From 0929321d46ef53429ae8d78a0f08e452e96b3cd2 Mon Sep 17 00:00:00 2001
From: hackbard <hackbard@hackdaworld.org>
Date: Tue, 7 Feb 2012 17:30:28 +0100
Subject: [PATCH] more math

---
 physics_compact/math.tex     | 11 ++++++-----
 physics_compact/math_app.tex | 16 +++++++++++++++-
 2 files changed, 21 insertions(+), 6 deletions(-)

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}
-- 
2.20.1