From: hackbard Date: Mon, 6 Feb 2012 14:22:44 +0000 (+0100) Subject: more math X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=df550a4ec6a24e44ceba6ccf4111722940040c1d;p=lectures%2Flatex.git more math --- diff --git a/physics_compact/math.tex b/physics_compact/math.tex index 0e49a40..5f437fa 100644 --- a/physics_compact/math.tex +++ b/physics_compact/math.tex @@ -1,6 +1,26 @@ \part{Mathematical foundations} +Reminder: Modern Quantum Chemistry \& Sakurai \& Group Theory \ldots + \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 --- a/physics_compact/math_app.tex +++ b/physics_compact/math_app.tex @@ -1,5 +1,37 @@ \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}