Formulaires/BA3 - Physique III/BA3 - Physique III.tex

\documentclass[fontsize=8pt, paper=a4, pagesize, DIV=calc]{scrartcl}

\input{../Base.tex}

\title{Formulaire de Physique III}

\begin{document}

\begin{tabu}to \textwidth{ |X|X| }
\hline

\textbf{Fluides} \newline
  $ \dif\vec{F} = - P \cdot \dif\vec{\sigma} $ \newline
  $ \frac{\partial\rho}{\partial t} + \nabla \bullet \left( \rho \cdot \vec{v} \right) = 0 $ \hfill Éq. de continuité \newline
  $ \frac{1}{2} \cdot \rho \cdot v^2 + \rho \cdot g \cdot z + P = const $ \hfill Éq. de Bernoulli \newline
  $ - \nabla P + \rho \cdot \vec{g} + \eta \cdot \nabla^2 \vec{v} = \rho \cdot \left( \frac{\partial \vec{v}}{\partial t} + \left( \vec{v} \bullet \nabla \right) \vec{v} \right) $ \hfill Éq. d'Euler \newline
  $ \dif \vec{x} \parallel \vec{v} \quad \Leftrightarrow \quad \frac{\dif x}{\dif y} = \frac{v_x}{v_y} $ \hfill Lignes de courant
&
\textbf{Fluides II} \newline
  $ \Delta P = \frac{8 \cdot \eta \cdot L \cdot D}{\pi \cdot R^4} $ \hfill Loi de Poiseuille \newline
  $ v \left( r \right) = \frac{\Delta P}{4 \cdot \eta \cdot L} \cdot \left( R^2 - r^2 \right) $ \hfill Profil de vitesse de Poiseuille \newline
  $ \vec{F}_{visc} = \eta \cdot \frac{S \cdot \left( \vec{v}_{sup} - \vec{v}_{inf} \right)}{d} $ \newline
  $ \dif \vec{F}_{visc} = \eta \cdot \nabla^2 \vec{v} \cdot \dif V $ \newline
  $ \frac{\dif E}{\dif t} = -\Phi_{en} + \frac{\dif W}{\dif t} $
\\\hline

\textbf{Équations de Maxwell} \newline
  $ \begin{array}{@{}l@{\qquad\qquad}l} \nabla \bullet \vec{E} = \frac{\rho}{\varepsilon_0} & \nabla \times \vec{E} = - \frac{\partial}{\partial t} \vec{B} \\ \nabla \bullet \vec{B} = 0  & \nabla \times \vec{B} = \mu_0 \cdot \vec{j} + \frac{1}{c^2} \cdot \frac{\partial}{\partial t} \vec{E} \end{array} $
&
\textbf{Formes intégrales} \newline
  $ \oiint_\Sigma \vec{E} \bullet \dif\vec{\sigma} = \frac{Q_{int}}{\varepsilon_0} = \Phi_E $ \hfill Th. de Gauss \newline
  $ \oint_\Gamma \vec{B} \bullet \dif\vec{l} = \mu_0 \cdot I + \frac{1}{c^2} \cdot \frac{\dif \Phi_E}{\dif t} \hspace{8mm} I_d = \varepsilon_0 \cdot \frac{\dif \Phi_E}{\dif t} $ \hfill Th. d'Ampère \newline
  $ V = \oint_\Gamma \vec{E} \bullet \dif\vec{l} = - \frac{\dif \Phi_M}{\dif t} $ \hfill Induction
\\\hline

\end{tabu}

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\begin{tabu}to \textwidth{ |X| }

\textbf{Électrostatique} \newline
  $ \vec{E} = \frac{1}{4 \cdot \pi \cdot \varepsilon_0} \cdot \left( \sum q_i \cdot \frac{\vec{r}-\vec{r_i}}{\abs{\vec{r}-\vec{r_i}}^3} + \int_\Gamma \frac{\lambda \left( \vec{r}' \right) \cdot \left( \vec{r}-\vec{r}' \right)}{\abs{\vec{r}-\vec{r}'}^3} \cdot \dif l  + \iint_\Sigma \frac{\sigma \left( \vec{r}' \right) \cdot \left( \vec{r}-\vec{r}' \right)}{\abs{\vec{r}-\vec{r}'}^3} \cdot \dif\sigma + \iiint_V \frac{\rho \left( \vec{r}' \right) \cdot \left( \vec{r}-\vec{r}' \right)}{\abs{\vec{r}-\vec{r}'}^3} \cdot \dif V \right) $ \newline

