more math

[lectures/latex.git] / physics_compact / math_app.tex

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 \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}