,
*Mathematical methods in the physical sciences*, 2nd edition,
Wiley.
+2. B. Lautrup,
+ *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition,
+ CRC Press.
diff --git a/source/know/concept/hilbert-space/index.md b/source/know/concept/hilbert-space/index.md
index 57926ce..42b9cb1 100644
--- a/source/know/concept/hilbert-space/index.md
+++ b/source/know/concept/hilbert-space/index.md
@@ -8,19 +8,20 @@ categories:
layout: "concept"
---
-A **Hilbert space**, also called an **inner product space**, is an
-abstract **vector space** with a notion of length and angle.
+A **Hilbert space**, also called an **inner product space**,
+is an abstract **vector space** with a notion of length and angle.
+
## Vector space
-An abstract **vector space** $$\mathbb{V}$$ is a generalization of the
-traditional concept of vectors as "arrows". It consists of a set of
-objects called **vectors** which support the following (familiar)
-operations:
+An abstract **vector space** $$\mathbb{V}$$ is a generalization
+of the traditional concept of vectors as "arrows".
+It consists of a set of objects called **vectors**
+which support the following (familiar) operations:
-+ **Vector addition**: the sum of two vectors $$V$$ and $$W$$, denoted $$V + W$$.
-+ **Scalar multiplication**: product of a vector $$V$$ with a scalar $$a$$, denoted $$a V$$.
++ **Vector addition**: the sum of two vectors $$V$$ and $$W$$, denoted by $$V + W$$.
++ **Scalar multiplication**: product of a vector $$V$$ with a scalar $$a$$, denoted by $$a V$$.
In addition, for a given $$\mathbb{V}$$ to qualify as a proper vector
space, these operations must obey the following axioms:
@@ -34,24 +35,26 @@ space, these operations must obey the following axioms:
+ **Multiplication is distributive over scalars**: $$(a + b)V = aV + bV$$
+ **Multiplication is distributive over vectors**: $$a (U + V) = a U + a V$$
-A set of $$N$$ vectors $$V_1, V_2, ..., V_N$$ is **linearly independent** if
-the only way to satisfy the following relation is to set all the scalar coefficients $$a_n = 0$$:
+A set of $$N$$ vectors $$V_1, V_2, ..., V_N$$ is **linearly independent**
+if the only way to satisfy the following relation
+is to set all the scalar coefficients $$a_n = 0$$:
$$\begin{aligned}
\mathbf{0} = \sum_{n = 1}^N a_n V_n
\end{aligned}$$
-In other words, these vectors cannot be expressed in terms of each
-other. Otherwise, they would be **linearly dependent**.
+In other words, these vectors cannot be expressed in terms of each other.
+Otherwise, they would be **linearly dependent**.
-A vector space $$\mathbb{V}$$ has **dimension** $$N$$ if only up to $$N$$ of
-its vectors can be linearly indepedent. All other vectors in
-$$\mathbb{V}$$ can then be written as a **linear combination** of these $$N$$ **basis vectors**.
+A vector space $$\mathbb{V}$$ has **dimension** $$N$$
+if only up to $$N$$ of its vectors can be linearly indepedent.
+All other vectors in $$\mathbb{V}$$ can then be written
+as a **linear combination** of these $$N$$ **basis vectors**.
-Let $$\vu{e}_1, ..., \vu{e}_N$$ be the basis vectors, then any
-vector $$V$$ in the same space can be **expanded** in the basis according to
-the unique weights $$v_n$$, known as the **components** of $$V$$
-in that basis:
+Let $$\vu{e}_1, ..., \vu{e}_N$$ be the basis vectors,
+then any vector $$V$$ in the same space can be **expanded**
+in the basis according to the unique weights $$v_n$$,
+known as the **components** of $$V$$ in that basis:
$$\begin{aligned}
V = \sum_{n = 1}^N v_n \vu{e}_n
@@ -71,19 +74,20 @@ $$\begin{gathered}
\end{gathered}$$
+
## Inner product
-A given vector space $$\mathbb{V}$$ can be promoted to a **Hilbert space**
-or **inner product space** if it supports an operation $$\Inprod{U}{V}$$
-called the **inner product**, which takes two vectors and returns a
-scalar, and has the following properties:
+A given vector space $$\mathbb{V}$$ can be promoted to a **Hilbert space** or **inner product space**
+if it supports an operation $$\Inprod{U}{V}$$ called the **inner product**,
+which takes two vectors and returns a scalar,
+and has the following properties:
+ **Skew symmetry**: $$\Inprod{U}{V} = (\Inprod{V}{U})^*$$, where $${}^*$$ is the complex conjugate.
