<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">
\documentclass[10pt]{article} 

%\usepackage[latin2]{inputenc} 

\usepackage{latexsym} 
\usepackage{amsmath} 
%\usepackage{eufrak} 
\usepackage{amssymb} 
\usepackage{graphicx}

%\setlength{\sidemargin}{0in} 
\setlength{\oddsidemargin}{0in} 
\setlength{\evensidemargin}{0in} 
\setlength{\textwidth}{17cm} 
\setlength{\topmargin}{-1cm} 
\setlength{\textheight}{24cm} 
\setlength{\parindent}{0in} 

\pagestyle{empty} 

\newcommand{\lto}
{\mbox{\,\,\,\,\raisebox{-0.5mm}{\Large $\leadsto$}\,\,\,\,}}


\begin{document} 

%{\small 
\begin{center} 
        {\LARGE\bf  3. Vizsgaz\'{a}rthelyi megold\'asokkal}

     {\bf  2000/01 t\'{e}l II. Villamosm\'{e}rn\"{o}k 13.--18.tk.} 
\end{center} 

\vspace{-1mm}

{\bf\Large 1.} 
Legyen $ H $ az a h\'{a}romsz\"{o}gvonal, melynek  cs\'{u}csai  
a   (0,0)\,,\,     (0,2)\, ,\, (2,0)\,  pontok a s\'{\i}kban. 
Legyen  \linebreak \,$v(x,y) = (y^2 +2\,xy\,,\,\, x^2+yx)$\,  egy  
k\'{e}tdimenzi\'{o}s 
vektorf\"{u}ggv\'{e}ny.   
Sz\'{a}m\'{\i}tsuk ki $ v $  
vonalmenti  integr\'{a}lj\'{a}t a pozit\'\i van 
ir\'any\'\i tott $H$--n! 

\vspace{2mm} 

{\bf\large MO.} ${\rm rot} \, v = 
\frac{\partial v_2}{\partial x}-\frac{\partial v_1}{\partial y} 
= 2x +y - 2y-2x= -y \,$ , \'{\i}gy  Stokes 
 t\'{e}tellel\,:
($F$  a  $H$   \'{a}ltal bez\'{a}rt h\'{a}romsz\"{o}glap):

${\displaystyle \int_H v \; dr = \int_F {\rm rot} \,v \; dV = 
 \int_F \;-y\,dV =-\int_0^2\int_0^{2-y}y\,dx\,dy =
 -\int_0^2xy\,\big|_0^{2-y}\,dy =
-\int_0^2y(2-y)\,dy =}$
${\displaystyle  -\int_0^22y-y^2\,dx =}$

${\displaystyle=\rule[-6mm]{0.0mm}{15mm}
\left.\frac{y^3}{3}-x^2\,\right|_0^2=
\frac{8}{3}-4= -\frac{4}{3}}$

VAGY\,: a tengelyek ment\'en a vonalintegr\'al 0, mert 
az $x$ tengely ment\'en\,: $v(x,0)=(0,x^2)$\,, melynek csak 
$y$ ir\'any\'u, teh\'at az \'erint\H ore mer\H oleges komponense 
van \'es u.\'\i gy a m\'asik tengely eset\'en. Teh\'at csak az 
\'atfog\'ora kell kisz\'am\'\i tani a fel\"uleti integr\'alt. 
Ennek egyenlete ha $L$--et negat\'\i van ir\'any\'\i tjuk\,: 

${\displaystyle r(t)=(t, 2-t)\,,\,\,t\in [0,2]\lto }$
${\displaystyle 
 \dot{r}(t)=(1,-1) \lto}$
%{\rm CROSS}\,(\dot{r}(t))=(1,1) \lto}$\rule[-3mm]{0.0mm}{6mm}

${\displaystyle v(r(t))=\big((2-t)^2+2(2-t)t\,,\,\, 
t^2 +t(2-t)\big)=
(4-t^2,2t) \lto}$

${\displaystyle \hspace*{-2mm}\lto 
v(r(t))\cdot\,\dot{r}(t)=
-t^2-2t+4 \lto 
\int_{\hspace{-0.5mm}-L} v\,dr = 
\int_0^2\,v(r(t))\cdot\dot{r}(t)\,dt =}$

${\displaystyle = 
\int_0^2-t^2-2t+4\,dt = (\left.-\frac{t^3}{3}-t^2+
4t)\,\right|_0^2=\rule[-1mm]{0.0mm}{9mm}-\frac{8}{3}-4+8=
-\frac{8}{3}-\frac{12}{3}+\frac{24}{3}=\frac{4}{3}\lto 
\int_L v\,dr = -\frac{4}{3}
}$

\vspace{2mm}

{\bf\Large 2.} Legyen \,$m&gt;0$\, \'es \,$K$\, a 
h\'aromdimenzi\'os t\'erben 
  az a \,$z=m $\, s\'\i kban 
elhelyezked\H o \,$R\,$ sugar\'u felfel\'e ir\'any\'\i tott 
k\"orlap, melynek 
k\"oz\'eppontja a \,$z\,$ tengelyen van.  \ 
${\displaystyle \int_H r\;  df =\,? }$ 

{\bf\large MO.} 
Legyen $ v(r) = r  \,$,  $n=k$ a k\"orlap norm\'{a}lisa, 
\,$v_n$\, pedig \,$v$--nek \,$n$--re es\H{o} vet\"{u}lete. 
Ekkor a k\"orlapon minden\"utt \,$v_n=m$\, 
\'\i gy \ 
${\displaystyle \int_K v \;  df =  \int_K v_n \;  | df |  = 
\int_K m \;|df| =  m\,\int_K \;|df|=
m\cdot R^2\pi= R^2m\,\pi.}$

