Add lecture notes for coordinates

src/coordinates.md

@@ -10,13 +10,33 @@ title: Coordinates
 
 The most common coordinates are *Cartesian coordinates*, where we use a
 number $n$ of perpendicular axes. The coordinates corresponding to these
-axes are $x_j$ where $j=1, \ldots, n$. The distance $\Delta s$ between
-two points $(x_1, x_2, \ldots, x_n)$ and $(x'_1, x'_2, \ldots, x'_n)$ is
-given by
+axes are $x_j$ where $j=1, \ldots, n$.
+
+Cartesian coordinates are simple, as the coordinate axis are simply
+straight lines and perpendicular to each other. Due to this, it is
+very easy to do calculations in Cartesian coordinates. For example,
+the distance $\Delta s$ between two points $(x_1, x_2, \ldots, x_n)$
+and $(x'_1, x'_2, \ldots, x'_n)$ is easily computed as
+
 $$\Delta s^2 = (x'_1 - x_1)^2 + (x'_2 - x_2)^2 + \ldots + (x'_n - x_n)^2.$$
-The expression on the right hand side can be written using a summation
-sign $\sum$: $$\Delta s^2 = \sum_{i=1}^n (x'_i - x_i)^2.$$ A space with
-such a distance definition is called a *Euclidean space*.
+
+(A space with such a distance definition is called a *Euclidean
+space*.)
+
+In mathematics, we are often dealing with so-called *infinitesimally* small
+distances, for example in the definition of derivatives and integrals.
+In Cartesion coordinates the expressions for infinitesimal distances $ds$ and
+infinitesimal volumes $dV$ are given as
+
+$$ds = \sqrt{dx_1^2 + dx_2^2 + \ldots + dx_n^2}$$
+
+and
+
+$$dV = dx_1 dx_2 \ldots dx_N.$$
+
+The formula for $dV$ also indicates that in Cartesian coordinates, the integral
+over a volume can be expressed as individual integrals over all coordinate directions:
+$\int dV = \idotsint dx_1 dx_2 \ldots dx_N$.
 
 Cartesian coordinates are used a lot. They are particularly suitable for
 infinite spaces or for rectangular volumes.