+ **Positive semidefiniteness**: $$\Inprod{V}{V} \ge 0$$, and $$\Inprod{V}{V} = 0$$ if $$V = \mathbf{0}$$.
+ **Linearity in second operand**: $$\Inprod{U}{(a V + b W)} = a \Inprod{U}{V} + b \Inprod{U}{W}$$.
-The inner product describes the lengths and angles of vectors, and in
-Euclidean space it is implemented by the dot product.
+The inner product describes the lengths and angles of vectors,
+and in Euclidean space it is implemented by the dot product.
The **magnitude** or **norm** $$|V|$$ of a vector $$V$$ is given by
$$|V| = \sqrt{\Inprod{V}{V}}$$ and represents the real positive length of $$V$$.
@@ -123,6 +127,7 @@ $$\begin{aligned}
\end{aligned}$$
+
## Infinite dimensions
As the dimensionality $$N$$ tends to infinity, things may or may not
diff --git a/source/know/concept/material-derivative/index.md b/source/know/concept/material-derivative/index.md
index 7225053..d11287d 100644
--- a/source/know/concept/material-derivative/index.md
+++ b/source/know/concept/material-derivative/index.md
@@ -88,26 +88,6 @@ $$\begin{aligned}
}
\end{aligned}$$
-Where the advective term is to be evaluated in the following way in Cartesian coordinates:
-
-$$\begin{aligned}
- (\va{v} \cdot \nabla) \va{U}
- =
- \begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix}
- \cdot
- \begin{bmatrix}
- \displaystyle\pdv{U_x}{x} & \displaystyle\pdv{U_x}{y} & \displaystyle\pdv{U_x}{z} \\
- \displaystyle\pdv{U_y}{x} & \displaystyle\pdv{U_y}{y} & \displaystyle\pdv{U_y}{z} \\
- \displaystyle\pdv{U_z}{x} & \displaystyle\pdv{U_z}{y} & \displaystyle\pdv{U_z}{z}
- \end{bmatrix}
- =
- \begin{bmatrix}
- v_x \displaystyle\pdv{U_x}{x} & v_y \displaystyle\pdv{U_x}{y} & v_z \displaystyle\pdv{U_x}{z} \\
- v_x \displaystyle\pdv{U_y}{x} & v_y \displaystyle\pdv{U_y}{y} & v_z \displaystyle\pdv{U_y}{z} \\
- v_x \displaystyle\pdv{U_z}{x} & v_y \displaystyle\pdv{U_z}{y} & v_z \displaystyle\pdv{U_z}{z}
- \end{bmatrix}
-\end{aligned}$$
-
## References
diff --git a/source/know/concept/orthogonal-curvilinear-coordinates/index.md b/source/know/concept/orthogonal-curvilinear-coordinates/index.md
index 675b83a..c7299ee 100644
--- a/source/know/concept/orthogonal-curvilinear-coordinates/index.md
+++ b/source/know/concept/orthogonal-curvilinear-coordinates/index.md
@@ -910,131 +910,6 @@ Dot-multiplying by $$\vu{e}_j$$ isolates the $$c_j$$-component and gives the des
-## Divergence of a tensor
-
-It also possible to take the divergence of a 2nd-order tensor $$\overline{\overline{\mathbf{T}}}$$,
-yielding a vector with these components in $$(c_1, c_2, c_3)$$:
-
-$$\begin{aligned}
- \boxed{
- (\nabla \cdot \overline{\overline{\mathbf{T}}})_j
- = \sum_{k} \frac{1}{h_k} \pdv{T_{kj}}{c_k}
- + \sum_{k \neq j} \frac{T_{jk}}{h_j h_k} \pdv{h_j}{c_k}
- - \sum_{k \neq j} \frac{T_{kk}}{h_j h_k} \pdv{h_k}{c_j}
- + \sum_{k} \sum_{l \neq k} \frac{T_{lj}}{h_k h_l} \pdv{h_k}{c_l}
- }
-\end{aligned}$$
-
-{% include proof/start.html id="proof-div-tensor" -%}
-From our earlier calculation of $$\nabla f$$,
-we know how to express the del $$\nabla$$ in $$(c_1, c_2, c_3)$$.