  $ V = \frac{1}{4 \cdot \pi \cdot \varepsilon_0} \cdot \left( \sum q_i \cdot \frac{1}{\abs{\vec{r}-\vec{r_i}}} + \int_\Gamma \frac{\lambda \left( \vec{r}' \right)}{\abs{\vec{r}-\vec{r}'}} \cdot \dif l + \iint_\Sigma \frac{\sigma \left( \vec{r}' \right)}{\abs{\vec{r}-\vec{r}'}} \cdot \dif\sigma + \iiint_V \frac{\rho \left( \vec{r}' \right)}{\abs{\vec{r}-\vec{r}'}} \cdot \dif V \right) $
\\\hline

\textbf{Magnétostatique} \newline
  $ \vec{B} = \frac{\mu_0 \cdot I}{4 \cdot \pi} \cdot \oint_\Gamma \frac{\vec{u}_t \times \vec{u}_r}{r^2} \cdot \dif l = \frac{\mu_0}{4 \cdot \pi} \cdot \iiint_V \frac{\vec{j} \left( \vec{x}' \right) \times \vec{u}_r}{r^2} \cdot \dif^3 x' $ \hfill Loi de Biot-Savart
\\\hline

\end{tabu}

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\begin{tabu}to \textwidth{ |X|X| }

\textbf{Dipôle électrique} \newline
  $ \vec{p} = q \cdot \vec{r}_+ - q \cdot \vec{r}_- = q \cdot \vec{a} $ \newline
  $ \vec{\tau} = \vec{p} \times \vec{E}_{ext} $ \newline
  $ U_{\acute el} = - \vec{p} \bullet \vec{E}_{ext} $ \newline
  $ E_r = - \frac{\partial V}{\partial r} = \frac{2 \cdot p \cdot \cos \left( \theta \right)}{4 \cdot \pi \cdot \varepsilon_0 \cdot r^3} $ \newline
  $ E_\theta = - \frac{1}{r} \cdot \frac{\partial V}{\partial \theta} = \frac{p \cdot \sin \left( \theta \right)}{4 \cdot \pi \cdot \varepsilon_0 \cdot r^3} $
&
\textbf{Dipôle magnétique} \newline
  $ \vec{M} = I \cdot \vec{S} $ \newline
  $ \vec{\tau} = \vec{M} \times \vec{B}_{ext} $ \newline
  $ U_{mag} = - \vec{M} \bullet \vec{B}_{ext} $
\\\hline

\textbf{Polarisation} \newline
  $ \sigma_P = \vec{P} \bullet \vec{e}_n $ \newline
  $ \vec{\left\langle E \right\rangle} = \frac{E_{ext}}{\varepsilon_r} $ \newline
  $ \vec{P} = n \cdot \vec{\left\langle p \right\rangle} $
&
\textbf{Aimantation} \newline
  $ j_{li\acute e} = \vec{M} \bullet \vec{e}_n $ \newline
  $ \vec{\left\langle B \right\rangle} = \mu_r \cdot B_{ext} $
\\\hline

\textbf{Champ électrique D} \newline
  $ \vec{D} = \varepsilon_0 \cdot \vec{E} + \vec{P} $ \newline
  $ \nabla \bullet \vec{D} = \rho_{libre} $ \newline
  $ \vec{D} = \varepsilon_0 \cdot \vec{E} + \vec{P} = \varepsilon_0 \cdot \left( 1 + \chi \right) \cdot \vec{E} = \varepsilon \cdot \vec{E} $
&
\textbf{Champ magnétisant H} \newline
  $ \vec{H} = \frac{1}{\mu_0} \cdot \vec{B} - \vec{M} $ \newline
  $ \nabla \times \vec{H} = \vec{j}_{libre} $ \newline
  $ \vec{B} = \mu_0 \cdot \left( \vec{H} + \vec{M} \right) = \mu_0 \cdot \left( 1 + \chi \right) \cdot \vec{H} = \mu \cdot \vec{H} $
\\\hline

