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-rw-r--r--Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-1.tex7
-rw-r--r--Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-10.tex13
-rw-r--r--Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-11.tex15
-rw-r--r--Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-12.tex12
-rw-r--r--Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-13.tex27
-rw-r--r--Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-14.tex38
-rw-r--r--Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-15.tex12
-rw-r--r--Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-16.tex11
-rw-r--r--Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-17.tex19
-rw-r--r--Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-18.tex14
-rw-r--r--Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-19.tex9
-rw-r--r--Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-2.tex10
-rw-r--r--Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-20.tex12
-rw-r--r--Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-21a.tex74
-rw-r--r--Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-21b.tex76
-rw-r--r--Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-3.tex13
-rw-r--r--Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-4.tex11
-rw-r--r--Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-5.tex9
-rw-r--r--Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-6.tex11
-rw-r--r--Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-7.tex10
-rw-r--r--Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-8.tex10
-rw-r--r--Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-9.tex12
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}
+
+