%% This document created by Scientific Notebook (R) Version 3.5 %% Starting shell: article \documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amssymb} %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.00.0.2570} %TCIDATA{} %TCIDATA{Created=Wednesday, February 10, 1999 13:29:48} %TCIDATA{LastRevised=Sunday, February 13, 2005 17:16:21} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{Counters=arabic,1} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\QTR{small}{Matematick\U{e1} anal\U{fd}za II online - Plo\U{161}n\U{e9} integr\U{e1}ly - Z\U{e1}kladn\U{e9} fakty z vektorov\U{e9}ho po\U{10d}tu v }$\QTR{bf}{R}^{3}$\QTR{small}{\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{Plo\v{s}n\'{e} integr\'{a}ly} \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maiindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maiindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{Ma7.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{Ma7.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma72.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma72.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O7.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O7.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C7.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C7.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{Glos.tex}}% %BeginExpansion \msihyperref{Index}{}{}{Glos.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \paragraph{Ciele} Po pre\v{s}tudovan\'{\i} tejto \v{c}asti by ste mali by\v{t} schopn\'{\i}: \begin{itemize} \item definova\v{t} z\'{a}kladn\'{e} pojmy (vektorov\'{y} s\'{u}\v{c}in, gradient, divergenciu, rot\'{a}ciu) z vektorov\'{e}ho po\v{c}tu v $\mathbf{R}% ^{3}$a op\'{\i}sa\v{t} \ ich vlastnosti, \item definova\v{t} jednoduch\'{y} a hladk\'{y} plo\v{s}n\'{y} element, \item vypo\v{c}\'{\i}ta\v{t} plo\v{s}n\'{y} obsah hladk\'{e}ho elementu elementu, \item definova\v{t} aj vypo\v{c}\'{\i}ta\v{t} plo\v{s}n\'{y} integr\'{a}l zo skal\'{a}rneho aj z vektorov\'{e}ho po\v{l}a, \item formulova\v{t} a aplikova\v{t} Greenovu vetu, \item formulova\v{t} a aplikova\v{t} Gaussovu vetu, \item formulova\v{t} a aplikova\v{t} Stokesovu vetu. \end{itemize} \paragraph{Po\v{z}adovan\'{e} vedomosti:} \begin{itemize} \item znalos\v{t} jednorozmern\'{y}ch integr\'{a}lov a met\'{o}d ich v\'{y}po% \v{c}tov, \item znalos\v{t} viacrozmern\'{y}ch integr\'{a}lov a met\'{o}d ich v\'{y}po% \v{c}tov, \item znalos\v{t} krivkov\'{y}ch integr\'{a}lov. \end{itemize} \subsection{Z\'{a}kladn\'{e} fakty z vektorov\'{e}ho po\v{c}tu v $\mathbf{R}% ^{3}.$} \begin{definition} Nech $\mathbf{a},\,\mathbf{b}$ s\'{u} dva vektory z $\mathbf{R}^{3}.