\textbf{Conditions au bord} \newline
  $ E_{1t} = E_{2t} $ \newline
  $ D_{1n} = D_{2n} \quad \Rightarrow \quad \varepsilon_{r1} \cdot E_{1n} = \varepsilon_{r2} \cdot E_{2n} $ \hfill Isolant-Isolant \newline
  $ D_{1n} = \sigma_{libre} \quad \Rightarrow \quad E_{1n} = \frac{\sigma_{libre}}{\varepsilon_0 \cdot \varepsilon_{r1}} $ \hfill Isolant-Métal
&
\textbf{Conditions au bord} \newline
  $ H_{1t} = H_{2t} \quad \Rightarrow \quad \frac{B_{1t}}{\mu_{r1}} = \frac{B_{2t}}{\mu_{r2}} $ \newline
  $ B_{1n} = B_{2n} $
\\\hline

\textbf{Électrostatique} \newline
  $ \vec{F} = q \cdot \left( \vec{E} + \vec{v} \times \vec{B} \right) $ \hfill Force de Lorentz \newline
  $ \vec{E} = - \nabla V $ \newline
  $ V \left( \vec{r} \right) = V \left( \vec{r_0} \right) - \int_{\vec{r_0}}^{\vec{r}} \vec{E} \bullet \dif\vec{l} $ \newline
  $ \nabla^2 V \left( \vec{r} \right) = - \frac{\rho}{\varepsilon_0} $ \hfill Équation de Poisson \newline
  $ W_{AB} = \int_{\vec{r}_A}^{\vec{r}_B} q \cdot \vec{E} \cdot \dif \vec{l} = q \cdot V \left( \vec{r}_A \right) - q \cdot V \left( \vec{r}_B \right) $ \newline
  $ U_E = \frac{1}{4 \cdot \pi \cdot \varepsilon_0} \cdot \frac{1}{2} \cdot \sum_{i = 1}^N \sum_{j = 1, j \neq i}^N \frac{q_i \cdot q_j}{\abs{\vec{r}_i - \vec{r}_j}} $ \hfill Distribution discrète \newline
  $ U_E = \frac{1}{2} \cdot \iiint_V \rho \left( \vec{r} \right) \cdot V \left( \vec{r} \right) \cdot \dif V $ \hfill Distribution continue \newline
  $ \vec{j} = n \cdot q \cdot \vec{v} = \rho \cdot \vec{v} = \sigma \cdot \vec{E} $ \hfill Densité de courant, $ \sigma $ conductivité \newline
  $ \vec{E} = 0 \comma V = \cte $ \hfill Dans un conducteur
&
\textbf{Magnétostatique} \newline
  $ r = \frac{m \cdot v}{q \cdot B_0} $ \hfill Rayon de Larmor \newline
  $ \vec{\omega} = - \frac{q}{m} \cdot \vec{B_0} $ \hfill Fréquence de cyclotron \newline
  $ \vec{F} = I \cdot \int_\Gamma \dif\vec{l} \times \vec{B} $ \hfill Force de Laplace \newline
  $ \frac{F}{L} = \frac{\mu_0 \cdot I_1 \cdot I_2}{2 \cdot \pi \cdot d} $ \hfill Force entre deux conducteurs \newline
  $ B = \mu_0 \cdot I \cdot n $ \hfill Champ dans une bobine \newline
  $ \vec{B} \left( \vec{x} \right) = \frac{1}{c^2} \cdot \vec{v} \times \vec{E} \left( \vec{x} \right) $ \hfill Charge en mouvement \newline
  $ F_{\acute el} = \gamma \cdot F_{Lorentz} $ \hfill Effet relatif \newline
  $ \nabla^2 \vec{A} = - \mu_0 \cdot \vec{j} $ \hfill Potentiel Vecteur
\\\hline

