diff options
author | Karl Berry <karl@freefriends.org> | 2009-05-23 00:23:51 +0000 |
---|---|---|
committer | Karl Berry <karl@freefriends.org> | 2009-05-23 00:23:51 +0000 |
commit | a683c3d7e9fac38ec713f23fb6b9d2c7143aea82 (patch) | |
tree | 424ab223921f85fd3f167a4ccd0e2d37d05c2927 /Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7 | |
parent | 5beb5368a684995153c8566797ba054f21c666af (diff) |
move english latex doc out of texmf-doc
git-svn-id: svn://tug.org/texlive/trunk@13412 c570f23f-e606-0410-a88d-b1316a301751
Diffstat (limited to 'Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7')
22 files changed, 0 insertions, 425 deletions
diff --git a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-1.tex b/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-1.tex deleted file mode 100644 index fa73e5bbde9..00000000000 --- a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-1.tex +++ /dev/null @@ -1,7 +0,0 @@ -\documentclass{article} -\begin{document} -The derivative of the indirect function f[g(x)] is -$\{f[g(x)]\}' = f'[g(x)]g'(x)$. For the second derivative of the product -of $f(x)$ and $g(x)$ one has -$[f(x)g(x)]'' = f''(x)g(x) + 2f'(x)g'(x) + f(x)g''(x)$. -\end{document} diff --git a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-10.tex b/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-10.tex deleted file mode 100644 index 795eeb3c304..00000000000 --- a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-10.tex +++ /dev/null @@ -1,13 +0,0 @@ -\documentclass{article} -\begin{document} -$\alpha\vec{x} = \vec{x}\alpha$,\hfill$\alpha\beta\vec{x} -= \beta\alpha\vec{x}$,\hfill$(\alpha + \beta)\vec{x} = \alpha\vec{x} + -\beta\vec{x}$,\hfill $\alpha(\vec{x} + \vec{y}) -= \alpha\vec{x} + \alpha\vec{y}$.\\ -\hspace*{1.5em}$\vec{x}\vec{y} = \vec{y}\vec{x}$\hfill -but\hfill$\vec{x}\times\vec{y} = --\vec{y}\times\vec{x}$,\hfill\hfill$\vec{x}\vec{y} = 0$\hfill for\hfill% -$\vec{x}\perp\vec{y}$,\hfill\hfill$\vec{x}\times\vec{y} = 0$,\hfill -for\hfill$\vec{x}\parallel\vec{y}$. -\end{document} - diff --git a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-11.tex b/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-11.tex deleted file mode 100644 index afefe1e54a5..00000000000 --- a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-11.tex +++ /dev/null @@ -1,15 +0,0 @@ -\documentclass{article} -\begin{document} -\begin{equation} -\lim_{x\to0}\frac{\sqrt{1+x} - 1}{x} = -\lim_{x\to0}\frac{(\sqrt{1+x} - 1)(\sqrt{1+x} + 1)}{x(\sqrt{1+x} + 1)} = -\lim_{x\to0}\frac{1}{\sqrt{1+x} + 1} = \frac{1}{2} -\end{equation} -\begin{equation} -\frac{\partial^2 U}{\partial x^2} + \frac{\partial^2 U}{\partial y^2} = 0\quad -\Longrightarrow\quad -U_M = \frac{1}{4\pi}\oint\limits_\Sigma\frac{1}{r}\frac{\partial U}{\partial n} -\,ds - \frac{1}{4\pi}\oint\limits_\Sigma\frac{\partial\frac{1}{r}}{\partial n} -U\,ds -\end{equation} -\end{document} diff --git a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-12.tex b/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-12.tex deleted file mode 100644 index c8878796389..00000000000 --- a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-12.tex +++ /dev/null @@ -1,12 +0,0 @@ -\documentclass{article} -\setlength{\textwidth}{135mm} -\begin{document} -\setcounter{equation}{2} -\begin{equation} -S(z) = -\cos\left(\frac{\pi}{2} z^2\right) \sum_{n=0}^\infty - \frac{(-1)^n\pi^{2n+1}}{1\cdot3\cdots(4n+3)} z^{4n+3} - +\sin\left(\frac{\pi}{2} z^2\right) \sum_{n=0}^\infty - \frac{(-1)^n\pi^{2n}}{1\cdot3\cdots(4n+1)} z^{4n+1} -\end{equation} -\end{document} - diff --git a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-13.tex b/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-13.tex deleted file mode 100644 index 20d18995199..00000000000 --- a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-13.tex +++ /dev/null @@ -1,27 +0,0 @@ -\documentclass{article} -\begin{document} -\noindent\textbf{First solution example}:\\[1ex] -The solution for the system equation -\[ \renewcommand{\arraystretch}{1.2} - F(x,y) = 0\quad\mbox{and}\quad - \left|\begin{array}{ccc} - F''_{xx} & F''_{xy} & F'_x\\ - F''_{yx} & F''_{yy} & F'_y\\ - F'_x & F'_y & 0 - \end{array}\right| = 0 -\] -yields the coordinates for the possible inflection points of $F(x,y) = 0$. - -\bigskip -\noindent\textbf{Second solution example}:\\[1ex] -The solution for the system equation -\[ \renewcommand{\arraystretch}{1.2} - F(x,y) = 0\quad\mbox{and}\quad\begin{array}{|ccc|} - F''_{xx} & F''_{xy} & F'_x\\ - F''_{yx} & F''_{yy} & F'_y\\ - F'_x & F'_y & 0 - \end{array} = 0 -\] -yields the coordinates for the possible inflection points of $F(x,y) = 0$. -\end{document} - diff --git a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-14.tex b/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-14.tex deleted file mode 100644 index f4154ad0179..00000000000 --- a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-14.tex +++ /dev/null @@ -1,38 +0,0 @@ -\documentclass{article} -\begin{document} -\noindent\textbf{First solution example}:\\[1ex] -The shortest distance between two straight lines represented by the equations -\[ \frac{x-x_1}{l_1} = \frac{y-y_1}{m_1} = \frac{z-z_1}{n_1}\quad\mbox{und}% -\quad% - \frac{x-x_2}{l_2} = \frac{y-y_2}{m_2} = \frac{z-z_2}{n_2} \] -is given by the expression -\[ \frac{\pm\;\begin{array}{|ccc|} - x_1 - x_2 & y_1 - y_2 & z_1 - z_2 \\ - l_1 & m_1 & n_1 \\ - l_2 & m_2 & n_2 - \end{array}}{ - \sqrt{\left|\begin{array}{cc} l_1 & m_1 \\ l_2 & m_2 \end{array}\right|^2 - + \left|\begin{array}{cc} m_1 & n_1 \\ m_2 & n_2 \end{array}\right|^2 - + \left|\begin{array}{cc} n_1 & l_1 \\ n_2 & l_2 \end{array}\right|^2} - } \] -If the numerator is zero, the two lines meet somewhere. - -\bigskip -\noindent\textbf{Second solution example}:\\[1ex] -The shortest distance between two straight lines represented by the equations -\[ \frac{x-x_1}{l_1} = \frac{y-y_1}{m_1} = \frac{z-z_1}{n_1}\quad\mbox{und}% -\quad% - \frac{x-x_2}{l_2} = \frac{y-y_2}{m_2} = \frac{z-z_2}{n_2} \] -is given by the expression -\[ \frac{\pm\;\begin{array}{|ccc|} - x_1 - x_2 & y_1 - y_2 & z_1 - z_2 \\ - l_1 & m_1 & n_1 \\ - l_2 & m_2 & n_2 - \end{array}}{ - \sqrt{\begin{array}{|cc|} l_1 & m_1 \\ l_2 & m_2 \end{array}^2 - + \begin{array}{|cc|} m_1 & n_1 \\ m_2 & n_2 \end{array}^2 - + \begin{array}{|cc|} n_1 & l_1 \\ n_2 & l_2 \end{array}^2} - } \] -If the numerator is zero, the two lines meet somewhere. -\end{document} - diff --git a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-15.tex b/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-15.tex deleted file mode 100644 index 6383f732bab..00000000000 --- a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-15.tex +++ /dev/null @@ -1,12 +0,0 @@ -\documentclass{article} -\begin{document} -\noindent -Laurent expansion using $c_n = \frac{1}{2\pi i} -\oint (\zeta-a)^{-n-1}f(\zeta)\,d\zeta$, for every function $f(z)$ the -following representation is valid ($n=0$, $\pm1$, $\pm2$, \ldots) -\[ f(x) = \sum_{n=-\infty}^{+\infty} c_n(z-a)^n - = \left\{\begin{array}{r} - c_0 + c_1(z-a) + c_2(z-a)^2 +\cdots+ c_n(z-a)^n+\cdots\\ - \mbox{}+c_{-1}(z-a)^{-1} + c_{-2}(z-a)^{-2}+\cdots\\ - \mbox{}+c_{-n}(z-a)^{-n}+\cdots \end{array}\right. \] -\end{document} diff --git a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-16.tex b/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-16.tex deleted file mode 100644 index e603633e6a6..00000000000 --- a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-16.tex +++ /dev/null @@ -1,11 +0,0 @@ -\documentclass{article} -\begin{document} -\noindent -The total number of permutations of $n$ elements taken $m$ at a time -(symbol $V_n^m$) is -\vspace{-0.5ex} -\[ V_n^m = \prod_{i=0}^{m-1}(n-i) = - \underbrace{n(n-1)(n-2)\ldots(n-m+1)}_{\mbox{total of $m$ factors}} = - \frac{n!}{(n-m)!} \] -\end{document} - diff --git a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-17.tex b/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-17.tex deleted file mode 100644 index 7743bee0f28..00000000000 --- a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-17.tex +++ /dev/null @@ -1,19 +0,0 @@ -\documentclass{article} -\begin{document} -\setcounter{equation}{5} -\begin{eqnarray} -\arcsin x & = & -\arcsin(-x) = \frac{\pi}{2} - \arccos x = - \left[\arccos\sqrt{1-x^2}\,\right]\nonumber\\ - & = & \arctan\frac{x}{\sqrt{1-x^2}} = - \left[\mbox{\rm arccot}\frac{\sqrt{1-x^2}}{x}\right] -\end{eqnarray} -\begin{eqnarray} -\lefteqn{f(x+h,y+k) = f(x,y) + \left\{\frac{\partial f(x,y)}{\partial x}h + - \frac{\partial f(x,y)}{\partial y}k\right\}}\nonumber\\ - &&\mbox{} + \frac{1}{2}\left\{\frac{\partial^2f(x,y)}{\partial x^2}h^2 - + 2\frac{\partial^2f(x,y)}{\partial x\partial y}kh - + \frac{\partial^2f(x,y)}{\partial y^2}k^2\right\}\\ - &&\mbox{} + \frac{1}{6}\{\cdots\} + \cdots + \frac{1}{n!}\{\cdots\} + R_n - \nonumber -\end{eqnarray} -\end{document} diff --git a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-18.tex b/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-18.tex deleted file mode 100644 index 7abf3d9221d..00000000000 --- a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-18.tex +++ /dev/null @@ -1,14 +0,0 @@ -\documentclass{article} -\begin{document} -\noindent -The inverse function of the polynomial expansion ~~$y = f(x) = ax+bx^2+cx^3 -+dx^4 + ex^5 + fx^6+\cdots$ $(a\ne0)$ begins with the elements -\setlength{\arraycolsep}{2pt} -\begin{eqnarray*} -x = \varphi(y) = \frac{1}{a}y &-& \frac{b}{a^3}y^2 + \frac{1}{a^5}(2b^2-ac)y^3\\ - &+&\frac{1}{a^7}(5abc - z^2d -fb^3)y^4\\ - &+&\frac{1}{a^9}(6a^2bd + 3a^2c^2+14b^4 - a^3e - 21ab^2c)y^5\\ - &+&\frac{1}{a^{11}}(7a^3be + 7a^3cd + 84ab^3c - a^4f -\\ - &&28a^2b^2d -28 a^2bc^2 - 43b^5)y^6 + \cdots -\end{eqnarray*} -\end{document} diff --git a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-19.tex b/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-19.tex deleted file mode 100644 index fb9c620f598..00000000000 --- a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-19.tex +++ /dev/null @@ -1,9 +0,0 @@ -\documentclass[12pt]{article} -\newcommand{\D}{\displaystyle} -\begin{document} -\[ a_0 + \frac{1\hfill}{\D a_1 - + \frac{1\hfill}{\D a_2 - + \frac{1\hfill}{\D a_3 - + \frac{1\hfill}{a_4}}}} \] -\end{document} - diff --git a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-2.tex b/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-2.tex deleted file mode 100644 index 56b238f25b8..00000000000 --- a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-2.tex +++ /dev/null @@ -1,10 +0,0 @@ -\documentclass{article} -\begin{document} -The reduced cubic equation $y^3 + 3py +2q = 0$ has one real and two complex -solutions when $D = q^2 + p^3 > 0$. These are given by Cardan's formula as -\[ y_1 = u + v,\quad y_2 = -\frac{u+v}{2} + \frac{i}{2}\sqrt{3}(u - v),\quad - y_3 = -\frac{u+v}{2} - \frac{i}{2}\sqrt{3}(u-v) \] -where -\[ u = \sqrt[3]{-q + \sqrt{q^2+p^3}},\qquad v = \sqrt[3]{-q - \sqrt{q^2+p^3}} \] -\end{document} - diff --git a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-20.tex b/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-20.tex deleted file mode 100644 index 515b958a9b0..00000000000 --- a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-20.tex +++ /dev/null @@ -1,12 +0,0 @@ -\documentclass{article} -%\setlength{\textwidth}{140mm} -\begin{document} -\[ \begin{array}{@{}r@{\;=\;}l@{\qquad}r@{\;=\;}l@{}} - \sin2\alpha & 2\sin\alpha\cos\alpha, & - \cos2\alpha & \cos^2\alpha - \sin^2\alpha\\ - \sin3\alpha & 3\sin\alpha-4\sin^3\alpha & - \cos3\alpha & 3\cos^3\alpha - 3\cos\alpha\\ - \sin4\alpha & 8\cos^3\alpha\sin\alpha - 4\cos\alpha\sin\alpha & - \cos4\alpha & 8\cos^4\alpha - 8\cos^2\alpha + 1 - \end{array} \] -\end{document} diff --git a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-21a.tex b/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-21a.tex deleted file mode 100644 index 91e95077b77..00000000000 --- a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-21a.tex +++ /dev/null @@ -1,74 +0,0 @@ -\documentclass{article} -\begin{document} -\newcommand{\D}{\displaystyle} -\newcommand{\bm}{\boldmath} -\[ \begin{array}{@{}|c|c|c|@{}}\hline -\multicolumn{3}{@{}|c|@{}}{\rule[-0.125cm]{0mm}{0.5cm}% -\mbox{Equations for the tangential plane and surface normal}}\\ -\hline -\mbox{Equation}&&\\ -\mbox{for the} & \mbox{Tangential plane} & \mbox{Surface normal}\\ -\mbox{surface} & & \\ \hline -\rule{0mm}{0.583cm}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[-0.42cm]{0mm}{1cm}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 - & \begin{array}{c@{=}c} - \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|} & - \D \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|} \\ - & \rule{0mm}{4ex} - \D \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|} - \end{array} \\ -\rule[-0.42cm]{0mm}{1.17cm}\mbox{\boldmath$r=r$}(u,v) - & \begin{array}{r} - \mbox{\boldmath$(R-r)(r_1\times r_2) = \mbox{\unboldmath$0$}$}\\ - \mbox{or\qquad\boldmath$(R-r)N = \mbox{\unboldmath$0$}$} - \end{array} - & \begin{array}{r@{\;=\;}l} - \mbox{\boldmath$R$} & \mbox{\boldmath$r + - \mbox{\unboldmath$\lambda$}(r_1\times r_2$)}\\ - \mbox{or\quad\boldmath$R$} & - \mbox{\boldmath$r + \mbox{\unboldmath$\lambda$}N$} - \end{array}\\ \hline -\multicolumn{3}{@{}|c|@{}}{\parbox{11.3cm}{\vspace*{0.5ex}In this table - $x,\,y,\,z$ and - \mbox{\boldmath$r$} are the coordinates and the radius vector of a fixed - point $M$ on the curve; $X,\,Y,\,Z$, and \mbox{\boldmath$R$} are the - coordinates and radius vector of a point on the tangential plane or surface - normal with reference to $M$; furthermore, - $p = \frac{\partial z}{\partial x}$, $q = \frac{\partial z}{\partial y}$ - and $\mbox{\boldmath$r_1$} = \partial\mbox{\boldmath$r$}/\partial u$, - $\mbox{\boldmath$r_2$} = \partial\mbox{\boldmath$r$}/\partial v$.