$ \emph{% Vektorov\'{y} s\'{u}\v{c}in }vektorov $\mathbf{a},\,\mathbf{b}$ je definovan% \'{y} \[ \mathbf{a\times }\,\mathbf{b=}\left( a_{2}b_{3}-a_{3}b_{2},a_{3}b_{1}-a_{1}b_{3},a_{1}b_{2}-a_{2}b_{1}\right) . \] \end{definition} Vektorov\'{y} s\'{u}\v{c}in odli\v{s}uje priestor $\mathbf{R}^{3}$ od ostatn% \'{y}ch euklidovsk\'{y}ch priestorov. Ak ozna\v{c}\'{\i}me \v{s}tandardn\'{e} b\'{a}zov\'{e} vektory v $\mathbf{R}^{3}$ ako $\mathbf{i,j,k}$ potom $% \forall \mathbf{a}\in \mathbf{R}^{3},\,\mathbf{a}=a_{1}\mathbf{i}+a_{2}% \mathbf{j}+a_{3}\mathbf{k}$ a vektorov\'{y} s\'{u}\v{c}in m\^{o}\v{z}eme zap% \'{\i}sa\v{t} ako form\'{a}lny v\'{y}raz% \[ \mathbf{a\times }\,\mathbf{b=}\left| \begin{tabular}{ccc} $\mathbf{i}$ & $\mathbf{j}$ & $\mathbf{k}$ \\ $a_{1}$ & $a_{2}$ & $a_{3}$ \\ $b_{1}$ & $b_{2}$ & $b_{3}$% \end{tabular}% \right| \] Pre vektorov\'{y} s\'{u}\v{c}in plat\'{\i}: \[ \mathbf{a\times }\,\mathbf{b=-}\,\mathbf{b\times a,} \]% \[ \mathbf{i}\times \mathbf{j}=\mathbf{k,\,j}\times \mathbf{k}=\mathbf{i,\,k}% \times \mathbf{i}=\mathbf{j}. \] Ak $\mathbf{a},\,\mathbf{b},\,\mathbf{c}\in \mathbf{R}^{3},$\textbf{\ }% definujeme trojn\'{y} s\'{u}\v{c}in: \[ \left[ \,\mathbf{c},\,\mathbf{a},\,\mathbf{b}\right] =\left| \begin{tabular}{ccc} $c_{1}$ & $c_{2}$ & $c_{3}$ \\ $a_{1}$ & $a_{2}$ & $a_{3}$ \\ $b_{1}$ & $b_{2}$ & $b_{3}$% \end{tabular}% \right| . \] \v{L}ahko mo\v{z}no overi\v{t}, \v{z}e \[ \left[ \,\mathbf{c},\,\mathbf{a},\,\mathbf{b}\right] =\left[ \,\mathbf{a},\,% \mathbf{b},\mathbf{c}\right] =\left[ \,\mathbf{c},\,\mathbf{b,\,a}\right] =-% \left[ \,\mathbf{b},\,\mathbf{a},\,\mathbf{c}\right] =-\left[ \,\mathbf{c},\,% \mathbf{b},\,\mathbf{a}\right] =-\left[ \,\mathbf{a},\,\mathbf{c},\,\mathbf{b% }\right] \] Okrem toho \ \[ 0=\left[ \,\mathbf{a},\,\mathbf{a},\,\mathbf{b}\right] =\mathbf{a\cdot }% \left( \mathbf{a\times }\,\mathbf{b}\right) , \] \[ 0=\left[ \,\,\mathbf{b},\,\mathbf{a},\,\mathbf{b}\right] =\,\mathbf{b\cdot }% \left( \mathbf{a\times }\,\mathbf{b}\right) , \] t.j. $\,\mathbf{a\times }\,\mathbf{b}$ je otrogon\'{a}lne na $\mathbf{a}$ aj na $\,\mathbf{b.}$ Je to zauj\'{\i}mav\'{e}, ke\v{d} $\mathbf{a\times }\,% \mathbf{b\neq 0.}$ Ak $\mathbf{n}$ je vektor, potom mno\v{z}ina \[ \left\{ \mathbf{x}\in \mathbf{R}^{3};\mathbf{n\cdot }\,\mathbf{x=}\alpha \right\} \] sa naz\'{y}va \emph{rovina. }Jej rovnica je $% n_{1}x_{1}+n_{2}x_{2}+n_{3}x_{3}=\alpha .$ Vektor $\mathbf{n}$ naz\'{y}vame norm\'{a}lov\'{y} vektor na rovinu, preto\v{z}e pre ka\v{z}d\'{e} dva body $% A,\,X$ z roviny $\mathbf{x}\cdot \,\mathbf{n}=\alpha ,\,\mathbf{a}\cdot \mathbf{n}=\alpha $ plat\'{\i}, \v{z}e $\left( \mathbf{x-a}\right) \cdot \,% \mathbf{n}=0,$ teda $\overrightarrow{AX}=\mathbf{x}-\mathbf{a}$ je ortogon% \'{a}lny na $\mathbf{n}.$ Ak $\mathbf{a},\,\mathbf{b}$ s\'{u} dva vektory, tak\'{e} \v{z}e $\mathbf{a\times }\,\mathbf{b}\neq \mathbf{0},$ potom rovina s rovnicou \[ \mathbf{x}\cdot \left( \mathbf{a\times }\,\mathbf{b}\right) =0 \] obsahuje aj vektor $\mathbf{a}$ aj vektor $\,\mathbf{b}$ (teda obsahuje tri body $0,A,B$) sa naz\'{y}va \emph{rovinou ur\v{c}enou vektormi} $\mathbf{a,\,% }\,\mathbf{b}$ a jej norm\'{a}lov\'{y} vektor je $\mathbf{a\times }\,\mathbf{% b}.