VAGY\,: a k\"orlap egyenlete\,: \rule[-3mm]{0.0mm}{6mm}
${\displaystyle r(u,v)=(u\cos v,u\sin v, m)
\,,\,\,(u\in [0,R]\,,\,\,v\in [0,2\pi]) \lto }$

${\displaystyle \hspace*{-2mm}\lto r_u \times r_v = 
(0,0,u)\,, \,\, v(r(u,v))=r(u,v)=(u\cos v,u\sin v, m)\lto}$

${\displaystyle \hspace*{-2mm}\lto 
v(r(u,v))\cdot r_u \times r_v = m\,u\lto }$
${\displaystyle \int_H r \;  df = 
\int_0^R\int_0^{2\pi} m\,u \,dv\,du =
 m\,2\pi\, \frac{u^2}{2}\bigg|_0^R = 
m\,2\pi\, \frac{R^2}{2}= R^2m\,\pi.}$

\vspace{2mm} 

{\bf\Large 3.} Adjunk meg egy olyan ${\mathbb R}^4$--en \'ertelmezett 
\,$u = u(r)$\,  
skal\'{a}rf\"{u}ggv\'{e}nyt, hogy 
\,$ {\rm grad}\,u = r | r |^2 $\,  legyen minden \linebreak 
 $r\in{\mathbb R}^4\,,\,\, r \neq 0$ 
eset\'{e}n\,!


{\bf\large MO.} $v(r)= r | r |^2$ \ potenci\'alja\,: 
${\displaystyle u(r) = \int_0^1  rt | rt |^2 \cdot r \; dt = 
 | r |^4 \int_0^1 t^3 \; dt = \frac{ | r |^4}{4} \,.}$ \ \ 
Val\'oban\,:  ${\displaystyle 
{\rm grad}\,u = 4\,\frac{|r|^3}{4}\cdot \frac{r}{|r|}= r | r |^2 }$\,. 

\vspace{2mm} 

{\bf\Large 4.} 
Sz\'{a}m\'{\i}tsuk ki a \ 
$  j\,{\rm sh\,} (j\frac{\pi}{2}) -  {\rm ch\,} (j\frac{\pi}{2}) $ 
\ \'{e}rt\'{e}k\'{e}t \,!

{\bf\large MO.} 
$ {\displaystyle \,\,  j\,{\rm sh}\, (j\frac{\pi}{2})  - 
{\rm ch\,}  (j\frac{\pi}{2})  = 
j\,\frac{e^{j\frac{\pi}{2}} - e^{-j\frac{\pi}{2}}}{2} - 
\,\frac{e^{j\frac{\pi}{2} } + e^{j\frac{\pi}{2}}}{2} =  
\frac{1}{2}(j(j - (-j))-(j+ -j))=-1}$\,.

\vspace{2mm}

{\bf\Large 5.}  
Legyen minden \,$z\neq 0\,$ eset\'en  \ 
{\large ${\displaystyle f(z) = 
\frac{\frac{e^z - 1}{z} -  1}{z}
 }$\,.} \ \rule[-5mm]{0.0mm}{5mm}
Folytonoss\'a tehet\H o--e az \,$f$\, f\"uggv\'eny 
az orig\'oban\,? Ha igen, a folytonos\'\i tott v\'altozat 
deriv\'alhat\'o--e az orig\'oban\,? Ha igen, mennyi a deriv\'alt 
\'ert\'eke\,?  

\vspace{2mm}

{\bf\large MO.} $e^z$ Taylor--sora: \ 
${\displaystyle e^z = 1 + z + \frac{z^2}{2} + 
\frac{z^3}{6} + \ldots  \  
\lto f(z) = \frac{\frac{e^z - 1}{z} - 1}{z} = 
\frac{1}{2} + \frac{z}{6} + \ldots  \mbox{ \ ha \ } 
z \neq 0 \lto f(z) \circeq \frac{1}{2} }$ \ 
eset\'en \,$f\,$\, egy minden\"utt konvergens 
hatv\'anysor hat\'arf\"uggv\'enye, \'\i gy minden\"utt 
ak\'arh\'anyszor deriv\'alhat\'o, Taylor--sora az \H ot 
el\H o\'all\'\i t\'o hatv\'anysor 
${\displaystyle \lto f'(0) = \frac{1}{6}}$\,. 

\vspace{2mm}

{\bf\large 6.} 
Legyen  $K$  az orig\'ok\"{o}z\'{e}ppont\'{u} $R = 2$ 
 sugar\'{u} k\"{o}r. Mennyi az 
$ {\displaystyle \int_K \frac{1}{z^2-4z+3} \; dz }$ integr\'{a}l 
\'{e}rt\'{e}ke\,?  

{\bf\large MO.} Cauchy integr\'alformul\'aval\,: 

${\displaystyle \int_K \frac{1}{z^2-4z+3} \; dz = 
\int_K \frac{1}{(z-1)(z-3)} \; dz = 
\int_K \frac{\frac{1}{z-3}}{z-1} \; dz = 
2\pi\,j\,\left.{\frac{1}{z-3}}\right|_{z=1}=
 2\pi\,j(-\frac{1}{2})=-\pi\,j\,.}$ 
%}
\end{document}
</pre></body></html>