-Now we simply take the dot product and evaluate:
-
-$$\begin{aligned}
- \nabla \cdot \overline{\overline{\mathbf{T}}}
- &= \bigg( \vu{e}_1 \frac{1}{h_1} \pdv{}{c_1} + \vu{e}_2 \frac{1}{h_2} \pdv{}{c_2} + \vu{e}_3 \frac{1}{h_3} \pdv{}{c_3} \bigg)
- \\
- &\quad\:\:\: \cdot \Big( T_{11} \vu{e}_1 \vu{e}_1 + T_{12} \vu{e}_1 \vu{e}_2 + T_{13} \vu{e}_1 \vu{e}_3
- \\
- &\qquad + T_{21} \vu{e}_2 \vu{e}_1 + T_{22} \vu{e}_2 \vu{e}_2 + T_{23} \vu{e}_2 \vu{e}_3
- \\
- &\qquad + T_{31} \vu{e}_3 \vu{e}_1 + T_{32} \vu{e}_3 \vu{e}_2 + T_{33} \vu{e}_3 \vu{e}_3 \Big)
- \\
- &= \bigg( \sum_{j} \vu{e}_j \frac{1}{h_j} \pdv{}{c_j} \bigg) \cdot \bigg( \sum_{kl} T_{kl} \vu{e}_k \vu{e}_l \bigg)
- \\
- &= \sum_{jkl} \vu{e}_j \cdot \frac{1}{h_j} \pdv{}{c_j} (T_{kl} \vu{e}_k \vu{e}_l)
-\end{aligned}$$
-
-We apply the product rule of differentiation
-and use that $$\vb{c} \cdot (\vb{a} \vb{b}) = (\vb{c} \cdot \vb{a}) \vb{b}$$:
-
-$$\begin{aligned}
- \nabla \cdot \overline{\overline{\mathbf{T}}}
- &= \sum_{jkl} \bigg( (\vu{e}_j \cdot \vu{e}_k) \frac{1}{h_j} \pdv{T_{kl}}{c_j} \vu{e}_l
- + (\vu{e}_j \cdot \vu{e}_k) \frac{T_{kl}}{h_j} \pdv{\vu{e}_l}{c_j}
- + \Big( \vu{e}_j \cdot \pdv{\vu{e}_k}{c_j} \Big) \frac{T_{kl}}{h_j} \vu{e}_l \bigg)
- \\
- &= \sum_{jkl} \bigg( \delta_{jk} \frac{1}{h_j} \pdv{T_{kl}}{c_j} \vu{e}_l + \delta_{jk} \frac{T_{kl}}{h_j} \pdv{\vu{e}_l}{c_j}
- + \Big( \vu{e}_j \cdot \pdv{\vu{e}_k}{c_j} \Big) \frac{T_{kl}}{h_j} \vu{e}_l \bigg)
- \\
- &= \sum_{jl} \bigg( \frac{1}{h_j} \pdv{T_{jl}}{c_j} \vu{e}_l + \frac{T_{jl}}{h_j} \pdv{\vu{e}_l}{c_j}
- + \sum_{k} \Big( \vu{e}_j \cdot \pdv{\vu{e}_k}{c_j} \Big) \frac{T_{kl}}{h_j} \vu{e}_l \bigg)
-\end{aligned}$$
-
-Inserting our expressions for the derivatives of the basis vectors
-in the last term, we find:
-
-$$\begin{aligned}
- \nabla \cdot \overline{\overline{\mathbf{T}}}
- &= \sum_{jl} \bigg( \frac{1}{h_j} \pdv{T_{jl}}{c_j} \vu{e}_l + \frac{T_{jl}}{h_j} \pdv{\vu{e}_l}{c_j}
- + \sum_{k} \vu{e}_j \cdot
- \Big( \frac{1}{h_k} \pdv{h_j}{c_k} \vu{e}_j - \delta_{jk} \sum_{m} \frac{1}{h_m} \pdv{h_k}{c_m} \vu{e}_m \Big) \frac{T_{kl}}{h_j} \vu{e}_l \bigg)
- \\
- &= \sum_{jl} \bigg( \frac{1}{h_j} \pdv{T_{jl}}{c_j} \vu{e}_l + \frac{T_{jl}}{h_j} \pdv{\vu{e}_l}{c_j}
