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KoordinatenSysteme.tex
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KoordinatenSysteme.tex
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\subsubsection{Kartesische Koordinaten}
\footnotesize{
\begin{flalign*}
&\text{Variablen:}\quad x,y,z \qquad \qquad \text{Einheitsvektoren:}\quad \vec{e}_x, \vec{e}_y,\vec{e}_z \qquad \qquad
\text{Rechtssystem:}\quad \vec{e}_x \times \vec{e}_y=\vec{e}_z &&&\\
&\text{Linienelemente:} \quad ds=\sqrt{d x^{2}+d y^{2}+d z^{2}} = dx \cdot \vec{e}_x + dy \cdot \vec{e}_y + dz \cdot \vec{e}_z \\
&\text{Volumenelemente:} \quad dV = dx \, dy \, dz &&&\\
&\text{Flächenelemente:} \quad dA_{xy}= dx \, dy \, \vec{e}_z \quad dA_{yz} = dy \, dz \, \vec{e}_x \quad dA_{xz} = dx \, dz \, \vec{e}_y
&&&\\
&\text{Skalarfeld:}\quad \phi = \phi(x;y;z) \qquad \text{Vektorfeld:} \quad \boxed{\vec{F} = \vec{F}(x;y;z) = F_x\vec{e}_x+F_y\vec{e}_y+F_z\vec{e}_z} &&&\\
&\text{\textbf{Gradient}:}\quad \opgrad \phi \equiv \nabla \phi=\frac{\partial \phi}{\partial x} \vec{e}_x+\frac{\partial \phi}{\partial y} \vec{e}_y+\frac{\partial \phi}{\partial z} \vec{e}_z
\qquad\qquad \text{\textbf{Divergenz}:}
\quad \opdiv \vec{D} \equiv \nabla \cdot \vec{D}=\frac{\partial D_{x}}{\partial x}+\frac{\partial D_{y}}{\partial y}+\frac{\partial D_{z}}{\partial z}
&&&\\
&\text{\textbf{Rotation}:} \quad \operatorname{rot} \vec{E} \equiv \nabla \times \vec{E} =
\left[\frac{\partial E_z}{\partial y}-\frac{\partial E_y}{\partial z}\right] \vec{e}_x+\left[\frac{\partial E_x}{\partial z}-\frac{\partial E_z}{\partial x}\right] \vec{e}_y+\left[\frac{\partial E_y}{\partial x}-\frac{\partial E_x}{\partial y}\right] \vec{e}_z&&&\\
&\text{\textbf{La-Place}}: \quad \Delta \vec{E} \equiv \nabla\cdot\nabla\cdot\vec{E} =\frac{\partial^{2}\vec{E}}{\partial x^{2}}+\frac{\partial^{2}\vec{E}}{\partial y^{2}}+\frac{\partial^{2}\vec{E}}{\partial z^{2}} \qquad
\Delta \vec{E} = \opgrad \opdiv \vec{E}-\operatorname{rot} \operatorname{rot} \vec{E} = \Delta E_x \vec{e}_x+\Delta E_y \vec{e}_y+\Delta E_z \vec{e}_z \\
&\Delta \vec{E} = \left[\frac{\partial^{2} E_x}{\partial x^{2}}+\frac{\partial^{2} E_x}{\partial y^{2}}+\frac{\partial^{2} E_x}{\partial z^{2}}\right] \vec{e}_x+\left[\frac{\partial^{2} E_y}{\partial x^{2}}+\frac{\partial^{2} E_y}{\partial y^{2}}+\frac{\partial^{2} E_y}{\partial z^{2}}\right] \vec{e}_y+ \left[\frac{\partial^{2} E_z}{\partial x^{2}}+\frac{\partial^{2} E_z}{\partial y^{2}}+\frac{\partial^{2} E_z}{\partial z^{2}}\right] \vec{e}_z
&&&\\
\end{flalign*}
}
\subsubsection{Zylinderkoordinaten}
Polarkoordinaten siehe S.386, Papula S.387,
\begin{flalign*}
&\text{Variablen:}\quad r,\varphi \text{(alternativ $\alpha$)},z \qquad \qquad \text{Einheitsvektoren:}\quad \vec{e}_r, \vec{e}_\varphi,\vec{e}_z \qquad \qquad
\text{Rechtssystem:}\quad \vec{e}_r \times \vec{e}_\varphi=\vec{e}_z &&&\\
