diff --git a/src/2_coordinates.md b/src/2_coordinates.md
index 1d7e964d703426f1beed86f499fe8ba85efe3e33..6022b3c0511ac8634f05aecacaa5734085b599f5 100644
--- a/src/2_coordinates.md
+++ b/src/2_coordinates.md
@@ -62,7 +62,7 @@ system makes mathematics easier. For example, if you want to describe vibrations
circular drum, polar coordinates become very convenient. These are
defined for a two-dimensional space (a plane). The position on this plane is characterised by two
coordinates: the *distance* $r$ between the point and the origin, and by the
-angle ($\varphi$) between the line connecting the point to the origin and the $x$-axis.
+angle ($\varphi$) between the line connecting the point to the origin and the $x$-axis. The radius is therefore always a non-negative number $r \geq 0$, and the range for the polar angle is $\varphi \in \left< 0,2\pi \right)$
Note that each Cartesian coordinate has a *dimension* of length.
In polar coordinates, the radius $r$ has a dimension of *length*, but
@@ -247,7 +247,7 @@ problem.
Cylindrical coordinates are defined straightforwardly: we use polar
coordinates $r$ and $\varphi$ in the $xy$ plane, and the distance $z$
-along the symmetry-axis as the third coordinate. If the axis system is
+along the symmetry-axis as the third coordinate. The radius is therefore again always defined as a non-negative number $r \in \left<0, \infty \right)$, and the range for the azimuthal angle is analogically $\varphi \in \left< 0,2\pi \right)$. The *height* $z$ along the cylinder axis can take any real value, hence $z \in \mathbb{R}$ . If the axis system is
chosen in physical space, we have two coordinates which have the
dimension of a distance: $r$ and $z$. The other coordinate,
$\varphi$ is of course dimensionless.
@@ -283,30 +283,35 @@ sphere which is centered at the origin:
<figure markdown>
- <figcaption>The position of a point on the sphere is specified using the radius $r$ and two angles
-$\theta$ and $\phi$</figcaption>
+ <figcaption>The position of a point on the sphere is specified using the radius $r$ and two angles (azimuthal)
+$\varphi$ and (polar) $\theta$ in the given order </figcaption>
</figure>
+!!! warning "Parameter ranges in spherical coordinates"
+ - The radius ($r$) is defined for $r \in \left<0, \infty \right)$ </br>
+ - The azimuthal angle ($\varphi$) has the range of $\varphi \in \left< 0, 2\pi \right)$ </br>
+ - The polar angle ($\theta$) has the range $\theta \in \left<0, \pi \right>$
+
!!! warning
In mathematics, the angles are often labeled the other way
- around: there, $\phi$ is used for the angle between a line running from
+ around: there, $\varphi$ is used for the angle between a line running from
the origin to the point of interest and the $z$-axis, and $\theta$ for
the angle of the projection of that line with the $x$-axis. The
convention used here is customary in physics.
The relation between Cartesian and spherical coordinates is defined by:
!!! info "The relation between Cartesian and spherical coordinates"
- $$x = r \cos \varphi \sin \vartheta$$
- $$y = r \sin\varphi \sin \vartheta$$ $$z = r \cos\vartheta$$
+ $$x = r \cos \varphi \sin \theta$$
+ $$y = r \sin\varphi \sin \theta$$ $$z = r \cos\theta$$
The inverse transformation is easy to find:
!!! info "The inverse relation between Cartesian and spherical coordinates"
- $$r = \sqrt{x^2+y^2+z^2}$$
- $$\theta = \arccos(z/\sqrt{x^2+y^2+z^2})$$
- $$\phi = \begin{cases} \arctan(y/x) &{\rm for ~} x>0; \\
+ $$r = \sqrt{x^2+y^2+z^2}, \qquad r \in \left<0, \infty \right)$$
+ $$\varphi = \begin{cases} \arctan(y/x) &{\rm for ~} x>0; \\
\pi + \arctan(y/x) & {\rm for ~} x<0 {\rm ~ and ~} y>0;\\
-\pi + \arctan(y/x) &{\rm for ~} x<0 {\rm ~ and ~} y<0.
- \end{cases}$$
+ \end{cases}, \qquad \varphi \in \left< 0,2\pi \right)$$
+ $$\theta = \arccos(z/\sqrt{x^2+y^2+z^2}), \qquad \theta \in \left< 0,\pi \right> $$
These relations can be derived from the following figure:
@@ -318,15 +323,15 @@ These relations can be derived from the following figure:
The distance related to a change in the spherical coordinates is
calculated using Pythagoras’ theorem. The length $ds$ of a short segment
on the sphere with radius $r$ corresponding to the changes in the polar
-angles of $d\vartheta$ and $d\varphi$ is given as
-$$dl^2 = r^2 \left(\sin^2 \vartheta d\varphi^2 + d\vartheta^2\right).$$
+angles of $d\theta$ and $d\varphi$ is given as
+$$dl^2 = r^2 \left(\sin^2 \theta d\varphi^2 + d\theta^2\right).$$
In order to verify this, it is important to realize that all points with
-*the same* coordinate $\vartheta$ span a circle in a horizontal plane
-with a radius $r\sin\vartheta$ as shown in the figure below.
+*the same* coordinate $\theta$ span a circle in a horizontal plane
+with a radius $r\sin\theta$ as shown in the figure below.
