First major update of src/2_coordinates.md

src/2_coordinates.md

@@ -351,35 +351,33 @@ We have discussed four different coordinate systems:
     $${\bf r} = (x_1, \ldots, x_n).$$ This systems can be
     used for any dimension $n$. It is particularly convenient for: infinite spaces, systems
     with rectangular symmetry.
-
     Distance between two points ${\bf r} = (x_1, \ldots, x_n)$ and
     ${\bf r}' = (x'_1, \ldots, x'_n)$:
     $$\Delta s^2 = (x'_1 - x_1)^2 + (x'_2 - x_2)^2 + \ldots + (x'_n - x_n)^2.$$
 
-2.  *Polar coordinates*: $${\bf r} = (r, \phi).$$ This system can be used in two
+2.  !!! tip "Polar coordinates" 
+    $${\bf r} = (r, \phi).$$ This system can be used in two
     dimensions. It is particularly suitable for systems with circular symmetry or functions
-    given in terms of these coordinates.
-
+    given in terms of these coordinates. <br/>
     Infinitesimal distance: $$ds^2 = dr^2 + r^2 d\phi^2.$$
     Infinitesimal area: $$dA = r dr d\varphi.$$
 
-3.  *Cylindrical coordinates*: $${\bf r} = (r, \phi, z).$$ Can be
-    used in three dimensions. Suitable for systems with axial symmetry
-    or functions given in terms of these coordinates.
-
+3.  !!! tip "Cylindrical coordinates" 
+    $${\bf r} = (r, \phi, z).$$ This system can be
+    used in three dimensions. It is particularly suitable for systems with axial symmetry
+    or functions given in terms of these coordinates. <br/>
     Infinitesimal distance: $$ds^2 = dr^2 + r^2 d\phi^2 + dz^2.$$
-    Infinitesimal volume:: $$dV = r dr d\varphi dz.$$
+    Infinitesimal volume: $$dV = r dr d\varphi dz.$$
 
-4.  *Spherical coordinates*: $${\bf r} = (r, \theta, \phi).$$ Can be
-    used in three dimensions. Suitable for systems with spherical
+4.  !!! tip "Spherical coordinates" 
+    $${\bf r} = (r, \theta, \phi).$$ This sysytem can be
+    used in three dimensions. It is particularly suitable for systems with spherical
     symmetry or functions given in terms of these coordinates.
-
-    Infinitesimal distance:
+    Infinitesimal distance: <br/>
     $$ds^2 =r^2 (\sin^2 \theta d\phi^2 + d\theta^2) +  dr^2 .$$
     Infinitesimal volume:
     $$dV = r^2 \sin(\theta) dr d\theta d\varphi.$$ 
 
-
 ## 2.5. Problems
 
 1.  [:grinning:] *Warm-up*
@@ -404,7 +402,8 @@ We have discussed four different coordinate systems:
     5. $r=1$ and $\theta=\pi/4$ in spherical coordinates,
     6. $\varphi=\pi/2$ and $\theta=\pi/2$ in spherical coordinates.
 
-3.  [:smirk:]
+3.  [:smirk:] *Partial derivatives*
+
     (a) Consider the function $f(r,\varphi,\theta)=\frac{1}{r^2}$ defined
         using spherical coordinates.
         Compute $\frac{\partial}{\partial z} f(r, \varphi, \theta)$.
@@ -414,26 +413,32 @@ We have discussed four different coordinate systems:
         question).
         Compute again $\frac{\partial}{\partial z} f(r, \varphi, z)$.
  
-4.  [:smirk:] From the transformation from polar to Cartesian
+4.  [:smirk:] *Chain rule practice* 
+
+    From the transformation from polar to Cartesian
     coordinates, show that
     $$\frac{\partial}{\partial x} = \cos\varphi \frac{\partial}{\partial r} - \frac{\sin\varphi}{r} \frac{\partial}{\partial \varphi}$$
     and
     $$\frac{\partial}{\partial y} = \sin\varphi \frac{\partial}{\partial r} + \frac{\cos\varphi}{r} \frac{\partial}{\partial \varphi}.$$
     (Use the chain rule for differentiation).
 
-5.  [:sweat:] Using the result of problem 4, show that the Laplace
+5.  [:sweat:] *Laplace operator in spherical coordinates*
+
+    Using the result of problem 4, show that the Laplace
     operator acting on a function $\psi({\bf r})$ in polar coordinates
     takes the form
     $$\nabla^2 \psi({\bf r}) =\left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right) \psi({\bf r}) = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial \psi(r,\varphi)}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 \psi(r,\varphi)}{\partial \varphi^2}.$$
 
     In a similar fashion it can be shown that for spherical coordinates,
     the Laplace operator acting on a function $\psi({\bf r})$ becomes:
-    $$\nabla^2 \psi (r,\vartheta,\varphi) = 
-    \frac{1}{r^2} \frac{\partial}{\partial r^2} \left( r^2 \frac{\partial \psi(r,\vartheta,\varphi)}{\partial r} \right) + \frac{1}{r^2\sin^2\vartheta} \frac{\partial^2 \psi(r,\vartheta, \varphi)}{\partial \varphi^2} + \frac{1}{r^2\sin\vartheta} 
-    \frac{\partial}{\partial \vartheta}\left( \sin\vartheta \frac{\partial\psi(r,\vartheta, \varphi)}{\partial \vartheta}\right).$$
+    $$\begin{align} \nabla^2 \psi (r,\vartheta,\varphi) &= 
+    \frac{1}{r^2} \frac{\partial}{\partial r^2} \left( r^2 \frac{\partial \psi(r,\vartheta,\varphi)}{\partial r} \right) \\ &+ \frac{1}{r^2\sin^2\vartheta} \frac{\partial^2 \psi(r,\vartheta, \varphi)}{\partial \varphi^2} \\ &+ \frac{1}{r^2\sin\vartheta} 
+    \frac{\partial}{\partial \vartheta}\left( \sin\vartheta \frac{\partial\psi(r,\vartheta, \varphi)}{\partial \vartheta}\right).
+    \end{align}$$
     This is however even more tedious (you do not have to show this).
 
 6.  [:grinning:] *Integration and coordinates I*
+
     We define $f(r, \varphi) = \frac{1}{r}$ in polar coordinates. Explain how
     a circular region, centered at the origin and with radius $r_0$, can be described
     using polar coordinates. Then compute the integral of $f(r,\varphi)$ over
@@ -445,7 +450,7 @@ We have discussed four different coordinate systems:
 
 8.  [:smirk:] *Integration and coordinates III*
 
-    In 2D we can define a shape by specifying a function $r(\varphi)$:
+    In 2D, we can define a shape by specifying a function $r(\varphi)$:
 
     ![image](figures/shape_polar.svg)