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Commit 9d111886 authored by Maciej Topyla's avatar Maciej Topyla
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Update src/2_coordinates.md

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......@@ -305,12 +305,15 @@ The inverse transformation is easy to find:
$$\theta = \arccos(z/\sqrt{x^2+y^2+z^2})$$
$$\phi = \begin{cases} \arctan(y/x) &{\rm for ~} x>0; \\
\pi + \arctan(y/x) & {\rm for ~} x<0 {\rm ~ and ~} y>0;\\
-\pi + \arctan(y/x) &{\rm ~ for ~} x<0 {\rm ~ and ~} y<0.
-\pi + \arctan(y/x) &{\rm for ~} x<0 {\rm ~ and ~} y<0.
\end{cases}$$
These relations can be derived from the following figure:
![image](figures/Coordinates_17_0.svg)
<figure markdown>
![image](figures/Coordinates_17_0.svg)
<figcaption></figcaption>
</figure>
The distance related to a change in the spherical coordinates is
calculated using Pythagoras’ theorem. The length $ds$ of a short segment
......@@ -328,7 +331,11 @@ $$ds^2 = r^2 \left(\sin^2 \vartheta d\varphi^2 + d\vartheta^2\right) + dr^2.$$
The picture below shows the geometry behind the calculation of this
displacement.
![image](figures/Coordinates_19_0.svg)
<figure markdown>
![image](figures/Coordinates_19_0.svg)
<figcaption></figcaption>
</figure>
From these arguments we can again also find the volume element, it is
here given as
......
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