\documentclass{article} \usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Created=Monday, June 25, 2001 17:45:57} %TCIDATA{LastRevised=Monday, March 24, 2003 14:31:38} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\QTR{small}{Matematick\U{e1} anal\U{fd}za II online - D\U{f4}kazy\dotfill \thepage }} %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \author{A. U. Thor} \title{Lab Report} \date{The Date } \maketitle \begin{abstract} A Laboratory report created with Scientific Notebook \end{abstract} \section{Integr\'{a}lne vety - vz\v{t}ahy medzi krivkov\'{y}m a plo\v{s}n\'% {y}m integr\'{a}lom} \subsection{D\^{o}kaz vety} \textbf{D\^{o}kaz: \ }Nech $S=\left\{ \mathbf{r}\left( u,v\right) ;\,\left( u,v\right) \in D\right\} $ a ozna\v{c}me body na hrani\v{c}nej krivke $% B^{+}\subset D$ ako $\left\{ \left( u\left( t\right) ,v\left( t\right) \right) ;\,a\leq t\leq b\right\} .$ Odkia\v{l} \[ \int_{C}\mathbf{F\cdot }d\mathbf{s=}\int_{a}^{b}\mathbf{F}\left( \mathbf{c}% \left( t\right) \right) \mathbf{\cdot c}^{\prime }\left( t\right) dt\mathbf{,% } \]% pri\v{c}om \[ \mathbf{c}\left( t\right) =\left( x\left( u\left( t\right) ,v\left( t\right) \right) ,y\left( u\left( t\right) ,v\left( t\right) \right) ,z\left( u\left( t\right) ,v\left( t\right) \right) \right) ;\,a\leq t\leq b \]% a \[ \mathbf{c}^{\prime }\left( t\right) =\left( \frac{\partial x}{\partial u}% u^{\prime }\left( t\right) +\frac{\partial x}{\partial v}v^{\prime }\left( t\right) ,\frac{\partial y}{\partial u}u^{\prime }\left( t\right) +\frac{% \partial y}{\partial v}v^{\prime }\left( t\right) ,\frac{\partial z}{% \partial u}u^{\prime }\left( t\right) +\frac{\partial z}{\partial v}% v^{\prime }\left( t\right) \right) . \]% Uva\v{z}ujme v\'{y}raz \[ \int_{C}F_{1}dx=\int_{a}^{b}F_{1}\left( \frac{\partial x}{\partial u}% u^{\prime }\left( t\right) +\frac{\partial x}{\partial v}v^{\prime }\left( t\right) \right) dt=\int_{B^{+}}\left( F_{1}\frac{\partial x}{\partial u}% du+F_{1}\frac{\partial x}{\partial v}dv\right) , \]% kde $B^{+}$ je hranica $D$. Pod\v{l}a Greenovej vety: \[ \iint_{D}\left[ \frac{\partial }{\partial u}\left( F_{1}\frac{\partial x}{% \partial v}\right) -\frac{\partial }{\partial v}\left( F_{1}\frac{\partial x% }{\partial u}\right) \right] dudv=\iint_{D}\left( \frac{\partial F_{1}}{% \partial u}\frac{\partial x}{\partial v}-\frac{\partial F_{1}}{\partial v}% \frac{\partial x}{\partial u}\right) dudv, \]% ale \[ \frac{\partial F}{\partial u}=\frac{\partial F}{\partial x}\frac{\partial x}{% \partial u}+\frac{\partial F}{\partial y}\frac{\partial y}{\partial u}+\frac{% \partial F}{\partial z}\frac{\partial z}{\partial u}, \]% \[ \frac{\partial F}{\partial v}=\frac{\partial F}{\partial x}\frac{\partial x}{% \partial v}+\frac{\partial F}{\partial y}\frac{\partial y}{\partial v}+\frac{% \partial F}{\partial z}\frac{\partial z}{\partial v}, \]% odkia\v{l} \[ \frac{\partial F}{\partial u}\frac{\partial x}{\partial v}-\frac{\partial F}{% \partial v}\frac{\partial x}{\partial u}=-\frac{\partial F}{\partial y}J_{3}+% \frac{\partial F}{\partial z}J_{2}, \]% kde% \[ J_{1}=\frac{\partial \left( y,z\right) }{\partial \left( u,v\right) }% ,\,J_{2}=\frac{\partial \left( z,x\right) }{\partial \left( u,v\right) }% ,\,J_{3}=\frac{\partial \left( x,y\right) }{\partial \left( u,v\right) }. \]% $\,$Tak integr\'{a}l prejde na tvar \[ \int_{C}F_{1}dx=\iint_{S}\left( -\frac{\partial F_{1}}{\partial y}J_{3}+% \frac{\partial F_{1}}{\partial z}J_{2}\right) dudv=\iint_{S}\left( -\frac{% \partial F_{1}}{\partial y}n_{3}+\frac{\partial F_{1}}{\partial z}% n_{2}\right) dS, \]% kde \[ \mathbf{n}=\left( n_{1},n_{2},n_{3}\right) =\frac{\left( J_{1},J_{2},J_{3}\right) }{\sqrt{J_{1}^{2}+J_{2}^{2}+J_{3}^{2}}} \]% je jednotkov\'{a} norm\'{a}la a \ $dS=\sqrt{J_{1}^{2}+J_{2}^{2}+J_{3}^{2}}% dudv.$ Podobne vypo\v{c}\'{\i}tame druh\'{e} dva v\'{y}razy a dostaneme: \[ \int_{C}\mathbf{F\cdot }d\mathbf{s=}\iint_{S}\left( -\frac{\partial F_{1}}{% \partial y}n_{3}+\frac{\partial F_{1}}{\partial z}n_{2}-\frac{\partial F_{2}% }{\partial z}n_{1}+\frac{\partial F_{2}}{\partial x}n_{3}-\frac{\partial F_{3}}{\partial x}n_{2}+\frac{\partial F_{3}}{\partial y}n_{1}\right) dS= \]% \[ =\iint_{S}\left[ \left( \frac{\partial F_{3}}{\partial y}-\frac{\partial F_{2}}{\partial z}\right) n_{1}+\left( \frac{\partial F_{1}}{\partial z}-% \frac{\partial F_{3}}{\partial x}\right) n_{2}+\left( \frac{\partial F_{2}}{% \partial x}-\frac{\partial F_{1}}{\partial y}\right) n_{3}\right] dS=\iint_{S}\left( rot\mathbf{F}\right) \cdot \mathbf{n}dS.\,\blacksquare \] \begin{center} \begin{tabular}{|c|} \hline {\small \hyperref{Sp\"{a}\v{t}}{}{}{Ma74.tex#4}} \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za II} \end{document}