- + \sum_{k} \frac{T_{kl}}{h_j h_k} \pdv{h_j}{c_k} \vu{e}_l
- - \sum_{m} (\vu{e}_j \cdot \vu{e}_m) \frac{T_{jl}}{h_j h_m} \pdv{h_j}{c_m} \vu{e}_l \bigg)
- \\
- &= \sum_{jl} \bigg( \frac{1}{h_j} \pdv{T_{jl}}{c_j} \vu{e}_l + \frac{T_{jl}}{h_j} \pdv{\vu{e}_l}{c_j}
- + \sum_{k} \frac{T_{kl}}{h_j h_k} \pdv{h_j}{c_k} \vu{e}_l - \frac{T_{jl}}{h_j h_j} \pdv{h_j}{c_j} \vu{e}_l \bigg)
- \\
- &= \sum_{jl} \bigg( \frac{1}{h_j} \pdv{T_{jl}}{c_j} \vu{e}_l + \frac{T_{jl}}{h_j} \pdv{\vu{e}_l}{c_j}
- + \sum_{k \neq j} \frac{T_{kl}}{h_j h_k} \pdv{h_j}{c_k} \vu{e}_l \bigg)
-\end{aligned}$$
-
-Where we noticed that the latter two terms cancel out if $$k = j$$.
-Next, rewriting $$\ipdv{\vu{e}_l}{c_j}$$:
-
-$$\begin{aligned}
- \nabla \cdot \overline{\overline{\mathbf{T}}}
- &= \sum_{jl} \bigg( \frac{1}{h_j} \pdv{T_{jl}}{c_j} \vu{e}_l
- + \frac{T_{jl}}{h_j} \Big( \frac{1}{h_l} \pdv{h_j}{c_l} \vu{e}_j - \delta_{jl} \sum_{m} \frac{1}{h_m} \pdv{h_l}{c_m} \vu{e}_m \Big)
- + \sum_{k \neq j} \frac{T_{kl}}{h_j h_k} \pdv{h_j}{c_k} \vu{e}_l \bigg)
- \\
- &= \sum_{jl} \bigg( \frac{1}{h_j} \pdv{T_{jl}}{c_j} \vu{e}_l
- + \frac{T_{jl}}{h_j h_l} \pdv{h_j}{c_l} \vu{e}_j - \delta_{jl} \sum_{m} \frac{T_{jl}}{h_j h_m} \pdv{h_l}{c_m} \vu{e}_m
- + \sum_{k \neq j} \frac{T_{kl}}{h_j h_k} \pdv{h_j}{c_k} \vu{e}_l \bigg)
- \\
- &= \sum_{jl} \frac{1}{h_j} \pdv{T_{jl}}{c_j} \vu{e}_l
- + \sum_{jl} \frac{T_{jl}}{h_j h_l} \pdv{h_j}{c_l} \vu{e}_j
- - \sum_{jm} \frac{T_{jj}}{h_j h_m} \pdv{h_j}{c_m} \vu{e}_m
- + \sum_{jl} \sum_{k \neq j} \frac{T_{kl}}{h_j h_k} \pdv{h_j}{c_k} \vu{e}_l
-\end{aligned}$$
-
-Renaming the indices such that each term contains $$\vu{e}_l$$,
-we arrive at the full result:
-
-$$\begin{aligned}
- \nabla \cdot \overline{\overline{\mathbf{T}}}
- &= \sum_{jl} \bigg( \frac{1}{h_j} \pdv{T_{jl}}{c_j}
- + \frac{T_{lj}}{h_j h_l} \pdv{h_l}{c_j}
- - \frac{T_{jj}}{h_j h_l} \pdv{h_j}{c_l}
- + \sum_{k \neq j} \frac{T_{kl}}{h_j h_k} \pdv{h_j}{c_k} \bigg) \vu{e}_l
-\end{aligned}$$
-
-To isolate the $$c_m$$-component, we dot-multiply by $$\vu{e}_m$$
-and resolve the Kronecker delta $$\delta_{lm}$$:
-
-$$\begin{aligned}
- (\nabla \cdot \overline{\overline{\mathbf{T}}})_m