&\text{Linienelemente:} \quad ds=\sqrt{d r^{2}+\mathbf{r} d \varphi^{2}+d z^{2}} = dr \cdot \vec{e}_r + \mathbf{r} \, d\varphi \cdot \vec{e}_\varphi + dz \cdot \vec{e}_z\\ &\text{Volumenelemente:} \quad dV = \mathbf{r} \, dr \, d\varphi \, dz &&&\\
&\text{Flächenelemente:} \quad dA_{r\varphi} = \mathbf{r} \, dr \, d\varphi \, \vec{e}_z \quad dA_{rz} = dr \, dz \, \vec{e}_\varphi \quad dA_{\varphi z} = \mathbf{r} \, d\varphi \, dz \, \vec{e}_r
&&&\\
&\text{Skalarfeld:}\quad \phi = \phi(x;\varphi;z) \qquad \text{Vektorfeld:} \quad \boxed{\vec{F} = \vec{F}(r;\varphi;z) = F_r\vec{e}_r+F_\varphi\vec{e}_\varphi+F_z\vec{e}_z} &&&\\
&\text{\textbf{Gradient}:}\quad \opgrad \phi \equiv \nabla \phi= \frac{\partial \phi}{\partial r} \vec{e}_r
+\frac{1}{r} \frac{\partial \phi}{\partial \varphi} \vec{e}_\varphi
+\frac{\partial \phi}{\partial z} \vec{e}_z &&&\\
&\text{\textbf{Divergenz}:}
\quad \opdiv \vec{D} \equiv \nabla \cdot \vec{D}=\frac{1}{r}\cdot\frac{\partial}{\partial r}\left(r\cdot\vec{D}_{r}\right)
+\frac{1}{r}\cdot\frac{\partial \vec{D}_{\varphi}}{\partial \varphi}
+\frac{\partial \vec{D}_{z}}{\partial z}
&&&\\
&\text{\textbf{Rotation}:} \quad \operatorname{rot} \vec{E} \equiv \nabla \times \vec{E}=
\left[\frac{1}{r}\cdot\frac{\partial E_z}{\partial \varphi}-\frac{\partial E_\varphi}{\partial z}\right] \vec{e}_r
+\left[\frac{\partial E_r}{\partial z}-\frac{\partial E_z}{\partial r}\right] \vec{e}_\varphi
+\frac{1}{r}\left[\frac{\partial}{\partial r}\left(r\cdot E_\varphi\right)-\frac{\partial E_r}{\partial \varphi}\right] \vec{e}_z&&&\\
&\text{\textbf{La-Place}}: \Delta\phi
= \frac{1}{r}\cdot\frac{\partial }{\partial r}\left(r\cdot\frac{\partial\phi}{\partial r}\right)
+ \frac{1}{r^2}\cdot\frac{\partial^2 \phi}{\partial \varphi^2}
+ \frac{\partial^2 \phi}{\partial z^2} \qquad
\Delta \vec{E} = \opgrad \opdiv \vec{E}-\operatorname{rot} \operatorname{rot} \vec{E} = \Delta E_r \vec{e}_r+\Delta E_\varphi \vec{e}_\varphi+\Delta E_z \vec{e}_z \\
& \Delta \vec{E} = \left[\Delta E_r-\frac{2}{r^{2}} \frac{\partial E_\varphi}{\partial \varphi}-\frac{E_r}{r^{2}}\right] \vec{e}_r
+\left[\Delta E_\varphi+\frac{2}{r^{2}} \frac{\partial E_r}{\partial \varphi}-\frac{E_\varphi}{r^{2}}\right] \vec{e}_\varphi+\left[\Delta E_z\right] \vec{e}_z
&&&\\
\end{flalign*}
\subsubsection{Kugelkoordinaten}
siehe Papula S.391/392
\begin{flalign*}
&\text{Variablen:}\quad r,\vartheta,\varphi \text{(alternativ $\alpha$)} \qquad \qquad \text{Einheitsvektoren:}\quad \vec{e}_r, \vec{e}_\vartheta,\vec{e}_\varphi \qquad \qquad
\text{Rechtssystem:}\quad \vec{e}_r \times \vec{e}_\vartheta=\vec{e}_\varphi &&&\\
&\text{Linienelemente:} \quad ds=\sqrt{d r^{2}+ \mathbf{r^2} \sin^2 \vartheta \, d \varphi^{2}+\mathbf{r^2} d \vartheta^{2}} = dr \cdot \vec{e}_r + r \, d\vartheta \cdot \vec{e}_\vartheta + r \, \sin \varphi \, d\varphi \cdot \vec{e}_\varphi \\
&\text{Volumenelemente:} \quad dV = \mathbf{r^2} \, \sin \vartheta \, dr \, d\vartheta \, d\varphi &&&\\
