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}
40 \label{math_app:product}
47 If $\vec{u}\in U$, $\vec{v}\in V$ and $\vec{v}^{\dagger}\in V^{\dagger}$ are vectors within the respective vector spaces and $V^{\dagger}$ is the dual space of $V$,
48 the outer product $\vec{u}\otimes\vec{v}$ is defined as the tensor product of $\vec{v}^{\dagger}$ and $\vec{u}$,
49 which constitutes a map $A:V\rightarrow U$ by
51 \vec{v}\mapsto\vec{v}^{\dagger}(\vec{v})\vec{u}
54 where $\vec{v}^{\dagger}(\vec{v})$ denotes the linear functional $\vec{v}^{\dagger}\in V^{\dagger}$ on $V$ when evaluated at $\vec{v}\in V$, a scalar that in turn is multiplied with $\vec{u}\in U$.
56 In matrix formalism, with respect to a given basis ${\vec{e}_i}$ of $\vec{u}$ and ${\vec{e}'_i}$ of $\vec{v}$,
57 if $\vec{u}=\sum_i^m \vec{e}_iu_i$ and $\vec{v}=\sum_i^n\vec{e}'_iv_i$,
58 the outer product can be written as matrix $A$ as
60 \vec{u}\otimes\vec{v}=A=\left(
61 \begin{array}{c c c c}
62 u_1v_1 & u_1v_2 & \cdots & u_1v_n\\
63 u_2v_1 & u_2v_2 & \cdots & u_2v_n\\
64 \vdots & \vdots & \ddots & \vdots\\
65 u_mv_1 & u_mv_2 & \cdots & u_mv_n\\
72 The matrix can be equivalently obtained by matrix multiplication:
74 \vec{u}\otimes\vec{v}=\vec{u}\vec{v}^{\dagger} \text{ ,}
76 if $\vec{u}$ and $\vec{v}$ are represented as $m\times 1$ and $n\times 1$ column vectors, respectively.
77 Here, $\vec{v}^{\dagger}$ represents the conjugate transpose of $\vec{v}$.
78 By definition, and as can be easily seen in the matrix representation, the following identity holds:
80 (\vec{u}\otimes\vec{v})\vec{w}=\vec{u}(\vec{v},\vec{w})
84 \section{Spherical coordinates}
86 \section{Fourier integrals}