\documentclass{article} \usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Created=Monday, June 25, 2001 17:45:57} %TCIDATA{LastRevised=Monday, March 24, 2003 12:49:07} %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{Deriv\'{a}cie} \subsection{Veta o vz\v{t}ahu medzi diferencovate\v{l}nos\v{t}ou funkcie a existenciou parci\'{a}lnych deriv\'{a}ci\'{\i}.} \textbf{D\^{o}kaz: }Funkcia $\mathbf{f}:G\left( \subset \mathbf{R}% ^{n}\right) \longrightarrow \mathbf{R}^{m}$ je diferencovate\v{l}n\'{a} v bode $\mathbf{a}\in G.$ Potom pre ka\v{z}d\'{y} smer $\mathbf{e}% _{j},\,j=1,\dots ,n$ \ m\'{a}me \textbf{\ }% \[ \mathbf{f}(\mathbf{a}+t\mathbf{e}_{j})-\mathbf{f}(\mathbf{a})=tD\mathbf{f}_{% \mathbf{a}}(\mathbf{e}_{j})+\mathbf{R}(t), \]% kde \[ \lim_{t\longrightarrow 0}\frac{\left\| \mathbf{R}(t)\right\| }{\left\| \mathbf{a}+t\mathbf{e}_{j}\mathbf{-a}\right\| }=\lim_{t\longrightarrow 0}% \frac{\left\| \mathbf{R}(t)\right\| }{\left| t\right| \left\| \mathbf{e}% _{j}\right\| }=\lim_{t\longrightarrow 0}\frac{\left\| \mathbf{R}(t)\right\| }{\left| t\right| }=\lim_{t\longrightarrow 0}\frac{\left\| \mathbf{R}% (t)\right\| }{t}=\lim_{t\longrightarrow 0}\frac{\left\| \mathbf{f}(\mathbf{a}% +t\mathbf{e}_{j})-\mathbf{f}(\mathbf{a})-tD\mathbf{f}_{\mathbf{a}}(\mathbf{e}% _{j})\right\| }{t}=0. \]% Ak ozna\v{c}\'{\i}me $\mathbf{L=}D\mathbf{f}_{\mathbf{a}},$ potom pre ka\v{z}% d\'{u} komponentu m\'{a}me \[ L_{i}(\mathbf{e}_{j})=\lim_{t\longrightarrow 0}\frac{f_{i}(\mathbf{a}+t% \mathbf{e}_{j})-f_{i}(\mathbf{a})}{t}=\frac{\partial f_{i}}{\partial x_{j}}% \left( \mathbf{a}\right) . \]% \textbf{\ }Toto dokazuje vetu, preto\v{z}e oper\'{a}tor $\mathbf{L=}D\mathbf{% f}_{\mathbf{a}}$ je pre \v{s}tandardn\'{u} b\'{a}zu definovan\'{y} pomocou matice. $\blacksquare $ \begin{center} \begin{tabular}{|c|} \hline {\small \hyperref{Sp\"{a}\v{t}}{}{}{Ma32.tex#1}} \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za II} \end{document}