&\text{Flächenelemente:} \quad dA_{r\vartheta} = \mathbf{r}\, dr \, d\vartheta \, \vec{e}_\varphi \quad dA_{r\varphi} = \mathbf{r}\, \sin \, \vartheta \, dr \, d\varphi \, \vec{e}_\vartheta \quad dA_{\vartheta \varphi} = \mathbf{r^2} \, \sin \vartheta \, d\vartheta \, d\varphi \, \vec{e}_r
&&&\\
&\text{Skalarfeld:}\quad \phi = \phi(r;\vartheta;\varphi) \qquad \text{Vektorfeld:} \quad \boxed{\vec{F} = \vec{F}(r;\vartheta;\varphi) = F_r\vec{e}_r+F_\vartheta\vec{e}_\vartheta+F_\varphi\vec{e}_\varphi} &&&\\
&\text{\textbf{Gradient}:}\quad \quad \opgrad \phi \equiv \nabla \phi=\frac{\partial \phi}{\partial r} \vec{e}_r+\frac{1}{r} \frac{\partial \phi}{\partial \vartheta} \vec{e}_\vartheta+\frac{1}{r \sin \vartheta} \frac{\partial \phi}{\partial \varphi} \vec{e}_\varphi &&&\\
&\text{\textbf{Divergenz}:}
\quad \opdiv \vec{D} \equiv \nabla \cdot \vec{D}=\frac{1}{r^{2}} \frac{\partial\left(r^{2} D_{r}\right)}{\partial r}+\frac{1}{r \sin \vartheta} \frac{\partial\left(\sin \vartheta \cdot D_{\vartheta}\right)}{\partial \vartheta}+\frac{1}{r \sin \vartheta} \frac{\partial D_{\varphi}}{\partial \varphi}
&&&\\
&\text{\textbf{Rotation}:} \quad \operatorname{rot} \vec{E} = \frac{1}{r \sin \vartheta}\left[\frac{\partial\left(\sin \vartheta \cdot E_\varphi\right)}{\partial \vartheta}-\frac{\partial E_\vartheta}{\partial \varphi}\right] \vec{e}_r
+ \frac{1}{r}\left[\frac{1}{\sin \vartheta} \frac{\partial E_r}{\partial \varphi}-\frac{\partial (r E_\varphi)}{\partial r}\right] \vec{e}_\vartheta+\frac{1}{r}\left[\frac{\partial\left(r E_\vartheta\right)}{\partial r}-\frac{\partial E_r}{\partial \vartheta}\right] \vec{e}_\varphi&&&\\
&\text{\textbf{La-Place}}: \Delta\phi=\frac{1}{r^{2}}\left\{\frac{\partial}{\partial r}\left(r^2\cdot\frac{\partial\phi}{\partial r}\right)
+\frac{1}{\sin\vartheta}\cdot\frac{\partial}{\partial\vartheta}\left(\sin\vartheta\cdot\frac{\partial\phi}{\partial\vartheta}\right)
+\frac{1}{\sin^{2}\vartheta}\cdot\frac{\partial^{2}\phi}{\partial \varphi^{2}}\right\}\\
\end{flalign*}
Laplace Operator in Kugelkoordinaten, angewandt auf einen Vektor:
\footnotesize{
\begin{multline*}
\Delta \vec{E} =\left[\Delta E_r-\frac{2}{r^{2}} E_r-\frac{2}{r^{2} \sin \vartheta} \frac{\partial\left(\sin \vartheta \cdot E_\vartheta\right)}{\partial \vartheta}-\frac{2}{r^{2} \sin \vartheta} \frac{\partial E_\varphi}{\partial \varphi}\right] \vec{e}_r \\
+\left[\Delta E_\vartheta-\frac{E_\vartheta}{r^{2} \sin ^{2} \vartheta}+\frac{2}{r^{2}} \frac{\partial E_r}{\partial \vartheta}-\frac{2 \cot \vartheta}{r^{2} \sin \vartheta} \frac{\partial E_\varphi}{\partial \varphi}\right] \vec{e}_\vartheta \\
+\left[\Delta E_\varphi-\frac{E_\varphi}{r^{2} \sin ^{2} \vartheta}+\frac{2}{r^{2} \sin \vartheta} \frac{\partial E_r}{\partial \varphi}+\frac{2 \cot \vartheta}{r^{2} \sin \vartheta} \frac{\partial E_\vartheta}{\partial \varphi}\right] \vec{e}_\varphi
\end{multline*}
}
%\end{description}