diff --git a/src/2_coordinates.md b/src/2_coordinates.md index dedcd244f491d549dcf73a56c10d12e8972629da..4cd5a3ac12bf053b3696967acc5a67cd370e8b37 100644 --- a/src/2_coordinates.md +++ b/src/2_coordinates.md @@ -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: - +<figure markdown> +  + <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. - +<figure markdown> +  + <figcaption></figcaption> +</figure> + From these arguments we can again also find the volume element, it is here given as