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 f1ac777..4f7822e 100644 (file)
--- a/physics_compact/math_app.tex
+++ b/physics_compact/math_app.tex
@@ -165,7 +165,7 @@ u_mv_1^* & u_mv_2^* & \cdots & u_mv_n^*\\
 \right)
 \text{ ,}
 \end{equation}
-which can be equivalently obtained by the rulrs of 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}
@@ -180,5 +180,65 @@ holds.
 
 \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}