\right)
\text{ ,}
\end{equation}
-which can be equivalently obtained by the rulrs of matrix multiplication
+which can be equivalently obtained by the rules of matrix multiplication
\begin{equation}
\vec{u}\otimes\vec{v}=\vec{u}\vec{v}^{\dagger} \text{ ,}
\end{equation}
\section{Spherical coordinates}
+Cartesian coordinates $\vec{r}(x,y,z)$ are related to spherical coordinates $\vec{r}(r,\theta,\phi)$ by
+\begin{eqnarray}
+x&=&r\sin\theta\cos\phi\textrm{ ,}\\
+y&=&r\sin\theta\sin\phi\textrm{ ,}\\
+z&=&r\cos\theta\textrm{ .}
+\end{eqnarray}
+Infinitesimal translations $dq_i$ and $dq'_i$ of the two coordinate systems are related by the partial derivatives.
+\begin{equation}
+dq_i=\sum_j \frac{\partial q_i}{\partial q'_j}dq'_j
+\end{equation}
+\begin{definition}[Jacobi matrix]
+The matrix J with components
+\begin{equation}
+J_{ij}=\frac{\partial q_i}{\partial q'_j}
+\end{equation}
+is called the Jacobi matrix.
+\end{definition}
+
+For cartesian and spherical coordinates the relation of the translations are presented in detail
+\begin{eqnarray}
+dx&=&\frac{\partial x}{\partial r}dr +
+ \frac{\partial x}{\partial \theta}d\theta +
+ \frac{\partial x}{\partial \phi}d\phi\\
+dy&=&\frac{\partial y}{\partial r}dr +
+ \frac{\partial y}{\partial \theta}d\theta +
+ \frac{\partial y}{\partial \phi}d\phi\\
+dz&=&\frac{\partial z}{\partial r}dr +
+ \frac{\partial z}{\partial \theta}d\theta +
+ \frac{\partial z}{\partial \phi}d\phi\\
+\end{eqnarray}
+and the vector consisting of all or using the Jacobi matrix
+
+
+\begin{equation}
+ =\sin\theta\cos\phi dr + \\
+\end{equation}
+
+To obtain infinitesimal
+\begin{definition}[Jacobi matrix]
+
+\end{definition}
+
\section{Fourier integrals}
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