}} -\\[0.8ex] \hline -\end{array} \] -\end{document} diff --git a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-21b.tex b/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-21b.tex deleted file mode 100644 index 42a54e46336..00000000000 --- a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-21b.tex +++ /dev/null @@ -1,76 +0,0 @@ -%======================================================================== -% With an slightly increased \textwidth a better presentation for the -% whole math. table can be made together with an even simpler solution. -%======================================================================== -\documentclass{article} -\setlength{\textwidth}{135mm} -\begin{document} -\newcommand{\D}{\displaystyle} -\newcommand{\bm}{\boldmath} -\[ \begin{array}{@{}|c|c|c|@{}}\hline -\multicolumn{3}{@{}|c|@{}}{\rule[-0.125cm]{0mm}{0.5cm}% -\mbox{Equations for the tangential plane and surface normal}}\\ -\hline -\mbox{Equation}&&\\ -\mbox{for the} & \mbox{Tangential plane} & \mbox{Surface normal}\\ -\mbox{surface} & & \\ \hline -\rule{0mm}{0.583cm}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[-0.42cm]{0mm}{1cm}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|} - \D \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|} - \D \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[-0.42cm]{0mm}{1.17cm}\mbox{\boldmath$r=r$}(u,v) - & \begin{array}{r} - \mbox{\boldmath$(R-r)(r_1\times r_2) = \mbox{\unboldmath$0$}$}\\ - \mbox{or\qquad\boldmath$(R-r)N = \mbox{\unboldmath$0$}$} - \end{array} - & \begin{array}{r@{\;=\;}l} - \mbox{\boldmath$R$} & \mbox{\boldmath$r + - \mbox{\unboldmath$\lambda$}(r_1\times r_2$)}\\ - \mbox{or\quad\boldmath$R$} & - \mbox{\boldmath$r + \mbox{\unboldmath$\lambda$}N$} - \end{array}\\ \hline -\multicolumn{3}{@{}|c|@{}}{\parbox{12.5cm}{\vspace*{0.5ex}In this table - $x,\,y,\,z$ and - \mbox{\boldmath$r$} are the coordinates and the radius vector of a fixed - point $M$ on the curve; $X,\,Y,\,Z$, and \mbox{\boldmath$R$} are the - coordinates and radius vector of a point on the tangential plane or surface - normal with reference to $M$; furthermore, - $p = \frac{\partial z}{\partial x}$, $q = \frac{\partial z}{\partial y}$ - and $\mbox{\boldmath$r_1$} = \partial\mbox{\boldmath$r$}/\partial u$, - $\mbox{\boldmath$r_2$} = \partial\mbox{\boldmath$r$}/\partial v$.}} -\\[0.8ex] \hline -\end{array} \] -\end{document} diff --git a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-3.tex b/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-3.tex deleted file mode 100644 index 4b15e9b0566..00000000000 --- a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-3.tex +++ /dev/null @@ -1,13 +0,0 @@ -\documentclass[fleqn]{article} -\begin{document} -\setlength{\mathindent}{2cm} -The reduced cubic equation $y^3 + 3py +2q = 0$ has one real and two complex -solutions when $D = q^2 + p^3 > 0$. These are given by Cardan's formula as -\begin{equation} y_1 = u + v \end{equation} -\begin{equation} y_2 = -\frac{u+v}{2} + \frac{i}{2}\sqrt{3}(u-v) \end{equation} -\begin{equation} y_3 = -\frac{u+v}{2} - \frac{i}{2}\sqrt{3}(u-v) \end{equation} -where -\[ u = \sqrt[3]{-q + \sqrt{q^2+p^3}},\qquad v = \sqrt[3]{-q - \sqrt{q^2+p^3}} \] -\end{document} - - diff --git a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-4.tex b/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-4.tex deleted file mode 100644 index 47e66f15dd1..00000000000 --- a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-4.tex +++ /dev/null @@ -1,11 +0,0 @@ -\documentclass{article} -\usepackage{german} -\begin{document} -Each of the measurements $x_1 < x_2 < \cdots < x_r$ occures -$p_1$-, $p_2$-$,\ldots,p_r$ times. The mean value and standard deviation -are then -\[ x = \frac{1}{n}\sum_{i=1}^r p_i x_i,\qquad s= \sqrt{\frac{1}{n}\sum_{i=1}^r - p_i(x_i - x)^2} \] -where $ n = p_1 + p_2 +\cdots+p_r $. -\end{document} - diff --git a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-5.tex