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add latex math exercise equation numbering subexercise
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| \documentclass{scrartcl} | ||
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| \usepackage[aux]{rerunfilecheck} | ||
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| \usepackage{fontspec} | ||
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| \usepackage[ngerman]{babel} | ||
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| \usepackage{amsmath} | ||
| \usepackage{amssymb} | ||
| \usepackage{mathtools} | ||
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| \usepackage[ | ||
| math-style=ISO, | ||
| bold-style=ISO, | ||
| sans-style=italic, | ||
| nabla=upright, | ||
| partial=upright, | ||
| mathrm=sym, | ||
| ]{unicode-math} | ||
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| \usepackage[unicode]{hyperref} | ||
| \usepackage{bookmark} | ||
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| \begin{document} | ||
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| \section{Biot--Savart} | ||
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| Das Magnetfeld $\symbf{B}$ am Ort $\symbf{r}$ eines stromdurchflossenen Leiters | ||
| ergibt sich zu | ||
| \begin{equation*} | ||
| \symbf{B}(\symbf{r}) = \frac{\mu_0}{4 \symup{\pi}} | ||
| \int_V \symbf{j}(\symbf{r}') \times | ||
| \frac{\symbf{r} - \symbf{r}'}{\lvert \symbf{r} - \symbf{r}' \rvert^3} | ||
| \, \symup{d}V' \, . | ||
| \end{equation*} | ||
| Hierbei bezeichnet $\symbf{j}$ die Stromdichte am Ort $\symbf{r}'$ und $\mu_0$ | ||
| die magnetische Feldkonstante. | ||
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| \section{Fehlerfortpflanzung} | ||
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| \begin{equation*} | ||
| \sigma_f = \sqrt{ | ||
| \sum_{i = 1}^N | ||
| \left( \frac{\partial f}{\partial x_i} \sigma_i \right)^{\!\! 2} | ||
| } | ||
| \end{equation*} | ||
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| \section{Die vier Maxwellgleichungen} | ||
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| \begin{minipage}{.48\textwidth} | ||
| \begin{align} | ||
| \nabla \cdot \symbf{E} &= \frac{\rho}{\varepsilon_0} \\ | ||
| \addtocounter{equation}{+1} | ||
| \nabla \times \symbf{E} &= - \partial_t \symbf{B} | ||
| \end{align} | ||
| \end{minipage} | ||
| \hfill | ||
| \begin{minipage}{.48\textwidth} | ||
| \begin{align} | ||
| \addtocounter{equation}{-2} | ||
| \nabla \cdot \symbf{B} &= 0 \\ | ||
| \addtocounter{equation}{+1} | ||
| \nabla \times \symbf{B} &= \mu_0 \symbf{j} + \mu_0 \varepsilon_0 \partial_t \symbf{E} | ||
| \end{align} | ||
| \end{minipage} | ||
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| \section{Wellengleichung} | ||
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| Im Vakuum gelten $\rho = 0$ und $\symbf{j} = 0$, womit sich die Maxwellgleichungen zu | ||
| \begin{align*} | ||
| \nabla \cdot \symbf{E} = 0 \label{eqn:max1} \stepcounter{equation}\tag{\theequation} \\ | ||
| \nabla \cdot \symbf{B} &= 0 \\ | ||
| \nabla \times \symbf{E} &= - \partial_t \symbf{B} | ||
| \label{eqn:max3} \stepcounter{equation}\tag{\theequation} \\ | ||
| \nabla \times \symbf{B} &= \mu_0 \varepsilon_0 \partial_t \symbf{E} | ||
| \label{eqn:max4} \stepcounter{equation}\tag{\theequation} | ||
| \intertext{reduzieren. | ||
| Nach erneuter Anwendung der Rotation auf~\eqref{eqn:max3} ergibt sich} | ||
