First major update of src/2_coordinates.md

src/2_coordinates.md

@@ -35,7 +35,7 @@ space*.)
 In mathematics, we are often dealing with so-called *infinitesimally* small
 distances, for example in the definition of derivatives and integrals.
 In Cartesian coordinates, the expressions for infinitesimal distances $ds$ and
-infinitesimal volumes $dV$ are given as
+infinitesimal volumes $dV$ are given as:
 
 !!! info "Segment and volume element in n-dimensional Cartesian coordinates"
     $$ds = \sqrt{dx_1^2 + dx_2^2 + \ldots + dx_n^2}$$
@@ -43,35 +43,37 @@ infinitesimal volumes $dV$ are given as
 
 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:
-$\int dV = \idotsint dx_1 dx_2 \ldots dx_N$.
+$$\int dV = \idotsint dx_1 dx_2 \ldots dx_N$$.
 
-Cartesian coordinates are used a lot. They are particularly suitable for
+Cartesian coordinates are used a lot and they are particularly suitable for
 infinite spaces or for rectangular volumes.
 
-![image](figures/Coordinates_5_1.svg)
-
+<figure markdown>
+  ![image](figures/Coordinates_5_1.svg)
+  <figcaption>An example of a vector drawn in a 2D Cartesian plane</figcaption>
+</figure>
 
 ### 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
-circular drum, polar coordinates turn out very convenient. These are
-defined for a two-dimensional space (a plane). When using such
-coordinates, a position on the plane is characterised by two
-coordinates: the *distance* $r$ from the point to the origin and the
-angle ($\varphi$). This is the angle between the line connecting the
-point to the origin and the $x$-axis.
-
-Note that each Cartesian coordinate has a *dimension* of length; in
-polar coordinates, the radius $r$ has a dimension of *length*, whereas
+It often turns out that a change to a different type of coordinate
+system makes mathematics easier. For example, if you want to describe vibrations of a
+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.
+
+Note that each Cartesian coordinate has a *dimension* of length. 
+In polar coordinates, the radius $r$ has a dimension of *length*, but
 the angular coordinate $\varphi$ is dimensionless.
 
-![image](figures/Coordinates_7_0.svg)
+<figure markdown>
+  ![image](figures/Coordinates_7_0.svg)
+  <figcaption>In this example of a polar plot, you can distinguish the radial coordinate (0.2, 0.4 etc.)
+from the angular one expressed in degrees ($0^\circ$, $45^\circ$ etc.).</figcaption>
+</figure>
 
-In this plot you can distinguish the radial coordinate (0.2, 0.4 etc.)
-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*
@@ -81,19 +83,23 @@ follows:
 $$\begin{equation} x = r \cos\varphi; \end{equation}$$
 $$\begin{equation} y = r \sin \varphi.\end{equation}$$
 
-![image](figures/Coordinates_9_0.svg)
+<figure markdown>
+  ![image](figures/Coordinates_9_0.svg)
+  <figcaption></figcaption>
+</figure>
 
-The inverse relation is given as
+The inverse relation is given as:
 
-$$\begin{equation} r=\sqrt{x^2 + y^2}; \label{rxy}\end{equation}$$
-$$\begin{equation} \varphi=\begin{cases}
-  \arctan(y/x) & \text{$x>0$,}\\
-  \pi + \arctan(y/x) & \text{$x<0$ and $y>0$,}\\
-  -\pi + \arctan(y/x) & \text{$x<0$ and $y<0$.}
-  \end{cases} \label{phixy}\end{equation}$$
+!!! info "Inverse relation between polar and Cartesian coordinate systems"
+    $$\begin{equation} r=\sqrt{x^2 + y^2}; \label{rxy}\end{equation}$$
+    \begin{equation} \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} \label{phixy}\end{equation}
 
 The last formula for $\varphi$ warrants a closer explanation: It is easy
-to see that $\tan(\varphi)=y/x$ - but this is not a unique relation, due to
+to see that $\tan(\varphi)=y/x$, but this is not a unique relation, due to
 the fact that the $\tan$ has different branches. Convince yourself that
 the expression above is correct for all the four sectors!
 
@@ -128,7 +134,8 @@ We can use the same arguments also for the area: since the different
 segments are approximately perpendicular, we find the area by simply
 multiplying them:
 
-$$dA = r dr d\varphi.$$
+!!! info "Surface element in polar coordinates
+    	$$dA = r dr d\varphi.$$
 
 This is an important formula to remember for integrating in polar
 coordinates!  The extra $r$ that appears here can be intuitively
@@ -150,7 +157,7 @@ understood: the area swept by an angle difference $d\varphi$
 
     which is indeed the area of a circle with radius $r_0$.
 
-## Converting derivatives between coordinate systems
+## 2.2. Converting derivatives between coordinate systems
 
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