From this, we can also infer that for a segment with a radial component
$dr$ in addition to the displacement on the surface of the sphere, the combined displacement is:
-$$ds^2 = r^2 \left(\sin^2 \vartheta d\varphi^2 + d\vartheta^2\right) + dr^2.$$
+$$ds^2 = r^2 \left(\sin^2 \theta d\varphi^2 + d\theta^2\right) + dr^2.$$
The picture below shows the geometry behind the calculation of this
displacement.
@@ -348,35 +353,39 @@ here given as
We have discussed four different coordinate systems:
1. !!! tip "Cartesian coordinates"
- $${\bf r} = (x_1, \ldots, x_n).$$ This systems can be
- used for any dimension $n$. It is particularly convenient for: infinite spaces, systems
+ $${\bf r} = (x_1, \ldots, x_n)$$
+ $$ x_{n} \in \mathbb{R}$$
+ This systems can be used for any dimension $n$. It is particularly convenient for: infinite spaces, systems
with rectangular symmetry.
Distance between two points ${\bf r} = (x_1, \ldots, x_n)$ and
${\bf r}' = (x'_1, \ldots, x'_n)$:
$$\Delta s^2 = (x'_1 - x_1)^2 + (x'_2 - x_2)^2 + \ldots + (x'_n - x_n)^2.$$
2. !!! tip "Polar coordinates"
- $${\bf r} = (r, \phi).$$ This system can be used in two
- dimensions. It is particularly suitable for systems with circular symmetry or functions
+ $${\bf r} = (r, \varphi)$$
+ $$ r \in \left<0, \infty \right), \quad \varphi \in \left< 0,2\pi \right) $$
+ This system can be used in two dimensions. It is particularly suitable for systems with circular symmetry or functions
given in terms of these coordinates. <br/>
- Infinitesimal distance: $$ds^2 = dr^2 + r^2 d\phi^2.$$
+ Infinitesimal distance: $$ds^2 = dr^2 + r^2 d\varphi^2.$$
Infinitesimal area: $$dA = r dr d\varphi.$$
3. !!! tip "Cylindrical coordinates"
- $${\bf r} = (r, \phi, z).$$ This system can be
- used in three dimensions. It is particularly suitable for systems with axial symmetry
+ $${\bf r} = (r, \varphi, z)$$
+ $$ r \in \left<0, \infty \right), \quad \varphi \in \left< 0,2\pi \right), \quad z \in \mathbb{R} $$
+ This system can be used in three dimensions. It is particularly suitable for systems with axial symmetry
or functions given in terms of these coordinates. <br/>
- Infinitesimal distance: $$ds^2 = dr^2 + r^2 d\phi^2 + dz^2.$$
+ Infinitesimal distance: $$ds^2 = dr^2 + r^2 d\varphi^2 + dz^2.$$
Infinitesimal volume: $$dV = r dr d\varphi dz.$$
4. !!! tip "Spherical coordinates"
- $${\bf r} = (r, \theta, \phi).$$ This system can be
- used in three dimensions. It is particularly suitable for systems with spherical
+ $${\bf r} = (r, \varphi, \theta)$$
+ $$ r \in \left<0, \infty \right), \quad \varphi \in \left< 0,2\pi \right), \quad \theta \in \left< 0,\pi \right> $$
+ This system can be used in three dimensions. It is particularly suitable for systems with spherical
symmetry or functions given in terms of these coordinates. <br/>
Infinitesimal distance:
- $$ds^2 =r^2 (\sin^2 \theta d\phi^2 + d\theta^2) + dr^2 .$$
+ $$ds^2 =r^2 (\sin^2 \theta d\varphi^2 + d\theta^2) + dr^2 .$$
Infinitesimal volume:
- $$dV = r^2 \sin(\theta) dr d\theta d\varphi.$$
+ $$dV = r^2 \sin(\theta) dr d\varphi d\theta.$$
## 2.5. Problems
@@ -431,9 +440,9 @@ We have discussed four different coordinate systems:
In a similar fashion it can be shown that for spherical coordinates,
the Laplace operator acting on a function $\psi({\bf r})$ becomes:
- $$\begin{align} \nabla^2 \psi (r,\vartheta,\varphi) &=
- \frac{1}{r^2} \frac{\partial}{\partial r^2} \left( r^2 \frac{\partial \psi(r,\vartheta,\varphi)}{\partial r} \right) \\ &+ \frac{1}{r^2\sin^2\vartheta} \frac{\partial^2 \psi(r,\vartheta, \varphi)}{\partial \varphi^2} \\ &+ \frac{1}{r^2\sin\vartheta}
- \frac{\partial}{\partial \vartheta}\left( \sin\vartheta \frac{\partial\psi(r,\vartheta, \varphi)}{\partial \vartheta}\right).
+ $$\begin{align} \nabla^2 \psi (r,\varphi, \theta) &=
+ \frac{1}{r^2} \frac{\partial}{\partial r^2} \left( r^2 \frac{\partial \psi(r,\varphi,\theta)}{\partial r} \right) \\ &+ \frac{1}{r^2\sin^2\theta} \frac{\partial^2 \psi(r,\varphi, \theta)}{\partial \varphi^2} \\ &+ \frac{1}{r^2\sin\theta}
+ \frac{\partial}{\partial \theta}\left( \sin\theta \frac{\partial\psi(r,\varphi, \theta)}{\partial \theta}\right).
\end{align}$$
This is however even more tedious (you do not have to show this).