\documentclass{article} \usepackage{german} \newcommand{\D}{\displaystyle} \newcommand{\bm}{\boldmath} \setlength{\textwidth}{140mm} \begin{document} \[ \begin{array}{|c|c|c|}\hline \multicolumn{3}{|c|}{\rule[-1.25mm]{0mm}{5mm}\mbox{Gleichung der Tangentialebene und der Fl"achennormalen}}\\ \hline \mbox{Gleichungs-}&&\\ \mbox{form} & \mbox{Tangentialebene} & \mbox{Fl"achennormale}\\ \mbox{der Fl"ache} & & \\ \hline \rule{0mm}{7mm}F(x,y,z)=0 & \begin{array}[t]{r@{\:+\:}l} \D\frac{\partial F}{\partial x}(X-x) & \D\frac{\partial F}{\partial y}(Y-y) \\[2ex] & \D\frac{\partial F}{\partial z}(Z-z) = 0 \end{array} & \D\frac{X-x}{\D\frac{\partial F}{\partial x}} = \frac{Y-y}{\D\frac{\partial F}{\partial y}} = \frac{Z-z}{\D\frac{\partial F}{\partial z}}\\ \rule[-4.2mm]{0mm}{10mm}z=f(x,y) & Z-z = p(X-x) + q(Y-y) & \D\frac{X-x}{p} = \frac{Y-y}{q} = \frac{Z-z}{-1}\\ \begin{array}{c} x=x(u,v)\\y=y(u,v)\\z=z(u,v) \end{array} & \begin{array}{|ccc|} X-x & Y-y & Z-z\\[0.5ex] \D\frac{\partial x}{\partial u} & \D\frac{\partial y}{\partial u} & \D\frac{\partial z}{\partial u} \\[2.0ex] \D\frac{\partial x}{\partial v} & \D\frac{\partial y}{\partial v} & \D\frac{\partial z}{\partial v} \end{array} = 0 & \D\frac{X-x}{\left|\begin{array}{c} \frac{\partial y}{\partial u}\; \frac{\partial z}{\partial u}\\[0.8ex] \frac{\partial y}{\partial v}\;\frac{\partial z}{\partial v} \end{array}\right|} = \frac{Y-y}{\left|\begin{array}{c} \frac{\partial z}{\partial u}\; \frac{\partial x}{\partial u}\\[0.8ex] \frac{\partial z}{\partial v}\;\frac{\partial x}{\partial v} \end{array}\right|} = \frac{Z-z}{\left|\begin{array}{c} \frac{\partial x}{\partial u}\; \frac{\partial y}{\partial u}\\[0.8ex] \frac{\partial x}{\partial v}\;\frac{\partial y}{\partial v} \end{array}\right|} \\ \rule[-4.2mm]{0mm}{12mm}\mbox{\bm $r=r$}(u,v) & \begin{array}{r} \mbox{\bm $(R-r)(r_1\times r_2) = \mbox{\unboldmath$0$}$}\\ \mbox{oder\qquad\bm $(R-r)N = \mbox{\unboldmath$0$}$} \end{array} & \begin{array}{r@{\;=\;}l} \mbox{\bm $R$} & \mbox{\boldmath$r + \mbox{\unboldmath$\lambda$}(r_1\times r_2$)}\\ \mbox{oder\quad\bm $R$} & \mbox{\bm $r + \mbox{\unboldmath$\lambda$}N$} \end{array}\\ \hline \multicolumn{3}{|c|}{\parbox{125mm}{\vspace*{0.5ex}In dieser Tabelle sind $x,\,y,\,z$ und \mbox{\bm $r$} die Koordinaten und der Radiusvektor des Kurvenpunktes $M$; $X,\,Y,\,Z$ und \mbox{\bm $R$} sind die laufenden Koordinaten und der Radiusvektor eines Punktes der Tangentialebene oder der Fl"achennormalen im Punkt $M$; ferner ist $p = \frac{\partial z}{\partial x}$, $q = \frac{\partial z}{\partial y}$ und $\mbox{\bm $r_1$} = \partial\mbox{\bm $r$}/\partial u$, $\mbox{\bm $r_2$} = \partial\mbox{\bm$r$}/\partial v$.}} \\[0.8ex] \hline \end{array} \] \end{document}