$ Jednoduch\'{y}m v\'{y}po\v{c}tom dostaneme: \[ \left\| \mathbf{a\times }\,\mathbf{b}\right\| ^{2}=\left( a_{2}b_{3}-a_{3}b_{2}\right) ^{2}+\left( a_{3}b_{1}-a_{1}b_{3}\right) ^{2}+\left( a_{1}b_{2}-a_{2}b_{1}\right) ^{2}= \]% \[ =\left\| \mathbf{a}\right\| ^{2}\left\| \,\mathbf{b}\right\| ^{2}-\left( \mathbf{a}\cdot \mathbf{b}\right) ^{2}=\left\| \mathbf{a}\right\| ^{2}\left\| \,\mathbf{b}\right\| ^{2}\left( 1-\cos ^{2}\Theta \right) , \] odkia\v{l} \[ \left\| \mathbf{a\times }\,\mathbf{b}\right\| =\left\| \mathbf{a}\right\| \left\| \,\mathbf{b}\right\| \left| \sin \Theta \right| \] \ je obsah plochy rovnobe\v{z}n\'{\i}ka (kosod\'{l}\v{z}nika) so stranami \ $% \mathbf{a,\,}\,\mathbf{b}.$ Podobne m\^{o}\v{z}eme interpretova\v{t} aj $\pm % \left[ \,\mathbf{a},\,\mathbf{b},\mathbf{c}\right] $ - objem rovnobe\v{z}% nostena so stranami $\,\mathbf{a},\,\mathbf{b},\mathbf{c},\,$\ pri\v{c}om \ $% \pm \left[ \,\mathbf{a},\,\mathbf{b},\mathbf{c}\right] =\left\| \,\mathbf{a}% \times \,\mathbf{b}\right\| \left\| \mathbf{c}\right\| \cos \varphi ,$ kde $% \varphi =\sphericalangle \left( \,\mathbf{a}\times \,\mathbf{b},\,\mathbf{c}% \right) .$ \subsection{Gradient, divergencia, rot\'{a}cia.} Nech $G\subset \mathbf{R}^{3}$ je otvoren\'{a} mno\v{z}ina. Funkciu $\Phi :G\longrightarrow \mathbf{R}$ sa naz\'{y}vame \emph{skal\'{a}rne pole, }% funkciu $\mathbf{F}:G\longrightarrow \mathbf{R}^{3}$ sa naz\'{y}vame \emph{% vektorov\'{e} pole.} Nech $\Phi $ je diferencovate\v{l}n\'{e} skal\'{a}rne pole. \emph{Gradientom skal\'{a}rneho po\v{l}a }$\Phi $ je vektorov\'{e} pole, ktor\'{e} ozna\v{c}% ujeme \[ grad\Phi =\nabla \Phi =\left( \frac{\partial \Phi }{\partial x},\frac{% \partial \Phi }{\partial y},\frac{\partial \Phi }{\partial z}\right) . \] Gradient mo\v{z}no jednoducho form\'{a}lne zap\'{\i}sa\v{t} pomocou nabla oper\'{a}tora: \[ \nabla =\mathbf{i}\frac{\partial }{\partial x}+\mathbf{j}\frac{\partial }{% \partial y}+\mathbf{k}\frac{\partial }{\partial z}, \]% potom pre $C^{1}$skal\'{a}rne pole $\Phi $ m\'{a}me \[ \nabla \Phi =\mathbf{i}\frac{\partial \Phi }{\partial x}+\mathbf{j}\frac{% \partial \Phi }{\partial y}+\mathbf{k}\frac{\partial \Phi }{\partial z}% =\left( \frac{\partial \Phi }{\partial x},\frac{\partial \Phi }{\partial y},% \frac{\partial \Phi }{\partial z}\right) . \]% \ V\'{y}znam gradientu u\v{z} pozn\'{a}me z diferenci\'{a}lneho po\v{c}tu, preto nebudeme spom\'{\i}na\v{t} jeho \v{d}al\v{s}ie vlastnosti. Nech teraz $\mathbf{F}:G\longrightarrow \mathbf{R}^{3}$ je $C^{1}$ vektorov% \'{e} pole, $\mathbf{F}=\left( F_{1},F_{2},F_{3}\right) .