\documentclass{article} \usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Created=Monday, June 25, 2001 17:45:57} %TCIDATA{LastRevised=Sunday, March 23, 2003 16:35:23} %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{Krivkov\'{e} integr\'{a}ly} \subsection{D\^{o}kaz vety} \textbf{D\^{o}kaz: }Preto\v{z}e $\mathbf{p}(\tau )=\mathbf{c}\left( \Phi \left( \tau \right) \right) ,\tau \in \left\langle \alpha ,\beta \right\rangle ,$ potom $\mathbf{p\,}^{\prime }(\tau )=\mathbf{c\,\,}^{\prime }(\Phi (\tau ))\Phi ^{\prime }(\tau ).$ Teda $\left\| \mathbf{p}^{\prime }(\tau )\right\| =\left\| \mathbf{c\,\,}^{\prime }(\Phi (\tau ))\right\| \left| \Phi ^{\prime }(\tau )\right| $ a plat\'{\i} \begin{equation} \int_{C}f\left( s\right) ds=\int_{\alpha }^{\beta }f\left( \mathbf{p}\left( \tau \right) \right) \left\| \mathbf{p\,}^{\prime }(\tau )\right\| d\tau =\int_{\alpha }^{\beta }f\left( \mathbf{c}\left( \Phi \left( \tau \right) \right) \right) \left\| \mathbf{c\,}^{\prime }\left( \Phi \left( \tau \right) \right) \right\| \left| \Phi ^{\prime }\left( \tau \right) \right| d\tau \tag{(a)} \end{equation}% Ak teraz polo\v{z}\'{\i}me $t=\Phi (\tau )$ dostaneme: ak $\Phi $ je rast% \'{u}ca, potom $\Phi \left( \alpha \right) =a,\Phi \left( \beta \right) =b$ a $\left| \Phi ^{\prime }(t)\right| =\Phi ^{\prime }\left( t\right) ,$ zatia% \v{l} \v{c}o pre $\Phi $ klesaj\'{u}cu m\'{a}me: $\Phi \left( \alpha \right) =b,\Phi \left( \beta \right) =a$ a $\left| \Phi ^{\prime }(t)\right| =-\Phi ^{\prime }\left( t\right) .$ V oboch pr\'{\i}padoch a preto\v{z}e $\left( \int_{\beta }^{\alpha }=-\int_{\alpha }^{\beta }\right) $ dost\'{a}vame, \v{z}e prav\'{a} strane (a) je rovn\'{a} \[ \int_{\alpha }^{\beta }f\left( \mathbf{c}\left( \Phi \left( \tau \right) \right) \right) \left\| \mathbf{c\,}^{\prime }\left( \Phi \left( \tau \right) \right) \right\| \left| \Phi ^{\prime }\left( \tau \right) \right| d\tau =\int_{a}^{b}f(\mathbf{c}(t))\left\| \mathbf{c\,\,}^{\prime }(t)\right\| dt.\blacksquare \] \begin{center} \begin{tabular}{|c|} \hline {\small \hyperref{Sp\"{a}\v{t}}{}{}{Ma62.tex#1}} \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za II} \end{document}