Add lecture notes for coordinates

src/coordinates.md

@@ -2,11 +2,7 @@
 title: Coordinates
 ---
 
-# Coordinates
-
-
-## Cartesian coordinates
-
+# Cartesian coordinates
 
 The most common coordinates are *Cartesian coordinates*, where we use a
 number $n$ of perpendicular axes. The coordinates corresponding to these
@@ -44,8 +40,9 @@ infinite spaces or for rectangular volumes.
 ![image](figures/Coordinates_5_1.svg)
 
 
-## Polar coordinates
+# Polar coordinates
 
+## Definition
 
 It often turns out useful to change to a different type of coordinate
 system. For example, if you want to describe the vibrations of a
@@ -68,10 +65,24 @@ from the angular one ($0^\circ$, $45^\circ$ etc.).
 The plot below shows a point on a curve with the polar coordinates
 $(r,\varphi)$ indicated. From this, we can see that the *Cartesian*
 coordinates $(x,y)$ of the point are related to the polar ones as
-follows: $$x = r \cos\varphi;$$ $$y = r \sin \varphi.$$
+follows:
+
+$$x = r \cos\varphi;$$
+$$y = r \sin \varphi.$$
 
 ![image](figures/Coordinates_9_0.svg)
 
+The inverse relation is given as
+
+$$r=\sqrt{x^2 + y^2};$$
+$$\varphi=\begin{cases}
+\arctan(y/x) & \text{$x>0$,}\\
+\pi + \arctan(y/x) & \text{$x<0$ and $y>0$,}\\
+-\pi + \arctan(y/x) & \text{$x<0$ and $y<0$.}
+\end{cases}$$
+
+## Distances and areas
+
 Now suppose we want to calculate the distance between two points, one
 with polar coordinates $(r_1, \varphi_1)$, and the other with
 $(r_2, \varphi_2)$. This looks like a difficult exercise. A convenient