\chapter{Mathematical tools}

\section{Vector algebra}

\subsection{Vector space}
\label{math_app:vector_space}

\begin{definition}
A vector space $V$ over a field $(K,+,\cdot)$ is an additive abelian group $(V,+)$ and an additionally defined scalar multiplication of $\vec{v}\in V$ by $\lambda\in K$, which fullfills:
\begin{itemize}
\item $\forall \vec{v} \, \exists 1$ with: $\vec{v}1=\vec{v}$
      (identity element of scalar multiplication)
\item $\vec{v}(\lambda_1+\lambda_2)=\vec{v}\lambda_1+\vec{v}\lambda_2$
      (distributivity of scalar multiplication)
\item $(\vec{v}_1+\vec{v}_2)\lambda=\vec{v}_1\lambda + \vec{v}_2\lambda$
      (distributivity of scalar multiplication)
\item $(\vec{v}\lambda_1)\lambda_2=\vec{v}(\lambda_1\lambda_2)$
      (compatibility of scalar multiplication with field multiplication)
\end{itemize}
The elements $\vec{v}\in V$ are called vectors.
\end{definition}
\begin{remark}
Due to the additive abelian group, the following properties are additionally valid:
\begin{itemize}
\item $\vec{u}+\vec{v}=\vec{v}+\vec{u}$ (commutativity of addition)
\item $\vec{u}+(\vec{v}+\vec{w})=(\vec{u}+\vec{v})+\vec{w}$
      (associativity of addition)
\item $\forall \vec{v} \, \exists \vec{0}$ with:
      $\vec{0}+\vec{v}=\vec{v}+\vec{0}=\vec{v}$
      (identity elemnt of addition)
\item $\forall \vec{v} \, \exists -\vec{v}$ with: $\vec{v}+(-\vec{v})=0$
      (inverse element of addition)
\end{itemize}
The addition of two vectors is called vector addition.
\end{remark}

\subsection{Dual space}

\subsection{Inner and outer product}
\label{math_app:product}

\begin{definition}
The inner product ...
\end{definition}

\begin{definition}
If $\vec{u}\in U$ and $\vec{v}\in V$ are vectors within the respective vector spaces and $V^{\dagger}$ is the dual space of $V$, the outer product of $\vec{u}$ and $\vec{v}$ is defined as the tensor product ...
\end{definition}

\section{Spherical coordinates}

\section{Fourier integrals}

