more math

diff --git a/physics_compact/math.tex b/physics_compact/math.tex
index 0e49a40..5f437fa 100644 (file)
--- a/physics_compact/math.tex
+++ b/physics_compact/math.tex
@@ -1,6 +1,26 @@
 \part{Mathematical foundations}
 
 \part{Mathematical foundations}
 

+Reminder: Modern Quantum Chemistry \& Sakurai \& Group Theory \ldots
+

 \chapter{Linear algebra}
 
 \chapter{Linear algebra}
 

-Reminder: Modern Quantum Chemistry \& Sakurai \& Group Theory \ldots
+\section{Vectors and bases}
+
+A vector $\vec{a}$ of an $N$-dimensional vector space (see \ref{math_app:vector_space} for mathematical details) is represented by its components $a_i$ with respect to a set of $N$ basis vectors ${\vec{e}_i}$.
+\begin{equation}
+\vec{a}=\sum_i \vec{e}_i a_i
+\label{eq:vec_sum}
+\end{equation}
+The scalar product for an $N$-dimensional vector space is defined as
+\begin{equation}
+(\vec{a},\vec{b})=\sum_i^N a_i b_i \text{ ,}
+\end{equation}
+which introduces a norm
+\begin{equation}
+||\vec{a}||=\sqrt{(\vec{a},\vec{a})}
+\end{equation}
+that correpsonds to the length of vector \vec{a}.
+Evaluating the scalar product $(\vec{a},\vec{b})$ by the sum representation of \eqref{eq:vec_sum} \ldots
+\begin{equation}
+\end{equation}

 
 

diff --git a/physics_compact/math_app.tex b/physics_compact/math_app.tex
index 1903ab1..e6e935e 100644 (file)
--- a/physics_compact/math_app.tex
+++ b/physics_compact/math_app.tex
@@ -1,5 +1,37 @@
 \chapter{Mathematical tools}
 
 \chapter{Mathematical tools}
 

+\section{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}
+

 \section{Spherical coordinates}
 
 \section{Fourier integrals}
 \section{Spherical coordinates}
 
 \section{Fourier integrals}