| \nabla \times \left( \nabla \times \symbf{E} \right) | ||
| &= \nabla \times \left( - \partial_t \symbf{B} \right) \, . | ||
| \intertext{Nach dem Satz von Schwarz lassen sich die partiellen Ableitungen vertauschen, | ||
| was zu} | ||
| \nabla \times \left( \nabla \times \symbf{E} \right) | ||
| &= - \partial_t \! \left( \nabla \times \symbf{B} \right) | ||
| \intertext{führt. Wir setzen auf der rechten Seite~\eqref{eqn:max4} ein:} | ||
| \nabla \times \left( \nabla \times \symbf{E} \right) | ||
| &= - \mu_0 \varepsilon_0 \partial_t^2 \symbf{E} \, . | ||
| \intertext{Aus der linken Seite wird mit} | ||
| \nabla \times \left( \nabla \times \symbf{E} \right) | ||
| &= \nabla \cdot \left( \nabla \cdot \symbf{E} \right) - \increment \symbf{E} | ||
| \shortintertext{und Ausnutzen von~\eqref{eqn:max1}} | ||
| - \increment\symbf{E} &= -\mu_0 \varepsilon_0 \partial_t^2 \symbf{E} . | ||
| \intertext{Dies ist die Wellengleichung für das elektrische Feld, | ||
| in der sich die Lichtgeschwindigkeit} | ||
| \symup{c} &= \frac{1}{\sqrt{\mu_0 \varepsilon_0}} | ||
| \intertext{identifizieren lässt. | ||
| Damit können wir} | ||
| \left(\increment - \frac{1}{\symup{c}^2} \frac{\partial^2}{\partial t^2} \right) | ||
| \! \symbf{E} &= 0 | ||
| \end{align*} | ||
| schreiben. | ||
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| \section{Wellengleichung} | ||
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| Ebene Welle: | ||
| \begin{equation} | ||
| \nabla^2 A - \frac{1}{\symup{c}^2} \frac{\partial^2}{\partial t^2} A = 0 | ||
| \end{equation} | ||
| Eine Lösung: | ||
| \begin{equation} | ||
| A = A_0 \exp(\mathrm{i} (\symbf{k} \symbf{x} - \omega t)) | ||
| \end{equation} | ||
| Gruppen- und Phasengeschwindigkeit: | ||
| \begin{align} | ||
| v_\text{Gr} &= \frac{\partial \omega}{\partial k} & | ||
| v_\text{Ph} &= \frac{\omega}{k} | ||
| \end{align} | ||
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| \section{Multipolentwicklung} | ||
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| \begin{align*} | ||
| \Phi(\symbf{r}) &= \frac{1}{4 \symup{\pi} \varepsilon_0} \left( | ||
| \frac{Q}{r} + \frac{\symbf{r} \cdot \symbf{p}}{r^3} | ||
| + \frac{1}{2} \sum_{k, l} Q_{k l} \frac{r_k r_l}{r^5} + \dotsb | ||
| \right) \, , | ||
| \shortintertext{wobei} | ||
| Q_{k l} &= \sum_{i\,=\,1}^n q_i | ||
| \left( 3 r_{i k} r_{i l} - r_i^2 \symup{\delta}_{k l} \right) \, . | ||
| \end{align*} | ||
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| \section{Jacobi-Matrix} | ||
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| \begin{equation} | ||
| \symbf{J} = | ||
| \begin{pmatrix} | ||
| \frac{\partial f_1}{\partial x_1} & \cdots & \frac{\partial f_1}{\partial x_n} \\ | ||
| \vdots & \ddots & \vdots \\ | ||
| \frac{\partial f_m}{\partial x_1} & \cdots & \frac{\partial f_m}{\partial x_n} | ||
| \end{pmatrix} | ||
| \end{equation} | ||
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| \section{Harmonischer Oszillator} | ||
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| \begin{equation} | ||
| \ddot{x} + 2 \gamma \dot{x} + \omega_0^2 x = 0 | ||
| \end{equation} | ||
| Reelle Lösung: | ||
| \begin{align} | ||
| x(t) &= \symup{e}^{-\gamma t} (A \cos(\omega t) + B \sin(\omega t)) \, , | ||
| \shortintertext{mit} | ||
| \omega &= \sqrt{\omega_0^2 - \gamma^2} \, . | ||
| \end{align} | ||
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| \end{document} |
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