diff --git a/mkdocs.yml b/mkdocs.yml
index a588325568fbcafe4f532fc01e63ae824d2d6015..09bcd2f03bde7e44f627f433e58d39f5402eee3f 100644
--- a/mkdocs.yml
+++ b/mkdocs.yml
@@ -6,7 +6,8 @@ site_description: |
Lecture notes for TN3105 - Mathematics for Quantum Physics
nav:
- Intro: 'index.md'
-
+ - Coordinates: 'coordinates.md'
+
theme:
name: material
custom_dir: theme
diff --git a/src/1_complex_numbers.md b/src/1_complex_numbers.md
index 725c1ab992c2a1016012a6590357653756845712..3351552810b1eac582afee1a7d417090e7b62c9d 100644
--- a/src/1_complex_numbers.md
+++ b/src/1_complex_numbers.md
@@ -67,10 +67,7 @@ Complex numbers can be rendered on a two-dimensional (2D) plane, the
horizontal, which represents the real number 1, whereas the vertical
unit vector represents ${\rm i}$.
-
-
-[\
-]{}
+
Note that the norm of $z$ is the length of this vector.
@@ -80,10 +77,7 @@ Addition in the complex plane
Adding two numbers in the complex plane corresponds to adding the
horizontal and vertical components:
-
-
-[\
-]{}
+
We see that the sum is found as the diagonal of a parallelogram spanned
by the two numbers.
@@ -99,10 +93,7 @@ part. However, it is also possible to characterize that vector by its
*length* and *direction*, where the latter can be represented by the
angle the vector makes with the horizontal axis:
-
-
-[\
-]{}
+
The angle with the horizontal axis is denoted by $\varphi$, just as in
the case of polar coordinates. In the context of complex numbers, this
@@ -168,17 +159,13 @@ should be independent of $dz=dx + {\rm i} dy$! Thus, $f(z)$ is
differentiable only when
$$\frac{\partial u}{\partial x} + {\rm i} \frac{\partial v}{\partial x} = -{\rm i} \frac{\partial u}{\partial y} + \frac{\partial v}{\partial y}.$$
Equating the real and imaginary parts of the left and right hand side we
-obtain the
-
-TODO: Here was a remark environment
-
-[ *Cauchy Riemann* differential equations:
+obtain the *Cauchy Riemann* differential equations:
$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} {~~~ \rm and ~~~ } \frac{\partial v}{\partial x} = -
\frac{\partial u}{\partial y}.$$ The derivative is then given as
$$\frac{df}{dz} = \frac{\partial u}{\partial x} + {\rm i} \frac{\partial v}{\partial x}.$$
A complex function whose real and imaginary part ($u$ and $v$) obey the
Cauchy-Riemann differential equations in a point $z$, is complex
-differentiable at the point $z$. ]{}
+differentiable at the point $z$.
Note that differentiability is a property which not only pertains to a
function, but also to a point.
@@ -296,8 +283,6 @@ $$\tanh'(x) = 1 + \frac{\sinh^2 x}{\cosh^2 (x)} = - \frac{1}{\cosh^2(x)}.$$
Summary
=======
-TODO: Here was the beginning of a mdframed env
-
- A complex number $z$ has the form $$z = a + b \rm i$$ where $a$ and
$b$ are both real, and $\rm i^2 = 1$. The real number $a$ is called
the *real part* of $z$ and $b$ is the *imaginary part*. Two complex
@@ -345,22 +330,21 @@ TODO: Here was the beginning of a mdframed env
- Hyperbolic functions are defined as:
$$\sinh(z) = \frac{e^{z} - e^{-z}}{2}; \phantom{xxx} \cosh(z) = \frac{e^{z} + e^{-z}}{2}.$$
-TODO: Here was the end of a mdframed env
Problems
========
-1. [\[]{}D1[\]]{} Given $a=1+2\rm i$ and $b=4-2\rm i$, draw in the
+1. [:grinning:] Given $a=1+2\rm i$ and $b=4-2\rm i$, draw in the
complex plane the numbers $a+b$, $a-b$, $ab$, $a/b$, $e^a$ and
$\ln(a)$.
-2. [\[]{}D1[\]]{} Evaluate (i) $\rm i^{1/4}$, (ii)
+2. [:grinning:] Evaluate (i) $\rm i^{1/4}$, (ii)
$\left(-1+\rm i \sqrt{3}\right)^{1/2}$, (iii) $\exp(2\rm i^3)$.
