From: hackbard Date: Tue, 7 Feb 2012 16:30:28 +0000 (+0100) Subject: more math X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=0929321d46ef53429ae8d78a0f08e452e96b3cd2;p=lectures%2Flatex.git more math --- diff --git a/physics_compact/math.tex b/physics_compact/math.tex index ddaebc6..4c8e6f9 100644 --- a/physics_compact/math.tex +++ b/physics_compact/math.tex @@ -11,7 +11,7 @@ A vector $\vec{a}$ of an $N$-dimensional vector space (see \ref{math_app:vector_ \vec{a}=\sum_i^N \vec{e}_i a_i \label{eq:vec_sum} \end{equation} -The scalar product for an $N$-dimensional vector space is defined as +The scalar product for an $N$-dimensional real vector space is defined as \begin{equation} (\vec{a},\vec{b})=\sum_i^N a_i b_i \text{ ,} \label{eq:vec_sp} @@ -38,15 +38,16 @@ i.e.\ the basis vectors are mutually perpendicular (orthogonal) and have unit Such a basis set is called orthonormal. The component of a vector can be obtained by taking the scalar product with the respective basis vector. \begin{equation} -\vec{e}_j\vec{a}=\vec{e}_j \sum_i \vec{e}_ia_i=\sum_i \vec{e}_j\vec{e}_ia_i= +(\vec{e}_j,\vec{a})=(\vec{e}_j,\sum_i \vec{e}_ia_i)= +\sum_i (\vec{e}_j,\vec{e}_i)a_i= \sum_i\delta_{ij}a_i=a_j \end{equation} Inserting the expression for the coefficients into \eqref{eq:vec_sum}, the vector can be written as \begin{equation} \label{eq:complete} -\vec{a}=\sum_i \vec{e}_i (\vec{e}_i\vec{a}) \Leftrightarrow \sum_i\vec{e}_i\vec{e}_i=\vec{1} +\vec{a}=\sum_i \vec{e}_i (\vec{e}_i,\vec{a}) \Leftrightarrow +\sum_i\vec{e}_i\cdot \vec{e}_i=\vec{1} \end{equation} if the basis is complete. -Thus, the very important second part of \eqref{eq:complete} is known as the completeness relation or closure. +Indeed, the very important identity representation by the outer product ($\cdot$) in the second part of \eqref{eq:complete} is known as the completeness relation or closure. -Todo: outer product ... + explicitly mark scalar product diff --git a/physics_compact/math_app.tex b/physics_compact/math_app.tex index e6e935e..079d8d9 100644 --- a/physics_compact/math_app.tex +++ b/physics_compact/math_app.tex @@ -1,6 +1,8 @@ \chapter{Mathematical tools} -\section{Vector space} +\section{Vector algebra} + +\subsection{Vector space} \label{math_app:vector_space} \begin{definition} @@ -32,6 +34,18 @@ Due to the additive abelian group, the following properties are additionally val The addition of two vectors is called vector addition. \end{remark} +\subsection{Dual space} + +\subsection{Inner and outer product} + +\begin{definition} +The inner product ... +\end{definition} + +\begin{definition} +The outer product ... +\end{definition} + \section{Spherical coordinates} \section{Fourier integrals}