\textbf{Condensateur} \newline
  $ Q = C \cdot \Delta V $ \newline
  $ U = \frac{1}{2} \cdot C \cdot V^2 = \frac{Q^2}{2 \cdot C} $ \newline
  $ V = \frac{1}{C} \cdot \int I \cdot \dif t $ \newline
  $ C = \frac{\varepsilon_0 \cdot A}{d} $ \hfill Pour un condensateur plan \newline
  $ C = 4 \cdot \pi \cdot \varepsilon_0 \cdot \frac{R_b \cdot R_a}{R_b - R_a} $ \hfill Pour un condensateur sphère
&
\textbf{Inductance} \newline
  $ \Phi_M = L \cdot I $ \newline
  $ U = \frac{1}{2} \cdot L \cdot I^2 $ \newline
  $ V = L \cdot \frac{\dif I}{\dif t} $
\\\hline

\end{tabu}

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\begin{tabu}to \textwidth{ |X|X| }
\hline

\textbf{Ondes} \newline
  $ \frac{\partial^2 \xi}{\partial t^2} = v^2 \cdot \nabla^2 \xi $ \hfill Équation d'Alembert \newline
  $ \xi \left( x, t \right) = f \left( x - v \cdot t \right) +  g \left( x + v \cdot t \right) $ \newline
  $ \xi \left( x, t \right) = \xi_0 \cdot \sin \left( k \cdot x - \omega \cdot t \right) $ \newline
  $ v = \frac{\omega}{k} = \lambda \cdot \nu $ \newline
  $ v_g = v + k \cdot \frac{\dif v}{\dif t} $ \newline
  $ v_{tr} = - \omega \cdot \xi_0 \cdot \cos \left( k \cdot x - \omega \cdot t \right) $ \newline
  $ k \cdot \lambda = 2 \cdot \pi $ \newline
  $ \nu' = \left( \frac{v - v_O}{v - v_S} \right) \cdot \nu $ \hfill Effet Doppler \newline
  $ \nu' = \left( \frac{\sqrt{1 - v_R/c}}{\sqrt{1 + v_R/c}} \right) \cdot \nu $ \hfill Effet Doppler (lumière) \newline
  $ I = \frac{P}{A} = \frac{1}{A} \cdot \frac{\dif W}{\dif t} \propto \xi^2 $ \newline
  $ n = 10 \cdot \log_{10} \left( \frac{I}{I_0} \right) $
&
\textbf{Électromagnétisme} \newline
  $ E = c \cdot B $ \newline
  $ c^2 = \frac{1}{\mu_0 \cdot \varepsilon_0} $ \newline
  $ I = S = c \cdot u_{EM} $ \newline
  $ u_E = \frac{1}{2} \cdot \vec{E} \bullet \vec{D} = \frac{1}{2} \cdot \varepsilon_0 \cdot \abs{\vec{E}}^2 $ \newline
  $ u_M = \frac{1}{2} \cdot \vec{B} \bullet \vec{H} = \frac{1}{2 \cdot \mu_0} \cdot \abs{\vec{B}}^2 $ \newline
  $ u_E = u_M = \frac{1}{2} \cdot u_{EM} $ \newline
  $ \vec{S} = \frac{1}{\mu_0} \cdot \vec{E} \times \vec{B} $ \newline
  $ \frac{\partial u_{EM}}{\partial t} + \nabla \bullet \vec{S} = 0 $ \hfill Théorème de Poynting \newline
  $ P = \frac{I}{c} $ \hfill Pression de radiation (absorbtion) \newline
  $ P = \frac{2 \cdot I}{c} $ \hfill Pression de radiation (réflexion) \newline
  $ \vec{p} = \varepsilon_0 \cdot \vec{E} \times \vec{B} = \frac{\vec{S}}{c} $
\\\hline