-3. [\[]{}D1[\]]{} Evaluate $$\left| \frac{a+b\rm i}{a-b\rm i} \right|$$
+3. [:grinning:] Evaluate $$\left| \frac{a+b\rm i}{a-b\rm i} \right|$$
for real $a$ and $b$.
-4. [\[]{}D1[\]]{} Show that $\cos x = \cosh(\rm i x)$ and
+4. [:grinning:] Show that $\cos x = \cosh(\rm i x)$ and
$\cos(\rm i x) = \cosh x$. Derive similar relations for $\sinh$ and
$\sin$.
@@ -369,28 +353,28 @@ Problems
Also show that $\cosh x$ is a solution to the differential equation
$$y'' = \sqrt{1 + y'^2}.$$
-5. [\[]{}D1[\]]{} Calculate the real part of
+5. [:grinning:] Calculate the real part of
$\int_0^\infty e^{-\gamma t +\rm i \omega t} dt$ ($\omega$ and
$\gamma$ are real; $\gamma$ is positive).
-6. [\[]{}D1[\]]{} Is the function $f(z) = |z| = \sqrt{x^2 + y^2}$
+6. [:grinning:] Is the function $f(z) = |z| = \sqrt{x^2 + y^2}$
analytic on the complex plane or not? If not, where is the function
not analytic?
-7. [\[]{}D1[\]]{} Show that the Cauchy-Riemann equations imply that the
+7. [:grinning:] Show that the Cauchy-Riemann equations imply that the
real and imaginary part of a differentiable complex function both
represent solutions to the Laplace equation, i.e.
$$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0,$$
for the real part $u$ of the function, and similarly for the
imaginary part $v$.
-8. [\[]{}D3[\]]{} Show that the set of points $z$ obeying
+8. [:sweat:] Show that the set of points $z$ obeying
$$| z - \rm i a| = \lambda |z + \rm i a|,$$ with $a$ and $\lambda$
real, form a circle with radius $2|\lambda/(1-\lambda^2) a|$
centered on the point $\rm i a (1+\lambda^2)/(1-\lambda^2)$,
provided $\lambda \neq 1$. What is the set like for $\lambda = 1$?
-9. [\[]{}D2[\]]{} In two dimensions, the Coulomb potential is
+9. [:smirk:] In two dimensions, the Coulomb potential is
proportional to $\log |r|$. Viewing the 2D space as a complex plane,
this is $\log |z|$. Consider a system consisting of charges $q_i$
placed at ‘positions’ $z_i$, all close to the origin. The point $z$
@@ -406,7 +390,7 @@ Problems
This is called a *multipole expansion*. A similar expansion exist in
three dimensions.
-10. [\[]{}D2[\]]{} In this problem, we consider the function $1/z$ close
+10. [:smirk:] In this problem, we consider the function $1/z$ close
to the real axis: $z=x-\rm i \epsilon$ where $\epsilon$ is small.
Show that the imaginary part of this function approaches $\pi$ times
the Dirac delta-function $\delta(x)$ for $\epsilon\rightarrow 0$. Do
diff --git a/src/coordinates.md b/src/coordinates.md
new file mode 100644
index 0000000000000000000000000000000000000000..df6740eed106acbba98a4dbef5f6007da1dc658b
--- /dev/null
+++ b/src/coordinates.md
@@ -0,0 +1,392 @@
+---
+title: Coordinates
+---
+
+# Cartesian 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 $j=1, \ldots, n$.
+
+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
+
+$$\Delta s^2 = (x'_1 - x_1)^2 + (x'_2 - x_2)^2 + \ldots + (x'_n - x_n)^2.$$
+
+(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.
+
+
+
+
+# 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
+the angular coordinate $\varphi$ is dimensionless.
+
+
+
+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*
+coordinates $(x,y)$ of the point are related to the polar ones as
+follows:
+
+$$\begin{equation} x = r \cos\varphi; \end{equation}$$
+$$\begin{equation} y = r \sin \varphi.\end{equation}$$
+
+
+
+The inverse relation is given as
+
+$$\begin{equation} r=\sqrt{x^2 + y^2}; \label{rxy}\end{equation}$$
+$$\begin{equation} \varphi=\begin{cases}
+ \arctan(y/x) & \text{$x>0$,}\\
+ \pi + \arctan(y/x) & \text{$x<0$ and $y>0$,}\\
+ -\pi + \arctan(y/x) & \text{$x<0$ and $y<0$.}
+ \end{cases} \label{phixy}\end{equation}$$
+
+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
+the fact that the $\tan$ has different branches. Convince yourself that
+the expression above is correct for all the four sectors!