b/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-5.tex deleted file mode 100644 index 1a2fc238383..00000000000 --- a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-5.tex +++ /dev/null @@ -1,9 +0,0 @@ -\documentclass{article} -\begin{document} -Although this equation looks very complicated, it should not present -any great difficulties: -\[ \int\frac{\sqrt{(ax+b)^3}}{x}\,dx = \frac{2\sqrt{(ax+b)^3}}{3} - + 2b\sqrt{ax+b} + b^2\int\frac{dx}{x\sqrt{ax+b}} \] -The same applies to $\int^8_{-1}(dx/\sqrt[3]{x}) = \frac{3}{2}(8^{2/3} + - 1^{2/3}) = 15/2$ -\end{document} diff --git a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-6.tex b/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-6.tex deleted file mode 100644 index 2d353a59383..00000000000 --- a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-6.tex +++ /dev/null @@ -1,11 +0,0 @@ -\documentclass{article} -\begin{document} -The union of two sets $\mathcal{A}$ and $\mathcal{B}$ is the set of elements -that are in at least one of the two sets, and is designated as -$\mathcal{A\cup B}$. This operation is commutative -$\mathcal{A\cup B = B\cup A}$ and associative $\mathcal{(A\cup B)\cup C = -A\cup(B\cup C)}$. If $\mathcal{A\subseteq B}$, then -$\mathcal{A\cup B = B}$. It then follows that $\mathcal{A\cup A = A}$, -$\mathcal{A\cup\{\emptyset\} = A}$ and $\mathcal{J\cup A = J}$. - -\end{document} diff --git a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-7.tex b/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-7.tex deleted file mode 100644 index 8ee134a9963..00000000000 --- a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-7.tex +++ /dev/null @@ -1,10 +0,0 @@ -\documentclass{article} -\begin{document} -\noindent -Applying l'Hopital's rule, one has -\[ \lim_{x\to0}\frac{\ln\sin\pi x}{\ln\sin x} - = \lim_{x\to0}\frac{\pi\frac{\cos\pi x}{\sin\pi x}}{\frac{\cos x}{\sin x}} - = \lim_{x\to0}\frac{\pi\tan x}{\tan\pi x} - = \lim_{x\to0}\frac{\pi/\cos^2 x}{\pi/\cos^2 \pi x} - = \lim_{x\to0}\frac{\cos^2\pi x}{\cos^2 x} = 1 \] -\end{document} diff --git a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-8.tex b/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-8.tex deleted file mode 100644 index b163e13ded6..00000000000 --- a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-8.tex +++ /dev/null @@ -1,10 +0,0 @@ -\documentclass{article} -\setlength{\textwidth}{135mm} -\begin{document} -\noindent -The gamma function $\Gamma(x)$ is defined as -\[ \Gamma(x)\equiv\lim_{n\to\infty}\prod_{\nu=0}^{n-1}\frac{n!n^{x-1}}{x+\nu} - = \lim_{n\to\infty}\frac{n!n^{x-1}}{x(x+1)(x+2)\cdots(x+n-1)} - \equiv\int_0^\infty e^{-t}t^{x-1}\,dt \] -The integral definition is valid only for $x>0$ (2nd Euler integral). -\end{document} diff --git a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-9.tex b/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-9.tex deleted file mode 100644 index 3a104cd3ba0..00000000000 --- a/Master/texmf-doc/doc/english/guide-to-latex/exercises/chap7/exer7-9.tex +++ /dev/null @@ -1,12 +0,0 @@ -\documentclass{article} -\begin{document} -The reduced cubic equation $y^3 + 3py +2q = 0$ has one real and two complex -solutions when $D = q^2 + p^3 > 0$. These are given by Cardan's formula as -\begin{equation} y_1 = u + v \end{equation} -\begin{equation} y_2 = -\frac{u+v}{2} + \frac{i}{2}\sqrt{3}(u-v) \end{equation} -\begin{equation} y_3 = -\frac{u+v}{2} - \frac{i}{2}\sqrt{3}(u-v) \end{equation} -where -\[ u = \sqrt[3]{-q + \sqrt{q^2+p^3}},\qquad v = \sqrt[3]{-q - \sqrt{q^2+p^3}} \] -\end{document} - - |