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diff --git a/Master/texmf-doc/doc/german/kopka/uebungen/kapitel5/ueb5-21.tex b/Master/texmf-doc/doc/german/kopka/uebungen/kapitel5/ueb5-21.tex new file mode 100644 index 00000000000..6254d7f90fd --- /dev/null +++ b/Master/texmf-doc/doc/german/kopka/uebungen/kapitel5/ueb5-21.tex @@ -0,0 +1,74 @@ +\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} + |