From: hackbard <hackbard@hackdaworld.org>
Date: Wed, 8 Feb 2012 22:45:13 +0000 (+0100)
Subject: outer product
X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=69b350d82ab4382625484e3edbffd933f267aa50;p=lectures%2Flatex.git

outer product
---

diff --git a/physics_compact/math.tex b/physics_compact/math.tex
index c182a96..83e13cb 100644
--- a/physics_compact/math.tex
+++ b/physics_compact/math.tex
@@ -47,10 +47,10 @@ Inserting the expression for the coefficients into \eqref{eq:vec_sum}, the vecto
 \begin{equation}
 \label{eq:complete}
 \vec{a}=\sum_i \vec{e}_i (\vec{e}_i,\vec{a}) \Leftrightarrow
-\sum_i\vec{e}_i\cdot \vec{e}_i=\vec{1}
+\sum_i\vec{e}_i\otimes \vec{e}_i=\vec{1}
 \end{equation}
 if the basis is complete.
-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.
+Indeed, the very important identity representation by the outer product ($\otimes$, see \ref{math_app:product}) in the second part of \eqref{eq:complete} is known as the completeness relation or closure.
 
 \section{Operators, matrices and determinants}
 
diff --git a/physics_compact/math_app.tex b/physics_compact/math_app.tex
index 079d8d9..edee2ff 100644
--- a/physics_compact/math_app.tex
+++ b/physics_compact/math_app.tex
@@ -37,13 +37,14 @@ The addition of two vectors is called vector addition.
 \subsection{Dual space}
 
 \subsection{Inner and outer product}
+\label{math_app:product}
 
 \begin{definition}
 The inner product ...
 \end{definition}
 
 \begin{definition}
-The outer product ...
+If $\vec{u}\in U$ and $\vec{v}\in V$ are vectors within the respective vector spaces and $V^{\dagger}$ is the dual space of $V$, the outer product of $\vec{u}$ and $\vec{v}$ is defined as the tensor product ...
 \end{definition}
 
 \section{Spherical coordinates}