\documentclass{article} \usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Created=Monday, June 25, 2001 17:45:57} %TCIDATA{LastRevised=Monday, March 24, 2003 14:31:55} %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: \ }$\left( \Longrightarrow \right) $ Ak existuje skal\'{a}% rne pole $\Phi ,$ tak\'{e} \v{z}e $\mathbf{F}=\nabla \Phi ,$ potom plat\'{\i} \ $rot\mathbf{F}=rot\left( \nabla \Phi \right) =\mathbf{0}$ (\hyperref{Lema}{% }{}{Ma71.tex#2}). $\left( \Longleftarrow \right) $ Zafixujme $\mathbf{x}^{\left( 0\right) }\in A$ a $\forall \mathbf{x}\in A$ definujeme \[ \Phi \left( \mathbf{x}\right) =\int_{C}\mathbf{F}.d\mathbf{s}, \]% kde $C$ je \'{u}se\v{c}ka $\left\{ \left( 1-t\right) \mathbf{x}^{\left( 0\right) }+t\mathbf{x},\,0\leq t\leq 1\right\} .$ Nech $\mathbf{x}=\left( x,y,z\right) $ a $\mathbf{y}=\left( x+h,y,z\right) ,$ kde $h$ je tak\'{e}, \v{z}e $\mathbf{y}\in A.$ Ozna\v{c}me $T$ trojuholn\'{\i}k s vrcholmi $% \mathbf{x}^{\left( 0\right) },\mathbf{x},\mathbf{y}$. Nech $S$ je trojuholn% \'{\i}kov\'{a} oblas\v{t} ohrani\v{c}en\'{a} \'{u}se\v{c}kami $T_{0}:$ z $% \mathbf{x}^{\left( 0\right) }$ do $\mathbf{x},\;T_{1}:$ z $\mathbf{x}$ do $% \mathbf{y},\;T_{2}:$ z $\mathbf{y}$ do $\mathbf{x}^{\left( 0\right) }.$ Potom $\ $% \[ \int_{T}\mathbf{F}.d\mathbf{s=\pm }\;\iint_{S}\left( rot\mathbf{F}\cdot \mathbf{n}\right) .dS=0, \]% pod\v{l}a predpokladu, \v{z}e $rot\mathbf{F}=\mathbf{0}.$ Potom \ \[ \int_{T_{0}}\mathbf{F}.d\mathbf{s}+\int_{T_{1}}\mathbf{F}.d\mathbf{s}% +\int_{T_{2}}\mathbf{F}.d\mathbf{s=}0, \]% teda $\ \Phi \left( \mathbf{x}\right) -\Phi \left( \mathbf{y}\right) +\int_{T_{1}}\mathbf{F}.d\mathbf{s}=0.$ M\^{o}\v{z}eme p\'{\i}sa\v{t}, \v{z}% e $T_{1}=\left\{ \left( x+t,y,z\right) ;\,0\leq t\leq h\right\} $ a teda \[ \Phi \left( \mathbf{y}\right) -\Phi \left( \mathbf{x}\right) =\int_{0}^{h}F_{1}\left( x+t,y,z\right) dt=hF_{1}\left( x^{\prime },y,z\right) , \]% $\ $kde $x^{\prime }$ je nejak\'{y} bod medzi $x$ a $x+h,$ pod\v{l}a vety o strednej hodnote. Potom \ \[ \frac{\partial \Phi }{\partial x}=\lim_{h\longrightarrow 0}\frac{\left[ \Phi \left( x+h,y,z\right) -\Phi \left( x,y,z\right) \right] }{h}=F_{1}\left( x,y,z\right) . \]% Podobne% \[ \frac{\partial \Phi }{\partial y}=F_{2}\left( x,y,z\right) ,\,\frac{\partial \Phi }{\partial z}=F_{3}\left( x,y,z\right) .\blacksquare \] \begin{center} \begin{tabular}{|c|} \hline {\small \hyperref{Sp\"{a}\v{t}}{}{}{Ma74.tex#5}} \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za II} \end{document}