hopefully finished respective math stuff, finally!

diff --git a/physics_compact/math_app.tex b/physics_compact/math_app.tex
index f8361e8..f1ac777 100644 (file)
--- a/physics_compact/math_app.tex
+++ b/physics_compact/math_app.tex
@@ -139,7 +139,7 @@ and the usual rules of matrix multiplication.
 \end{remark}
 
 \begin{definition}[Outer product]
-If $\vec{u}\in U$, $\vec{v},\vec{w}\in V$ are vectors within the respective vector spaces and $\varphi_{\vec{v}}\in V^{\dagger}$ is a linear functional of the dual space $V^{\dagger}$ of $V$ determined in some way from $\vec{v}$ (e.g.\  as in \eqref{eq:ip_mapping}),
+If $\vec{u}\in U$, $\vec{v},\vec{w}\in V$ are vectors within the respective vector spaces and $\varphi_{\vec{v}}\in V^{\dagger}$ is a linear functional of the dual space $V^{\dagger}$ of $V$ (determined in some way by $\vec{v}$),
 the outer product $\vec{u}\otimes\vec{v}$ is defined as the tensor product of $\varphi_{\vec{v}}$ and $\vec{u}$,
 which constitutes a map $A:V\rightarrow U$ by
 \begin{equation}
@@ -150,9 +150,10 @@ where $\varphi_{\vec{v}}(\vec{w})$ denotes the linear functional $\varphi_{\vec{
 \end{definition}
 
 \begin{remark}
-In matrix formalism, with respect to a given basis ${\vec{e}_i}$ of $\vec{u}$ and ${\vec{e}'_i}$ of $\vec{v}$,
+
+In matrix formalism, if $\varphi_{\vec{v}}$ is defined as in \eqref{eq:ip_mapping} and
 if $\vec{u}=\sum_i^m \vec{e}_iu_i$ and $\vec{v}=\sum_i^n\vec{e}'_iv_i$,
-the outer product can be written as matrix $A$ as
+the standard form of the outer product can be written as the matrix
 \begin{equation}
 \vec{u}\otimes\vec{v}=A=\left(
 \begin{array}{c c c c}
@@ -162,19 +163,19 @@ u_2v_1^* & u_2v_2^* & \cdots & u_2v_n^*\\
 u_mv_1^* & u_mv_2^* & \cdots & u_mv_n^*\\
 \end{array}
 \right)
-\text{ .}
+\text{ ,}
 \end{equation}
-
-The matrix can be equivalently obtained by matrix multiplication:
+which can be equivalently obtained by the rulrs of matrix multiplication
 \begin{equation}
 \vec{u}\otimes\vec{v}=\vec{u}\vec{v}^{\dagger} \text{ ,}
 \end{equation}
 if $\vec{u}$ and $\vec{v}$ are represented as $m\times 1$ and $n\times 1$ column vectors, respectively.
-Here, $\vec{v}^{\dagger}$ represents the conjugate transpose of $\vec{v}$.
-By definition, and as can be easily seen in the matrix representation, the following identity holds:
+Here, again, $\vec{v}^{\dagger}$ represents the conjugate transpose of $\vec{v}$.
+By definition, and as can be easily seen in matrix representation, the identity
 \begin{equation}
 (\vec{u}\otimes\vec{v})\vec{w}=\vec{u}(\vec{v},\vec{w})
 \end{equation}
+holds.
 \end{remark}
 
 \section{Spherical coordinates}