Wrapping up

src/2_coordinates.md

@@ -359,6 +359,7 @@ We have discussed four different coordinate systems:
     $${\bf r} = (r, \phi).$$ 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.
+
     Infinitesimal distance: $$ds^2 = dr^2 + r^2 d\phi^2.$$
     Infinitesimal area: $$dA = r dr d\varphi.$$
 
@@ -366,6 +367,7 @@ We have discussed four different coordinate systems:
     $${\bf r} = (r, \phi, z).$$ 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.
+
     Infinitesimal distance: $$ds^2 = dr^2 + r^2 d\phi^2 + dz^2.$$
     Infinitesimal volume:: $$dV = r dr d\varphi dz.$$
 
@@ -373,6 +375,7 @@ We have discussed four different coordinate systems:
     $${\bf r} = (r, \theta, \phi).$$ 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:
     $$ds^2 =r^2 (\sin^2 \theta d\phi^2 + d\theta^2) +  dr^2 .$$
     Infinitesimal volume:
@@ -402,7 +405,8 @@ 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 $\frac{\partial}{\partial z} f(r, \varphi, \theta)$.
@@ -412,14 +416,18 @@ 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
     $$\nabla^2 \psi({\bf r}) =\left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right) \psi({\bf r}) = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial \psi(r,\varphi)}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 \psi(r,\varphi)}{\partial \varphi^2}.$$
@@ -433,6 +441,7 @@ We have discussed four different coordinate systems:
     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
@@ -444,7 +453,7 @@ 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)