+
+## Distances and areas
+
+Now suppose we want to calculate the distance between two points, one
+with polar coordinates $(r_1, \varphi_1)$, and the other with
+$(r_2, \varphi_2)$. This looks like a difficult exercise. One possible
+way to perform this is by translating the polar coordinates into
+Cartesian coordinates and using the expression given above for this
+distance: $$\Delta s^2 = (x_1 - x_2)^2 + (y_1 - y_2)^2,$$ so
+$$\Delta s^2 = (r_1\cos\varphi_1 - r_2 \cos \varphi_2)^2 + (r_1\sin\varphi_1 - r_2 \sin \varphi_2)^2$$
+which is not a very convenient expression.
+
+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).
+
+
+
+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
+from $r_1$ to $r_2$. Provided the difference between the angles
+$\varphi_1$ and $\varphi_2$ is (very) small, these paths are
+approximately perpendicular and we can use Pythagoras’ theorem to find
+the distance $d s$. Note that the arc is approximately straight –
+it has a length $r_1 d \varphi$, where
+$d \varphi = \varphi_2-\varphi_1$. So we have:
+$$d s^2 = (d r)^2 + (arc~length)^2 = (d r)^2 + r_1^2 (d \varphi)^2 .$$
+
+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:
+
+$$dA = r dr d\varphi.$$
+
+This is an important formula to remember for integrating in polar
+coordinates! The extra $r$ that appears here can be intuitively
+understood: the area swept by an angle difference $d\varphi$
+*increases* as we move further away from the origin.
+
+!!! check "Example: Integrating over a circular area"
+ To check the area element we just derived, let us compute a simple
+ integral. We compute the integral over a circle with radius $r_0$
+ with a very simple function that equals to $1$. In this case,
+ we expect to get as a result the are of the region we integrate over.
+
+ We find:
+
+ $$
+ \int_0^{2\pi} d\varphi \int_0^{r_0} r dr =\\
+ 2\pi \int_0^{r_0} r dr = 2 \pi \frac{1}{2} r_0^2 = \pi r_0^2,
+ $$
+
+ which is indeed the area of a circle with radius 0.
+
+## Converting derivatives between coordinate systems
+
+Often, in physics important equations involve derivatives given in terms
+of Cartesian coordinates. One prominent example are equations of the form
+$$\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right)
+f(x, y) = \ldots.$$
+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
+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?
+
+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
+ 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
+
+ $$\frac{\partial}{\partial x_i} = \sum_{j=1}^n
+ \frac{\partial f}{\partial y_j} \frac{\partial g_j}{\partial x_i}$$
+
+We start by replacing the function $f(x, y)$ by a function in polar coordinates
+$f(r, \varphi)$, and ask what is $\frac{\partial}{\partial x} f(r, \varphi)$. When
+we look at this expression, we need to understand what it *means* to take the derivative
+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,
+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,
+in the sense that
+
+$$f_\text{cart}(x, y) = f_\text{polar}(r(x,y), \varphi(x,y))$$
+
+In physics, we usually never write this down explicitly, but we are aware that these
+are two different functions from the fact that they use different coordinates.
+
+With this information, we can now apply the chain rule:
+
+$$ \frac{\partial}{\partial x} f(r, \varphi) =
+\frac{\partial f}{\partial r} \frac{\partial r(x, y)}{\partial x} +
+\frac{\partial f}{\partial \varphi} \frac{\partial \varphi(x,y)}{\partial x}
+$$
+
+and it is now a matter of (tedious) calculus to arrive at the right result.
+This is the task of exercises 3 and 4, which finally compute the Laplacian
+in polar coordinates.
+
+!!! warning Inverse function theorem
+ In this calculation one might be tempted to use the inverse
+ function theorem to compute derivatives like
+ $\frac{\partial \varphi}{\partial x}$ from the much simpler
+ $\frac{\partial x}{\partial \varphi}$. Note though that here we
+ are dealing with functions depending on several variables, so the
+ *Jacobian* has to be used (see [Wikipedia](https://en.wikipedia.org/wiki/Inverse_function_theorem)). A direct calculation is in this particular case more easy.