- &= (\nabla \cdot \overline{\overline{\mathbf{T}}}) \cdot \vu{e}_m
- \\
- &= \sum_{jl} \delta_{lm} \bigg( \frac{1}{h_j} \pdv{T_{jl}}{c_j}
- + \frac{T_{lj}}{h_j h_l} \pdv{h_l}{c_j}
- - \frac{T_{jj}}{h_j h_l} \pdv{h_j}{c_l}
- + \sum_{k \neq j} \frac{T_{kl}}{h_j h_k} \pdv{h_j}{c_k} \bigg)
- \\
- &= \sum_{j} \frac{1}{h_j} \pdv{T_{jm}}{c_j}
- + \sum_{j} \frac{T_{mj}}{h_j h_m} \pdv{h_m}{c_j}
- - \sum_{j} \frac{T_{jj}}{h_j h_m} \pdv{h_j}{c_m}
- + \sum_{j} \sum_{k \neq j} \frac{T_{km}}{h_j h_k} \pdv{h_j}{c_k}
-\end{aligned}$$
-
-The second and third terms cancel out for $$j = m$$,
-so we can sum over $$j \neq m$$ instead.
-{% include proof/end.html id="proof-div-tensor" %}
-
-
-
## Laplacian of a vector
The Laplacian $$\nabla^2 \vb{V}$$ of a vector $$\vb{V}$$
@@ -1168,6 +1043,131 @@ Which gives the desired formula after some simple index renaming and rearranging
+## Divergence of a tensor
+
+It also possible to take the divergence of a 2nd-order tensor $$\overline{\overline{\mathbf{T}}}$$,
+yielding a vector with these components in $$(c_1, c_2, c_3)$$:
+
+$$\begin{aligned}
+ \boxed{
+ (\nabla \cdot \overline{\overline{\mathbf{T}}})_j
+ = \sum_{k} \frac{1}{h_k} \pdv{T_{kj}}{c_k}
+ + \sum_{k \neq j} \frac{T_{jk}}{h_j h_k} \pdv{h_j}{c_k}
+ - \sum_{k \neq j} \frac{T_{kk}}{h_j h_k} \pdv{h_k}{c_j}
+ + \sum_{k} \sum_{l \neq k} \frac{T_{lj}}{h_k h_l} \pdv{h_k}{c_l}
+ }
+\end{aligned}$$
+
+{% include proof/start.html id="proof-div-tensor" -%}
+From our earlier calculation of $$\nabla f$$,
+we know how to express the del $$\nabla$$ in $$(c_1, c_2, c_3)$$.
+Now we simply take the dot product and evaluate:
+
+$$\begin{aligned}
+ \nabla \cdot \overline{\overline{\mathbf{T}}}
+ &= \bigg( \vu{e}_1 \frac{1}{h_1} \pdv{}{c_1} + \vu{e}_2 \frac{1}{h_2} \pdv{}{c_2} + \vu{e}_3 \frac{1}{h_3} \pdv{}{c_3} \bigg)
+ \\
+ &\quad\:\:\: \cdot \Big( T_{11} \vu{e}_1 \vu{e}_1 + T_{12} \vu{e}_1 \vu{e}_2 + T_{13} \vu{e}_1 \vu{e}_3
+ \\
+ &\qquad + T_{21} \vu{e}_2 \vu{e}_1 + T_{22} \vu{e}_2 \vu{e}_2 + T_{23} \vu{e}_2 \vu{e}_3
+ \\
+ &\qquad + T_{31} \vu{e}_3 \vu{e}_1 + T_{32} \vu{e}_3 \vu{e}_2 + T_{33} \vu{e}_3 \vu{e}_3 \Big)
+ \\
+ &= \bigg( \sum_{j} \vu{e}_j \frac{1}{h_j} \pdv{}{c_j} \bigg) \cdot \bigg( \sum_{kl} T_{kl} \vu{e}_k \vu{e}_l \bigg)