$ \begin{definition} \emph{Divergenciou vektorov\'{e}ho po\v{l}a} $\mathbf{F}$ je \emph{skal\'{a}% rne pole} \[ div\mathbf{F}=\frac{\partial F_{1}}{\partial x}+\frac{\partial F_{2}}{% \partial y}+\frac{\partial F_{3}}{\partial z}. \]% \emph{Rot\'{a}ciou vektorov\'{e}ho po\v{l}a} $\mathbf{F}$ je \emph{vektorov% \'{e} pole} \[ curl\mathbf{F}=rot\mathbf{F}=-\left( \frac{\partial F_{2}}{\partial z}-\frac{% \partial F_{3}}{\partial y},\frac{\partial F_{3}}{\partial x}-\frac{\partial F_{1}}{\partial z},\frac{\partial F_{1}}{\partial y}-\frac{\partial F_{2}}{% \partial x}\right) . \] \end{definition} Tieto defin\'{\i}cie mo\v{z}no pomocou oper\'{a}tora $\nabla $ zap\'{\i}sa% \v{t} v tvare: \[ div\mathbf{F}=\nabla \cdot \mathbf{F}, \]% \[ rot\mathbf{F}=\nabla \times \mathbf{F}=\left| \begin{tabular}{ccc} $\mathbf{i}$ & $\mathbf{j}$ & $\mathbf{k}$ \\ $\frac{\partial }{\partial x}$ & $\frac{\partial }{\partial y}$ & $\frac{% \partial }{\partial z}$ \\ $F_{1}$ & $F_{2}$ & $F_{3}$% \end{tabular}% \right| . \] \begin{lemma} \label{2}Nech $\varphi ,\psi $ s\'{u} $C^{1}$skal\'{a}rne polia a $\mathbf{F} $ je $C^{1}$vektorov\'{e} pole. Potom plat\'{\i} a) Ak $\mathbf{F}\in C^{2}\left( G,\mathbf{R}^{3}\right) ,\,div\left( rot% \mathbf{F}\right) =0,\,$alebo $\nabla \cdot \left( \nabla \times \mathbf{F}% \right) =0,$ b) Ak $\varphi \in C^{2}\left( G,\mathbf{R}\right) ,\,rot\left( div\varphi \right) =0,\,$alebo $\nabla \times \left( \nabla \cdot \varphi \right) =0,$ c) $grad\left( \varphi \psi \right) =\varphi grad\psi +\psi grad\varphi ,\,$% alebo $\nabla \left( \varphi \psi \right) =\varphi \left( \nabla \psi \right) +\psi \left( \nabla \varphi \right) ,\,$ d) $div\left( \varphi \mathbf{F}\right) =\varphi div\mathbf{F}+grad\varphi \cdot \mathbf{F},\,$alebo $\nabla \cdot \left( \varphi \mathbf{F}\right) =\varphi \left( \nabla \cdot \mathbf{F}\right) +\nabla \varphi \cdot \mathbf{% F},$ e) $rot\left( \varphi \mathbf{F}\right) =\varphi rot\mathbf{F}+grad\varphi \times \mathbf{F},\,$alebo $\nabla \times \left( \varphi \mathbf{F}\right) =\varphi \left( \nabla \times \mathbf{F}\right) +\nabla \varphi \times \mathbf{F.}$ \end{lemma} \begin{tabular}{|c|} \hline {\small %TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO711.tex}}% %BeginExpansion \msihyperref{D\^{o}kaz}{}{}{DO711.tex}% %EndExpansion } \\ \hline \end{tabular}% \label{1} \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maiindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maiindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{Ma7.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{Ma7.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma72.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma72.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O7.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O7.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C7.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C7.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{Glos.tex}}% %BeginExpansion \msihyperref{Index}{}{}{Glos.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za II} \section{Plo\v{s}n\'{e} integr\'{a}ly} \end{document}