+
+Note that this procedure also carries over to other coordinate systems,
+although the calculations can become quite tedious. In these cases,
+it's usually best to look up the correct form.
+
+
+# Cylindrical coordinates
+
+Three dimensional systems may have axial symmetry. An example is an
+electrically charged wire of which we would like to calculate the
+electric field, or a current-carrying wire for which we would like to
+calculate the magnetic field. For such problems the most convenient
+coordinates are *cylindrical coordinates*. For convenience, we choose
+the symmetry-axis as the $z$-axis. Note that this can be done as we can
+choose the coordinate system ourselves - this is not imposed by the
+problem.
+
+Cylindrical coordinates are defined straightforwardly: we use polar
+coordinates $r$ and $\varphi$ in the $xy$ plane, and the distance $z$
+along the symmetry-axis as the third coordinate. If the axis system is
+chosen in physical space, we have two coordinates which have the
+dimension of a distance: $r$ and $z$. The other coordinate,
+$\varphi$, is of course dimensionless.
+
+What is the distance travelled along a path when we express this in
+cylindrical coordinates? Let’s consider an example (Figure).
+
+
+
+We want to find the length of the (small) red segment $d s$. By
+inspecting the figure, we see that the horizontal (i.e. parallel to the
+$xy$-plane) segment $d l$ is perpendicular to the vertical segment
+$dz$. Using for $d l$ the length we obtained before for a line
+segment in the $xy$ plane, expressed in polar coordinates, we
+immediately find:
+$$d s^2 = d l^2 + d z^2 = d r^2 + r^2 d \varphi^2 + d z^2.$$
+The volume element is consequently given as
+$$dV = r dr d\varphi dz.$$
+
+# Spherical coordinates
+
+For problems with spherical symmetry, we use *spherical coordinates*.
+These work as follows. For a point $\bf r$ in 3D space, we can specify
+the position of that point by specifying its (1) distance to the origin
+and (2) the direction of the line connecting the origin to our point.
+The specification of this direction can be identified with a point on a
+sphere which is centered at the origin:
+
+
+
+The position of a point on the sphere is specified using the two angles
+$\theta$ and $\phi$ indicated in the figure.
+
+!!! warning
+ Note that in mathematics, often the angles are labelled the other way
+ round: there, $\phi$ is used for the angle between a line running from
+ the origin o the point of interest and the $z$-axis, and $\theta$ for
+ the angle of the projection of that line with the $x$-axis. The
+ convention used here is custom in physics.
+
+The relation between Cartesian and coordinates is defined by
+$$x = r \cos \varphi \sin \vartheta$$
+$$y = r \sin\varphi \sin \vartheta$$ $$z = r \cos\vartheta$$ The inverse
+transformation is easy to find: $$r = \sqrt{x^2+y^2+z^2}$$
+$$\theta = \arccos(z/\sqrt{x^2+y^2+z^2})$$
+$$\phi = \begin{cases} \arctan(y/x) &{\rm for ~} x>0; \\
+ \pi + \arctan(y/x) & {\rm for ~} x<0 {\rm ~ and ~} y>0;\\
+ -\pi + \arctan(y/x) &{\rm ~ for ~} x<0 {\rm ~ and ~} y<0.
+ \end{cases}$$ These relations can be derived from the following figure:
+
+
+
+The distance related to a change in the spherical coordinates is
+calculated using Pythagoras’ theorem. The length $ds$ of a short segment
+on the sphere with radius $r$ corresponding to changes in the polar
+angles of $d\vartheta$ and $d\varphi$ is given as
+$$dl^2 = r^2 \left(\sin^2 \vartheta d\varphi^2 + d\vartheta^2\right).$$
+In order to verify this, it is important to realize that all points with
+*the same* coordinate $\vartheta$ span a circle in a horizontal plane
+with a radius $r\sin\vartheta$ as shown in the figure below.
+
+From this we can also infer that, for a segment with a radial component
+$dr$ in addition to the displacement on the sphere, the displacement is:
+$$ds^2 = r^2 \left(\sin^2 \vartheta d\varphi^2 + d\vartheta^2\right) + dr^2.$$
+
+The picture below shows the geometry behind the calculation of this
+displacement.
+
+
+
+From these arguments we can again also find the volume element, it is
+here given as
+$$dV = r^2 \sin\theta dr d\theta d\varphi.$$
+
+### summary
+
+We have discussed four different coordinate systems:
+
+1. *Cartesian coordinates*: $${\bf r} = (x_1, \ldots, x_n).$$ Can be
+ used for any dimension $n$. Convenient for: infinite spaces, systems
+ with rectangular symmatry.