+ \\
+ &= \sum_{jkl} \vu{e}_j \cdot \frac{1}{h_j} \pdv{}{c_j} (T_{kl} \vu{e}_k \vu{e}_l)
+\end{aligned}$$
+
+We apply the product rule of differentiation
+and use that $$\vb{c} \cdot (\vb{a} \vb{b}) = (\vb{c} \cdot \vb{a}) \vb{b}$$:
+
+$$\begin{aligned}
+ \nabla \cdot \overline{\overline{\mathbf{T}}}
+ &= \sum_{jkl} \bigg( (\vu{e}_j \cdot \vu{e}_k) \frac{1}{h_j} \pdv{T_{kl}}{c_j} \vu{e}_l
+ + (\vu{e}_j \cdot \vu{e}_k) \frac{T_{kl}}{h_j} \pdv{\vu{e}_l}{c_j}
+ + \Big( \vu{e}_j \cdot \pdv{\vu{e}_k}{c_j} \Big) \frac{T_{kl}}{h_j} \vu{e}_l \bigg)
+ \\
+ &= \sum_{jkl} \bigg( \delta_{jk} \frac{1}{h_j} \pdv{T_{kl}}{c_j} \vu{e}_l + \delta_{jk} \frac{T_{kl}}{h_j} \pdv{\vu{e}_l}{c_j}
+ + \Big( \vu{e}_j \cdot \pdv{\vu{e}_k}{c_j} \Big) \frac{T_{kl}}{h_j} \vu{e}_l \bigg)
+ \\
+ &= \sum_{jl} \bigg( \frac{1}{h_j} \pdv{T_{jl}}{c_j} \vu{e}_l + \frac{T_{jl}}{h_j} \pdv{\vu{e}_l}{c_j}
+ + \sum_{k} \Big( \vu{e}_j \cdot \pdv{\vu{e}_k}{c_j} \Big) \frac{T_{kl}}{h_j} \vu{e}_l \bigg)
+\end{aligned}$$
+
+Inserting our expressions for the derivatives of the basis vectors
+in the last term, we find:
+
+$$\begin{aligned}
+ \nabla \cdot \overline{\overline{\mathbf{T}}}
+ &= \sum_{jl} \bigg( \frac{1}{h_j} \pdv{T_{jl}}{c_j} \vu{e}_l + \frac{T_{jl}}{h_j} \pdv{\vu{e}_l}{c_j}
+ + \sum_{k} \vu{e}_j \cdot
+ \Big( \frac{1}{h_k} \pdv{h_j}{c_k} \vu{e}_j - \delta_{jk} \sum_{m} \frac{1}{h_m} \pdv{h_k}{c_m} \vu{e}_m \Big) \frac{T_{kl}}{h_j} \vu{e}_l \bigg)
+ \\
+ &= \sum_{jl} \bigg( \frac{1}{h_j} \pdv{T_{jl}}{c_j} \vu{e}_l + \frac{T_{jl}}{h_j} \pdv{\vu{e}_l}{c_j}
+ + \sum_{k} \frac{T_{kl}}{h_j h_k} \pdv{h_j}{c_k} \vu{e}_l
+ - \sum_{m} (\vu{e}_j \cdot \vu{e}_m) \frac{T_{jl}}{h_j h_m} \pdv{h_j}{c_m} \vu{e}_l \bigg)
+ \\
+ &= \sum_{jl} \bigg( \frac{1}{h_j} \pdv{T_{jl}}{c_j} \vu{e}_l + \frac{T_{jl}}{h_j} \pdv{\vu{e}_l}{c_j}
+ + \sum_{k} \frac{T_{kl}}{h_j h_k} \pdv{h_j}{c_k} \vu{e}_l - \frac{T_{jl}}{h_j h_j} \pdv{h_j}{c_j} \vu{e}_l \bigg)
+ \\
+ &= \sum_{jl} \bigg( \frac{1}{h_j} \pdv{T_{jl}}{c_j} \vu{e}_l + \frac{T_{jl}}{h_j} \pdv{\vu{e}_l}{c_j}
+ + \sum_{k \neq j} \frac{T_{kl}}{h_j h_k} \pdv{h_j}{c_k} \vu{e}_l \bigg)
+\end{aligned}$$
+
+Where we noticed that the latter two terms cancel out if $$k = j$$.
+Next, rewriting $$\ipdv{\vu{e}_l}{c_j}$$:
+
+$$\begin{aligned}
+ \nabla \cdot \overline{\overline{\mathbf{T}}}
+ &= \sum_{jl} \bigg( \frac{1}{h_j} \pdv{T_{jl}}{c_j} \vu{e}_l
+ + \frac{T_{jl}}{h_j} \Big( \frac{1}{h_l} \pdv{h_j}{c_l} \vu{e}_j - \delta_{jl} \sum_{m} \frac{1}{h_m} \pdv{h_l}{c_m} \vu{e}_m \Big)
+ + \sum_{k \neq j} \frac{T_{kl}}{h_j h_k} \pdv{h_j}{c_k} \vu{e}_l \bigg)
+ \\
+ &= \sum_{jl} \bigg( \frac{1}{h_j} \pdv{T_{jl}}{c_j} \vu{e}_l
+ + \frac{T_{jl}}{h_j h_l} \pdv{h_j}{c_l} \vu{e}_j - \delta_{jl} \sum_{m} \frac{T_{jl}}{h_j h_m} \pdv{h_l}{c_m} \vu{e}_m
+ + \sum_{k \neq j} \frac{T_{kl}}{h_j h_k} \pdv{h_j}{c_k} \vu{e}_l \bigg)
+ \\
+ &= \sum_{jl} \frac{1}{h_j} \pdv{T_{jl}}{c_j} \vu{e}_l
+ + \sum_{jl} \frac{T_{jl}}{h_j h_l} \pdv{h_j}{c_l} \vu{e}_j
+ - \sum_{jm} \frac{T_{jj}}{h_j h_m} \pdv{h_j}{c_m} \vu{e}_m
+ + \sum_{jl} \sum_{k \neq j} \frac{T_{kl}}{h_j h_k} \pdv{h_j}{c_k} \vu{e}_l
+\end{aligned}$$
+
+Renaming the indices such that each term contains $$\vu{e}_l$$,
+we arrive at the full result:
+
+$$\begin{aligned}
+ \nabla \cdot \overline{\overline{\mathbf{T}}}
+ &= \sum_{jl} \bigg( \frac{1}{h_j} \pdv{T_{jl}}{c_j}
+ + \frac{T_{lj}}{h_j h_l} \pdv{h_l}{c_j}
+ - \frac{T_{jj}}{h_j h_l} \pdv{h_j}{c_l}
+ + \sum_{k \neq j} \frac{T_{kl}}{h_j h_k} \pdv{h_j}{c_k} \bigg) \vu{e}_l
+\end{aligned}$$
+
+To isolate the $$c_m$$-component, we dot-multiply by $$\vu{e}_m$$
+and resolve the Kronecker delta $$\delta_{lm}$$:
+
+$$\begin{aligned}
+ (\nabla \cdot \overline{\overline{\mathbf{T}}})_m
+ &= (\nabla \cdot \overline{\overline{\mathbf{T}}}) \cdot \vu{e}_m
+ \\
+ &= \sum_{jl} \delta_{lm} \bigg( \frac{1}{h_j} \pdv{T_{jl}}{c_j}
+ + \frac{T_{lj}}{h_j h_l} \pdv{h_l}{c_j}
+ - \frac{T_{jj}}{h_j h_l} \pdv{h_j}{c_l}
+ + \sum_{k \neq j} \frac{T_{kl}}{h_j h_k} \pdv{h_j}{c_k} \bigg)
+ \\
+ &= \sum_{j} \frac{1}{h_j} \pdv{T_{jm}}{c_j}
+ + \sum_{j} \frac{T_{mj}}{h_j h_m} \pdv{h_m}{c_j}
+ - \sum_{j} \frac{T_{jj}}{h_j h_m} \pdv{h_j}{c_m}
+ + \sum_{j} \sum_{k \neq j} \frac{T_{km}}{h_j h_k} \pdv{h_j}{c_k}
+\end{aligned}$$
+
+The second and third terms cancel out for $$j = m$$,
+so we can sum over $$j \neq m$$ instead.
+{% include proof/end.html id="proof-div-tensor" %}
+
+
+
## References
1. M.L. Boas,
*Mathematical methods in the physical sciences*, 2nd edition,
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.
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