after ice SO stuff

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

diff --git a/physics_compact/math.tex b/physics_compact/math.tex
index c182a96..bd2a888 100644 (file)
--- a/physics_compact/math.tex
+++ b/physics_compact/math.tex
@@ -1,9 +1,9 @@
 \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 ${\vec{e}_i}$
@@ -12,16 +12,16 @@ 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}
-that just corresponds to the length of vector \vec{a}.
+that just corresponds to the length of vector $\vec{a}$.
 Evaluating the scalar product $(\vec{a},\vec{b})$ by the sum representation of \eqref{eq:vec_sum} leads to
 \begin{equation}
 (\vec{a},\vec{b})=(\sum_i\vec{e}_ia_i,\sum_j\vec{e}_jb_j)=
@@ -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}