\textbf{Onde stationnaire} \newline
  $ \xi = 2 \cdot \xi_0 \cdot \sin \left( k \cdot x \right) \cdot \cos \left( \omega \cdot t \right) $ \newline
  $ L = m \cdot \frac{\lambda}{2} $ \hfill Corde fixée aux 2 ext. / Tuyeau ouvert \newline
  $ L = \left( 2 \cdot m + 1 \right) \cdot \frac{\lambda}{4} $ \hfill Corde fixée à 1 ext. / Tuyeau fermé \newline
  $ k \cdot x = m \cdot \pi $ \hfill Noeud ou Ventre \newline
  $ k \cdot x = \left( m + \frac{1}{2} \right) \cdot \pi $ \hfill Ventre ou Noeud
&
\textbf{Interférences} \newline
  $ \xi_0^2 = \xi_{01}^2 + \xi_{02}^2 + 2 \cdot \xi_{01} \cdot \xi_{02} \cdot \cos \left( \delta \right) $ \newline
  $ \xi_0^2 = 4 \cdot \xi_{01}^2 \cdot \cos^2 \left( \frac{\delta}{2} \right) $ \hfill Même amplitude \newline
  $ \xi \left( t \right) = \xi_0 \cdot \cos \left( \omega \cdot t - k \cdot r_1 + \delta/2 \right) $ \hfill Même amplitude \newline
  $ I = I_0 \cdot \cos^2 \left( \frac{\delta}{2} \right) $ \hfill Même amplitude \newline
  $ \delta = k \cdot \Delta r = k \cdot a \cdot \sin \left( \theta \right)  $ \newline
  $ \delta = 2 \cdot m \cdot \pi $ \hfill Max \newline
  $ \delta = \left( 2 \cdot m + 1 \right) \cdot \pi $ \hfill Min
\\\hline

\textbf{Diffraction} \newline
  $ I = I_0 \cdot \left( \frac{\sin \left( \pi \cdot b \cdot \sin \left( \theta / \lambda \right) \right)}{\pi \cdot b \cdot \sin \left( \theta / \lambda \right)} \right)^2 $ \newline
  $ b \cdot \sin \left( \theta \right) = \pm m \cdot \lambda \hspace{15mm} \left( m \neq 0 \right) $ \hfill Zéro \newline
  $ b \cdot \sin \left( \theta \right) = \pm \left( m + \frac{1}{2} \right) \cdot \lambda \hspace{5mm} \left( m \neq 0 \right) $ \hfill Max \newline
  $ \theta \geqslant \frac{\lambda}{b} $ \hfill Critère de Rayleigh (fente) \newline
  $ \theta \geqslant 1.22 \cdot \frac{\lambda}{D} $ \hfill Critère de Rayleigh (ouv. circ.) \newline
  $ 2 \cdot d \cdot \sin \left( \theta \right) = m \cdot \lambda $ \hfill Condition de Bragg
&
\textbf{Optique} \newline
  $ n_i \cdot \sin \left( \theta_i \right) = n_r \cdot \sin \left( \theta_r \right) $ \hfill Loi de Snell-Descartes \newline
  $ \sin \left( \theta_i \right) > \frac{n_r}{n_i} $ \hfill Réflexion totale \newline
  $ v = \frac{c}{n} $ \newline
  $ \lambda_n = \frac{\lambda}{n} $ \newline
  $ k_n = n \cdot k $ \newline
  $ n = \sqrt{\varepsilon_r \cdot \mu_r} \sim \sqrt{\varepsilon_r} $
\\\hline

\textbf{Polarisation} \newline
  $ \tan \left( \theta \right) = \frac{n_r}{n_i} $ \hfill Angle de Brewster \newline
  Angle de Brewster \hfill $ \quad \Rightarrow \quad $ \hfill Polarisation $ \pi $ 100\% transmise et 0\% réfléchie \newline
  $ I = I_m \cdot \cos^2 \left( \theta \right) $ \hfill Loi de Malus
&
\textbf{Interférences à N sources} \newline
  $ I = I_0 \cdot \left( \frac{\sin \left( N \cdot \pi \cdot a \cdot \sin \left( \theta / \lambda \right) \right)}{\sin \left( \pi \cdot a \cdot \sin \left( \theta / \lambda \right) \right)} \right) $ \newline
  $ a \cdot \sin \left( \theta \right) = m \cdot \lambda, \qquad I = N^2 \cdot I_0 $ \hfill Max \newline
  $ a \cdot \sin \left( \theta \right) = \frac{m'}{N} \cdot \lambda, \qquad \frac{m'}{N} \neq m $ \hfill Min
\\\hline

\textbf{Polarisation $ \boldsymbol{\sigma} $ et polarisation $ \boldsymbol{\pi} $} \newline
    \includegraphics[width=0.3\textwidth, keepaspectratio=true]{./Polarisation.png}
&
\textbf{Théorèmes} \newline
  $ \iiint_V \nabla f \cdot \dif V = \oiint_\Sigma f \cdot \dif\vec{\sigma} $ \hfill Th. du Gradient \newline
  $ \iiint_V \nabla \bullet \vec{F} \cdot \dif V = \oiint_\Sigma \vec{F} \bullet \dif\vec{\sigma} $ \hfill Th. de la Divergence \newline
  $ \iint_\Sigma \left( \nabla \times \vec{F} \right) \bullet \dif\vec{\sigma} = \oint_\Gamma \vec{F} \bullet \dif\vec{l} $ \hfill Th. de Stokes \newline
  $ \frac{\dif F}{\dif t}
    = \frac{\partial F}{\partial t}
    + \frac{\partial F}{\partial x} \cdot \frac{\dif x}{\dif t}
    + \frac{\partial F}{\partial y} \cdot \frac{\dif y}{\dif t}
    + \frac{\partial F}{\partial z} \cdot \frac{\dif z}{\dif t}
    = \frac{\partial F}{\partial t} + \left( \vec{v} \bullet \nabla \right) F $
\\\hline

\textbf{Opérateurs en coordonées cylindriques} \newline
\footnotesize{
  $ \nabla U =
    \begin{pmatrix}
      \frac{\partial U}{\partial \rho} \\
      \frac{1}{\rho} \nocdot \frac{\partial U}{\partial \phi} \\
      \frac{\partial U}{\partial z} \\
    \end{pmatrix}
  $ \newline

