First major update of src/2_coordinates.md

src/2_coordinates.md

@@ -70,7 +70,7 @@ the angular coordinate $\varphi$ is dimensionless.
 
 <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>
+  <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>
 
 
@@ -87,10 +87,10 @@ $$\begin{equation} y = r \sin \varphi.\end{equation}$$
   <figcaption></figcaption>
 </figure>
 
-The inverse relation is given as:
+#### The inverse relation
 
 !!! info "Inverse relation between polar and Cartesian coordinate systems"
-    $$\begin{equation} r=\sqrt{x^2 + y^2}; \label{rxy}\end{equation}$$
+    \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 $y>0$,}\\
@@ -117,7 +117,10 @@ If we consider two points which are *very close*, the analysis
 simplifies however. We can use the geometry of the problem to find the
 distance (see the figure below).
 
-![image](figures/Coordinates_11_0.svg)
+<figure markdown>
+  ![image](figures/Coordinates_11_0.svg)
+  <figcaption></figcaption>
+</figure>
 
 When going from point 1 to point 2, we first traverse a small circular
 arc of radius $r_1$ and then we move a small distance radially outward
@@ -161,7 +164,7 @@ understood: the area swept by an angle difference $d\varphi$
 <iframe width="100%" height=315 src="https://www.youtube-nocookie.com/embed/NGQWGx71w98?rel=0" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
 
 
-Often, in physics important equations involve derivatives given in terms
+Important equations in physics often involve derivatives given in terms
 of Cartesian coordinates. One prominent example are equations of the form
 $$\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right)
 f(x, y) = \ldots.$$
@@ -169,7 +172,7 @@ The derivative operator $\left(\frac{\partial^2}{\partial x^2} +
 \frac{\partial^2}{\partial y^2}\right)$ is so common it has its own name:
 the Laplacian (here for two-dimensional space).
 
-Such an equation is universal, but for particular situations it might be
+This equation is universal, but for particular situations it might be
 advantageous to use a different coordinate system, such as polar coordinates
 for a system with rotational symmetry. The question then is: How does the
 corresponding equation look like in a different coordinate system?
@@ -178,7 +181,7 @@ There are different ways to find the answer. Here, we will focus on
 directly deriving the transformed equation through an explicit calculation
 involving the chain rule for a function of several variables.
 
-!!! info Chain rule for a multi-variable function
+!!! info "Chain rule for a multi-variable function"
     Let $f$ be a function of $n$ variables: $f(y_1, y_2, \ldots, y_n)$,
     as well as $g_i(x_1, x_2, \ldots, x_n)$ for $i=1,2,\ldots, n$. Then
 
@@ -192,7 +195,7 @@ of a function of $r, \varphi$ in terms of $x$?
 
 For this, we need to realize that there are relations between the coordinate systems.
 In particular, $r = r(x,y)$ and $\varphi = \varphi(x, y)$ as defined in equations
-\ref{rxy} and \ref{phixy}. In fact, we have been rather sloppy in our notation above,
+of the [inverse relations](#The-inverse-relation). In fact, we have been rather sloppy in our notation above,
 as the functions $f(x,y)$ and $f(r, \varphi)$ do not mean that I substitute $x=r$
 and $y=\varphi$! It is more precise to state that there are two diferent
 functions $f_\text{cart}(x,y)$ and $f_\text{polar}(r, \varphi)$ that are equivalent,