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author | Karl Berry <karl@freefriends.org> | 2009-05-23 00:23:51 +0000 |
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committer | Karl Berry <karl@freefriends.org> | 2009-05-23 00:23:51 +0000 |
commit | a683c3d7e9fac38ec713f23fb6b9d2c7143aea82 (patch) | |
tree | 424ab223921f85fd3f167a4ccd0e2d37d05c2927 /Master/texmf-dist/doc/latex/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-dist/doc/latex/guide-to-latex/exercises/chap7')
22 files changed, 425 insertions, 0 deletions
diff --git a/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-1.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-1.tex new file mode 100644 index 00000000000..fa73e5bbde9 --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-1.tex @@ -0,0 +1,7 @@ +\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-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-10.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-10.tex new file mode 100644 index 00000000000..795eeb3c304 --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-10.tex @@ -0,0 +1,13 @@ +\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-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-11.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-11.tex new file mode 100644 index 00000000000..afefe1e54a5 --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-11.tex @@ -0,0 +1,15 @@ +\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-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-12.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-12.tex new file mode 100644 index 00000000000..c8878796389 --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-12.tex @@ -0,0 +1,12 @@ +\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-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-13.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-13.tex new file mode 100644 index 00000000000..20d18995199 --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-13.tex @@ -0,0 +1,27 @@ +\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-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-14.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-14.tex new file mode 100644 index 00000000000..f4154ad0179 --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-14.tex @@ -0,0 +1,38 @@ +\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-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-15.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-15.tex new file mode 100644 index 00000000000..6383f732bab --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-15.tex @@ -0,0 +1,12 @@ +\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-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-16.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-16.tex new file mode 100644 index 00000000000..e603633e6a6 --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-16.tex @@ -0,0 +1,11 @@ +\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-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-17.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-17.tex new file mode 100644 index 00000000000..7743bee0f28 --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-17.tex @@ -0,0 +1,19 @@ +\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-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-18.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-18.tex new file mode 100644 index 00000000000..7abf3d9221d --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-18.tex @@ -0,0 +1,14 @@ +\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-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-19.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-19.tex new file mode 100644 index 00000000000..fb9c620f598 --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-19.tex @@ -0,0 +1,9 @@ +\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-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-2.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-2.tex new file mode 100644 index 00000000000..56b238f25b8 --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-2.tex @@ -0,0 +1,10 @@ +\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-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-20.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-20.tex new file mode 100644 index 00000000000..515b958a9b0 --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-20.tex @@ -0,0 +1,12 @@ +\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-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-21a.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-21a.tex new file mode 100644 index 00000000000..91e95077b77 --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-21a.tex @@ -0,0 +1,74 @@ +\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-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-21b.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-21b.tex new file mode 100644 index 00000000000..42a54e46336 --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-21b.tex @@ -0,0 +1,76 @@ +%======================================================================== +% 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-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-3.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-3.tex new file mode 100644 index 00000000000..4b15e9b0566 --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-3.tex @@ -0,0 +1,13 @@ +\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-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-4.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-4.tex new file mode 100644 index 00000000000..47e66f15dd1 --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-4.tex @@ -0,0 +1,11 @@ +\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-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-5.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-5.tex new file mode 100644 index 00000000000..1a2fc238383 --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-5.tex @@ -0,0 +1,9 @@ +\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-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-6.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-6.tex new file mode 100644 index 00000000000..2d353a59383 --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-6.tex @@ -0,0 +1,11 @@ +\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-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-7.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-7.tex new file mode 100644 index 00000000000..8ee134a9963 --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-7.tex @@ -0,0 +1,10 @@ +\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-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-8.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-8.tex new file mode 100644 index 00000000000..b163e13ded6 --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-8.tex @@ -0,0 +1,10 @@ +\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-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-9.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-9.tex new file mode 100644 index 00000000000..3a104cd3ba0 --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-9.tex @@ -0,0 +1,12 @@ +\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} + + |