\documentclass{article} \usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Created=Monday, June 25, 2001 17:45:57} %TCIDATA{LastRevised=Monday, March 24, 2003 12:50:23} %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{Diferencovanie funkci\'{\i}} \subsection{Veta o diferencovan\'{\i} zlo\v{z}enej funkcie.} \textbf{D\^{o}kaz:} Preto\v{z}e\textbf{\ }$\mathbf{f}:E\longrightarrow \mathbf{R}^{m},\,\mathbf{g}:G\longrightarrow \mathbf{R}^{p}$ s\'{u} tak\'{e}% , \v{z}e $\mathbf{g}(G)\subset E,$ potom zlo\v{z}en\'{a} funkcia $\mathbf{h}% :G\longrightarrow \mathbf{R}^{m},\,\mathbf{h}(\mathbf{\,x\,})=\left( \mathbf{% f\circ g}\right) \left( \mathbf{x}\right) =\mathbf{f}(\mathbf{g}(\mathbf{% \,x\,})).$ Nech $\mathbf{L}=D\mathbf{g}_{\mathbf{x}},\mathbf{\,M}=D\mathbf{f}% _{\mathbf{y}},\mathbf{\,\,y\,}=\mathbf{g\,}\left( \mathbf{x}\right) .$ Chceme uk\'{a}za\v{t}, \v{z}e \[ \lim_{\left\| \mathbf{k}\right\| \longrightarrow 0}\frac{\left\| \mathbf{\,h}% \left( \mathbf{\,x\,}+\mathbf{\,k}\right) -\mathbf{h}\left( \mathbf{x}% \right) -\mathbf{MLk}\right\| }{\left\| \mathbf{k}\right\| }=0. \]% M\'{a}me% \[ \mathbf{h}\left( \mathbf{x\,+\,k}\right) -\mathbf{h}\left( \mathbf{x}\right) -\mathbf{ML\,k\,\,}=\mathbf{f}\left( \mathbf{g}\left( \mathbf{x+k}\right) \right) -\mathbf{f}(\mathbf{g}(\mathbf{\,x\,}))-\mathbf{M}\left( \mathbf{g}% \left( \mathbf{x+k}\right) \right) -\mathbf{g}(\mathbf{\,x\,}))+\mathbf{M}% \left( \mathbf{g}\left( \mathbf{x+k}\right) -\mathbf{g}(\mathbf{\,x\,})-% \mathbf{L\,k}\right) , \]% odkia\v{l} m\'{a}me% \[ \left\| \mathbf{h}\left( \mathbf{x\,+\,k}\right) -\mathbf{h}\left( \mathbf{x}% \right) -\mathbf{ML\,k}\right\| \leq \left\| \mathbf{f}\left( \mathbf{g}% \left( \mathbf{x+k}\right) \right) -\mathbf{f}(\mathbf{g}(\mathbf{\,x\,}))-% \mathbf{M}\left( \mathbf{g}\left( \mathbf{x+k}\right) \right) -\mathbf{g}(% \mathbf{\,x\,}))\right\| +\left\| \mathbf{M}\right\| \left\| \mathbf{g}% \left( \mathbf{x+k}\right) -\mathbf{g}(\mathbf{\,x\,})-\mathbf{L\,k}\right\| . \]% Preto\v{z}e $\mathbf{g}$ je diferencovate\v{l}n\'{a} v bode $\mathbf{\,x\,}$ $\forall \varepsilon >0$ $\exists \mathbf{\,}\delta _{1}>0;\,$ \[ \mathbf{\,\,}\left\| \mathbf{g}(\mathbf{\,x\,}+\mathbf{k})-\mathbf{g}(% \mathbf{\,x\,})-\mathbf{Lk}\right\| <\varepsilon \mathbf{\,\,}\left\| \mathbf{k}\right\| \mathbf{\,\ \ }\text{pre \ }\mathbf{\,\mathbf{\,}\left\| \mathbf{k}\right\| }<\mathbf{\,}\delta _{1}. \]% Pod\v{l}a \hyperref{vety}{}{}{Ma31.tex#6} ku $\varepsilon =1$ $\exists \delta _{2}>0;$ \[ \left\| \mathbf{g}(\mathbf{\,x\,}+\mathbf{\,\,k})-\mathbf{g}(\mathbf{\,x\,}% )\right\| \leq \left( \left\| \mathbf{L}\right\| +1\right) \mathbf{\,}% \left\| \mathbf{k}\right\| \mathbf{\,\,\ \ }\text{ak \ }\mathbf{\,\,\,}% \left\| \mathbf{k}\right\| <\delta _{2}, \]% Preto\v{z}e $\mathbf{f}$ je diferencovate\v{l}n\'{a} v $\mathbf{\,y,}$ tak $% \forall \varepsilon >0$ $\exists \delta _{3}>0;$ \[ \left\| \mathbf{f}\left( \mathbf{z}\right) -\mathbf{f}\left( \mathbf{y}% \right) -\mathbf{M}\left( \mathbf{z}-\mathbf{\,y}\right) \right\| <\varepsilon \left\| \mathbf{\,z}-\mathbf{\,y}\right\| \text{ \ ak \ }% \left\| \mathbf{\,z}-\mathbf{\,y}\right\| \mathbf{<\delta }_{3}. \]% Nech $\delta _{2}$ je tak\'{e}, \v{z}e $(\left\| \mathbf{L}\right\| +1)\delta _{2}<\delta _{3},$ potom m\^{o}\v{z}eme voli\v{t} $\mathbf{z}=% \mathbf{g}\left( \mathbf{x}+\mathbf{\,k}\right) $ a $\mathbf{\,y\,}=\mathbf{g% }(\mathbf{\,x\,})$ a m\'{a}me \[ \left\| \mathbf{f}(\mathbf{g}(\mathbf{\,x\,}+\mathbf{\,\,k}))-\mathbf{f}(% \mathbf{g}(\mathbf{\,x\,}))-\mathbf{\,M}(\mathbf{g}(\mathbf{\,x\,}+\mathbf{% \,\,k})-\mathbf{g}(\mathbf{\,x\,}))\right\| < \]% \[ <\varepsilon \left\| \mathbf{g}(\mathbf{\,x\,}+\mathbf{\,\,k})-\mathbf{g}(% \mathbf{\,x\,})\right\| <\varepsilon (\left\| \mathbf{L}\right\| +1)\mathbf{% \,}\left\| \mathbf{k}\right\| \mathbf{\,\ \ }\text{pre \ \textbf{\thinspace }% }\mathbf{\,\,}\left\| \mathbf{k}\right\| <\delta _{2}. \]% Ak $\delta =\min \{\mathbf{\,}\delta _{1},\delta _{2}\},$ potom $C=\mathbf{% \,\,}\left\| \mathbf{M\,}\right\| +\mathbf{\,}\left\| \mathbf{L}\right\| +1.$ $\blacksquare $ \begin{center} \begin{tabular}{|c|} \hline {\small \hyperref{Sp\"{a}\v{t}}{}{}{Ma33.tex#2}} \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za II} \end{document}