+
+ 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*: $${\bf r} = (r, \phi).$$ Can be used in two
+ dimensions. 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: $$dA = r dr d\varphi.$$
+
+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.
+
+ Infinitesimal distance: $$ds^2 = dr^2 + r^2 d\phi^2 + dz^2.$$
+ 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
+ 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:
+ $$dV = r^2 \sin(\theta) dr d\theta d\varphi.$$
+
+Problems
+========
+
+1. [:grinning:]
+
+ 1. Find the polar coordinates of the point with Cartesian
+ coordinates $${\bf r} = \sqrt{2} (1,1).$$
+
+ 2. Find the cylindrical coordinates of the point with Cartesian
+ coordinates $${\bf r} = \frac{3}{2} (\sqrt{3}, 1, 1).$$
+
+ 3. Find the spherical coordinates of the points
+ $${\bf r} = (3/2, \sqrt{3}/2, 1).$$
+
+2. [:grinning:] *Geometry and different coordinate systems}*
+
+ What geometric objects do the following boundary conditions describe?
+
+ 1. $r<1$ in cylindrical coordinates,
+ 2. $\varphi=0$ in cylindrical coordinates,
+ 3. $r=1$ in spherical coordinates,
+ 4. $\theta = \pi/4$ in spherical coordinates,
+ 5. $r=1$ and $\theta=\pi/4$ in spherical coordinates.
+
+3. [:smirk:] 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).
+
+3. [:sweat:] Using the result of problem 2, 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:
+ $$\nabla^2 \psi (r,\vartheta,\varphi) =
+ \frac{1}{r^2} \frac{\partial}{\partial r^2} \left( r^2 \frac{\partial \psi(r,\vartheta,\varphi)}{\partial r} \right) + \frac{1}{r^2\sin^2\vartheta} \frac{\partial^2 \psi(r,\vartheta, \varphi)}{\partial \varphi^2} + \frac{1}{r^2\sin\vartheta}
+ \frac{\partial}{\partial \vartheta}\left( \sin\vartheta \frac{\partial\psi(r,\vartheta, \varphi)}{\partial \vartheta}\right).$$
+ This is however even more tedious (you do not have to show this).
+
+5. [:grinning:] *Integration and coordinates I*
+
+ Compute the area of the spherical cap defined by $r=r_0$ and $\theta <\theta_0$.
+
+6. [:smirk:] *Integration and coordinates II*
+
+ In 2D we can define a shape by specifying a function $r(\varphi)$:
+
+ 
+
+ (Of course, here we need to have $r(0) = r(2\pi)$.)
+
+ Show that the area of this shape is given by
+ $$
+ \int_0^{2\pi} \frac{1}{2}\left[r(\varphi)\right]^2 d\varphi
+ $$
+
diff --git a/src/figures/Coordinates_11_0.svg b/src/figures/Coordinates_11_0.svg
new file mode 100644
index 0000000000000000000000000000000000000000..806d9562a13e561b74b6d8b4e8705189ab52306f
--- /dev/null
+++ b/src/figures/Coordinates_11_0.svg
@@ -0,0 +1,810 @@
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diff --git a/src/figures/shape_polar.svg b/src/figures/shape_polar.svg
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diff --git a/src/index.md b/src/index.md
index 28bb89128fdae0058ffc5284987646e77b0e54be..68b721539749ed59a2e1be4411fe87e6622947b5 100644
--- a/src/index.md
+++ b/src/index.md
@@ -4,8 +4,22 @@
After following this course you will be able to:
- - Solve awesome problems
- - Tell exciting stories
+ - reproduce elementary formulas from the topics covered.
+ - solve mathematical problems encountered in the follow-up courses of the minor.
+ - explain Hilbert spaces of (in)finite dimension.
+
+Mathematics for quantum mechanics ives you a compact introduction and review
+of the basic mathematical tools commonly used in quantum mechanics. Throughout
+the course we have directly quantum mechanics applications in mind, but at the
+core this is still a math course. For this reason, applying what you learned
+to examples and exercises is **crucial**!
+
+Each lecture note comes with an extensive set of exercises. Each exercise is labeled
+according to difficulty:
+
+- [:grinning:] easy
+- [:smirk:] intermediate
+- [:sweat:] difficult
In these notes our aim is to provide learning materials which are: