1 \chapter{Mathematical tools}
3 \section{Vector algebra}
5 \subsection{Vector space}
6 \label{math_app:vector_space}
9 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:
11 \item $\forall \vec{v} \, \exists 1$ with: $\vec{v}1=\vec{v}$
12 (identity element of scalar multiplication)
13 \item $\vec{v}(\lambda_1+\lambda_2)=\vec{v}\lambda_1+\vec{v}\lambda_2$
14 (distributivity of scalar multiplication)
15 \item $(\vec{v}_1+\vec{v}_2)\lambda=\vec{v}_1\lambda + \vec{v}_2\lambda$
16 (distributivity of scalar multiplication)
17 \item $(\vec{v}\lambda_1)\lambda_2=\vec{v}(\lambda_1\lambda_2)$
18 (compatibility of scalar multiplication with field multiplication)
20 The elements $\vec{v}\in V$ are called vectors.
23 Due to the additive abelian group, the following properties are additionally valid:
25 \item $\vec{u}+\vec{v}=\vec{v}+\vec{u}$ (commutativity of addition)
26 \item $\vec{u}+(\vec{v}+\vec{w})=(\vec{u}+\vec{v})+\vec{w}$
27 (associativity of addition)
28 \item $\forall \vec{v} \, \exists \vec{0}$ with:
29 $\vec{0}+\vec{v}=\vec{v}+\vec{0}=\vec{v}$
30 (identity elemnt of addition)
31 \item $\forall \vec{v} \, \exists -\vec{v}$ with: $\vec{v}+(-\vec{v})=0$
32 (inverse element of addition)
34 The addition of two vectors is called vector addition.
37 \subsection{Dual space}
39 \subsection{Inner and outer product}
49 \section{Spherical coordinates}
51 \section{Fourier integrals}