getting more and more worth, rewrite?!?

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

diff --git a/physics_compact/math.tex b/physics_compact/math.tex
index c182a96..57aef07 100644 (file)
--- a/physics_compact/math.tex
+++ b/physics_compact/math.tex
@@ -12,12 +12,12 @@ A vector $\vec{a}$ of an $N$-dimensional vector space (see \ref{math_app:vector_
 \label{eq:vec_sum}
 \end{equation}
 i.e., if the basis set is complete, any vector can be written as a linear combination of these basis vectors.
-The scalar product for an $N$-dimensional real vector space is defined as
+The scalar product in an $N$-dimensional Euclidean vector space is defined as
 \begin{equation}
 (\vec{a},\vec{b})=\sum_i^N a_i b_i \text{ ,}
 \label{eq:vec_sp}
 \end{equation}
-which enables to define a norm
+which satisfies the properties of an inner product (see \ref{math_app:product}) and enables to define a norm
 \begin{equation}
 ||\vec{a}||=\sqrt{(\vec{a},\vec{a})}
 \end{equation}
@@ -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}