pp + so started

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

diff --git a/physics_compact/math_app.tex b/physics_compact/math_app.tex
index f8361e8..4f7822e 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}
@@ -162,22 +163,82 @@ 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 rules 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}
 
+Cartesian coordinates $\vec{r}(x,y,z)$ are related to spherical coordinates $\vec{\tilde{r}}(r,\theta,\phi)$ by
+\begin{eqnarray}
+x&=&r\sin\theta\cos\phi\\
+y&=&r\sin\theta\sin\phi\\
+z&=&r\cos\theta
+\end{eqnarray}
+and
+\begin{eqnarray}
+r&=&(x^2+y^2+z^2)^{1/2}\\
+\theta&=&\arccos(z/r)\\
+\phi&=&\arctan(y/x)
+\end{eqnarray}
+The total differentials $dq_i$ and $dq'_i$ of two coordinate systems are related by partial derivatives.
+\begin{equation}
+dq_i=\sum_j \frac{\partial q_i}{\partial q'_j}dq'_j
+\end{equation}
+\begin{definition}[Jacobi matrix]
+The matrix J with components
+\begin{equation}
+J_{ij}=\frac{\partial q_i}{\partial q'_j}
+\end{equation}
+is called the Jacobi matrix.
+\end{definition}
+
+For cartesian and spherical coordinates the relation of the translations are
+\begin{eqnarray}
+dx&=&\frac{\partial x}{\partial r}dr +
+     \frac{\partial x}{\partial \theta}d\theta +
+     \frac{\partial x}{\partial \phi}d\phi\\
+dy&=&\frac{\partial y}{\partial r}dr +
+     \frac{\partial y}{\partial \theta}d\theta +
+     \frac{\partial y}{\partial \phi}d\phi\\
+dz&=&\frac{\partial z}{\partial r}dr +
+     \frac{\partial z}{\partial \theta}d\theta +
+     \frac{\partial z}{\partial \phi}d\phi\\
+\end{eqnarray}
+and
+\begin{eqnarray}
+dr&=&\frac{\partial r}{\partial x}dx +
+     \frac{\partial r}{\partial y}dy +
+     \frac{\partial r}{\partial z}dz\\
+d\theta&=&\frac{\partial \theta}{\partial x}dx +
+          \frac{\partial \theta}{\partial y}dy +
+          \frac{\partial \theta}{\partial z}dz\\
+d\phi&=&\frac{\partial \phi}{\partial x}dx +
+        \frac{\partial \phi}{\partial y}dy +
+        \frac{\partial \phi}{\partial z}dz\\
+\end{eqnarray}
+and vectorial translations using the Jacobi matrix are given by matrix multiplications
+\begin{equation}
+d\vec{r}(x,y,z)=Jd\vec{\tilde{r}}(r,\theta,\phi)
+\end{equation}
+and
+\begin{equation}
+d\vec{\tilde{r}}(r,\theta,\phi)=J^{-1}d\vec{r}(x,y,z) \text{ .}
+\end{equation}
+$J$ and $J^{-1}$ are explicitily given by
+\begin{equation}
+\end{equation}
+
 \section{Fourier integrals}