  $ \nabla \bullet \vec{A}
    = \frac{1}{\rho} \nocdot \frac{\partial \left( \rho \nocdot A_\rho \right)}{\partial \rho}
    + \frac{1}{\rho} \nocdot \frac{\partial A_\phi}{\partial \phi}
    + \frac{\partial A_z}{\partial z}
  $ \newline
  $ \nabla \times \vec{A} =
    \begin{pmatrix}
      \frac{1}{\rho} \nocdot \frac{\partial A_z}{\partial \phi} - \frac{\partial A_\phi}{\partial z} \\
      \frac{\partial A_\rho}{\partial z} - \frac{\partial A_z}{\partial \rho} \\
      \frac{1}{\rho} \nocdot \frac{\partial \left( \rho \nocdot A_\phi \right)}{\partial \rho} -  \frac{1}{\rho} \nocdot \frac{\partial A_\rho}{\partial \phi} \\
    \end{pmatrix}
  $ \newline
  $ \nabla^2 U
    = \frac{1}{\rho} \nocdot \frac{\partial}{\partial \rho} \left( \rho \nocdot \frac{\partial U}{\partial \rho} \right)
    + \frac{1}{\rho^2} \nocdot \frac{\partial^2 U}{\partial \phi^2}
    + \frac{\partial^2 U}{\partial z^2}
  $ \newline
  $ \hphantom{\nabla^2 U}
    = \frac{\partial^2 U}{\partial \rho^2}
    + \frac{1}{\rho} \nocdot \frac{\partial U}{\partial \rho}
    + \frac{1}{\rho^2} \nocdot \frac{\partial^2 U}{\partial \phi^2}
    + \frac{\partial^2 U}{\partial z^2}
  $ \newline
  $ \vec{\nabla}^2 \vec{A} =
    \begin{pmatrix}
      \nabla^2 A_\rho - \frac{A_\rho}{\rho^2} - \frac{2}{\rho^2} \nocdot \frac{\partial A_\phi}{\partial \phi} \\
      \nabla^2 A_\phi - \frac{A_\phi}{\rho^2} + \frac{2}{\rho^2} \nocdot \frac{\partial A_\rho}{\partial \phi} \\
      \nabla^2 A_z \\
    \end{pmatrix}
  $
}
&
\textbf{Opérateurs en coordonées sphériques} \newline
\footnotesize{
  $ \nabla U =
    \begin{pmatrix}
      \frac{\partial U}{\partial r} \\
      \frac{1}{r} \nocdot \frac{\partial U}{\partial \theta} \\
      \frac{1}{r \nocdot \sin \left( \theta \right)} \nocdot \frac{\partial U}{\partial \phi} \\
    \end{pmatrix}
  $ \newline
  $ \nabla \bullet \vec{A}
    = \frac{1}{r^2} \nocdot \frac{\partial \left( r^2 \nocdot A_r \right)}{\partial r}
    + \frac{1}{r \nocdot \sin \left( \theta \right)} \nocdot \frac{\partial \left( \sin \left( \theta \right) \nocdot A_\theta \right)}{\partial \theta}
    + \frac{1}{r \nocdot \sin \left( \theta \right)} \nocdot \frac{\partial A_\phi}{\partial \phi}
  $ \newline
  $ \nabla \times \vec{A} =
    \begin{pmatrix}
      \frac{1}{r \nocdot \sin \left( \theta \right)} \nocdot \left[ \frac{\partial \left( \sin \left( \theta \right) \nocdot A_\phi \right)}{\partial \theta} - \frac{\partial A_\theta}{\partial \phi} \right] \\
      \frac{1}{r \nocdot \sin \left( \theta \right)} \nocdot \frac{\partial A_r}{\partial \phi} - \frac{1}{r} \nocdot \frac{\partial \left( r \nocdot A_\phi \right)}{\partial r} \\
      \frac{1}{r} \nocdot \left[ \frac{\partial \left( r \nocdot A_\theta \right)}{\partial r} - \frac{\partial A_r}{\partial \theta} \right] \\
    \end{pmatrix}
  $ \newline
  $ \nabla^2 U
    = \frac{1}{r^2 \nocdot \sin \left( \theta \right)} \nocdot \left[ \frac{\partial}{\partial r} \left( r^2 \nocdot \sin \left( \theta \right) \nocdot \frac{\partial U}{\partial r} \right)
    + \frac{\partial}{\partial \theta} \left( \sin \left( \theta \right) \nocdot \frac{\partial U}{\partial \theta} \right)
    + \frac{\partial}{\partial \phi} \left( \frac{1}{\sin \left( \theta \right)} \nocdot \frac{\partial U}{\partial \phi} \right) \right]
  $ \newline
  $ \hphantom{\nabla^2 U}
    = \frac{1}{r^2} \nocdot \frac{\partial}{\partial r} \left( r^2 \nocdot \frac{\partial U}{\partial r} \right)
    + \frac{1}{r^2 \nocdot \sin \left( \theta \right)} \nocdot \frac{\partial}{\partial \theta} \left( \sin \left( \theta \right) \nocdot \frac{\partial U}{\partial \theta} \right)
    + \frac{1}{r^2 \nocdot \sin^2 \left( \theta \right)} \nocdot \frac{\partial^2 U}{\partial \phi^2}
  $ \newline
  $ \hphantom{\nabla^2 U}
    = \frac{\partial^2 U}{\partial r^2}
    + \frac{2}{r} \nocdot \frac{\partial U}{\partial r}
    + \frac{1}{r^2} \nocdot \frac{\partial^2 U}{\partial \theta^2}
    + \frac{1}{r^2} \nocdot \cot \theta \nocdot \frac{\partial U}{\partial \theta}
    + \frac{1}{r^2 \nocdot \sin^2 \left( \theta \right)} \nocdot \frac{\partial^2 U}{\partial \phi^2}
  $ \newline
  $ \vec{\nabla}^2 \vec{A} =
    \begin{pmatrix}
      \nabla^2 A_r - \frac{2}{r^2} \nocdot \left( A_r + \frac{1}{\sin \left( \theta \right)} \nocdot \frac{\partial}{\partial \theta} \left( \sin \left( \theta \right) \nocdot A_\theta \right) + \frac{1}{\sin \left( \theta \right)} \nocdot \frac{\partial A_\phi}{\partial \phi} \right) \\
      \nabla^2 A_\theta + \frac{2}{r^2} \nocdot \left( \frac{\partial A_r}{\partial \theta} - \frac{A_\theta}{2 \nocdot \sin^2 \left( \theta \right)} - \frac{\cot \theta}{\sin \left( \theta \right)} \nocdot \frac{\partial A_\phi}{\partial \phi} \right) \\
      \nabla^2 A_\phi + \frac{2}{r^2 \nocdot \sin \left( \theta \right)} \nocdot \left( \frac{\partial A_r}{\partial \phi} + \cot \theta \nocdot \frac{\partial A_\theta}{\partial \phi} - \frac{A_\phi}{2 \nocdot \sin \left( \theta \right)} \right) \\
    \end{pmatrix}
  $
}
\\\hline

\end{tabu}

\end{document}