%% 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 17:04:50} %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 - Integr\U{e1}lny po\U{10d}et - Viacrozmern\U{e9} integr\U{e1}ly ako iterovan\U{e9} integr\U{e1}ly\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}lny po\v{c}et} \begin{center} \begin{tabular}{|c|c|c|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maiindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maiindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{Ma5.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{Ma5.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{Ma52.tex}}% %BeginExpansion \msihyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{Ma52.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma54.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma54.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O5.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O5.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C5.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C5.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{Glos.tex}}% %BeginExpansion \msihyperref{Index}{}{}{Glos.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \subsection{Viacrozmern\'{e} integr\'{a}ly ako iterovan\'{e} integr\'{a}ly.} Teraz budeme uva\v{z}ova\v{t} o met\'{o}de v\'{y}po\v{c}tu integr\'{a}lov na $n$-kv\'{a}droch. Budeme sa koncentrova\v{t} na dvojn\'{e} integr\'{a}ly (integr\'{a}ly v $\mathbf{R}^{2}$), preto\v{z}e to ilustruje v\v{s}eobecn% \'{u} met\'{o}du. Uva\v{z}ujme teda $E=\left\langle a,b\right\rangle \times \left\langle c,d\right\rangle .$ Ak $f:E\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}$ je integrovate\v{l}n\'{a} na $E$, potom jej integr\'{a}l nap\'{\i}\v{s}eme v tvare $\int_{E}f$ alebo $\int_{E}f\left( x,y\right) dxdy$. D\^{o}vod pre druh\'{e} ozna\v{c}enie sa stane zrejm\'{y}% m, ak sa nau\v{c}\'{\i}me po\v{c}\'{\i}ta\v{t} tieto integr\'{a}ly. Ak $y$ je pevn\'{e} p\'{\i}\v{s}eme $f(.,y)$ pre tie funkcie, ktor\'{y}ch hodnoty s\'{u} v bode $x$ rovn\'{e} $f(x,y)$ t.j. $x\longmapsto f(x,y).$ Podobne pre $f(x,.).$ Pre obd\'{l}\v{z}nik $E=\left\langle a,b\right\rangle \times \left\langle c,d\right\rangle $ nech $\left\langle a,b\right\rangle $ je delen\'{e} na podintervaly $\left\{ I_{r};\,1\leq r\leq p\right\} $ a nech $\left\langle c,d\right\rangle $ sa del\'{\i} na podintervaly $\left\{ J_{s};\,1\leq s\leq q\right\} $ a nech $\mathcal{P}$ je odpovedaj\'{u}ce delenie $E$ na obd\'{l}% \v{z}niky $E_{rs}=I_{r}\times J_{s}.$ Ozna\v{c}me d\'{l}\v{z}ku intervalu $I$ ako $l(I).$ Potom plocha (obsah) $E_{rs}$ je $l(I_{r})l(J_{s}).$ Ak $% f:E\longrightarrow \mathbf{R}$ je integrovate\v{l}n\'{a} na $E,$ existuje delenie $\mathcal{P}$ tak\'{e}, \v{z}e integr\'{a}l sa d\'{a} dobre aproximova\v{t} sumou (t\'{a}to je zn\'{a}ma ako riemannov integr\'{a}lny s% \'{u}\v{c}et) \[ \sum_{r=1}^{p}\sum_{s=1}^{q}f(x_{r},y_{s})l(I_{r})l(J_{s}) \]% \v{c}o m\^{o}\v{z}eme p\'{\i}sa\v{t} ako $_{{}}$% \begin{equation} \sum_{s=1}^{q}\,l(J_{s})\left( \sum_{r=1}^{p}f(x_{r},y_{s})l(I_{r})\right) =\sum_{r=1}^{p}\,l(I_{r})\left( \sum_{s=1}^{q}f(x_{r},y_{s})l(J_{s})\right) \tag{(1)} \end{equation}% Prv\'{a} odpoved\'{a} vyjadreniu sumy cez v\v{s}etky obd\'{l}\v{z}niky najprv sumovan\'{e} cez horizont\'{a}lnu \v{c}iaru $y=y_{1},$ potom cez $% y=y_{2},\dots .$ Druh\'{e} vyjadrenie je cez $x$-y. Ak sa s\'{u}stred\'{\i}% me na prv\'{y} v\'{y}raz v (1), tak $\sum_{r=1}^{p}f(x_{r},y_{s})l(I_{r})$ aproximuje $\int_{a}^{b}f\left( x,y_{s}\right) dx.$ Ak nap\'{\i}\v{s}eme \[ G(y)=\int_{a}^{b}f\left( x,y\right) dx \]% potom \v{l}av\'{a} strana (1) je aproxim\'{a}ciou integr\'{a}lu \[ \int_{c}^{d}G\left( y\right) dy. \]% Podobne pre prav\'{u} stranu. O\v{c}ak\'{a}vame, \v{z}e bude plati\v{t}. \begin{equation} \int_{E}f=\int_{c}^{d}\left( \int_{a}^{b}f\left( x,y\right) dx\right) dy=\int_{a}^{b}\left( \int_{c}^{d}f\left( x,y\right) dy\right) dx \tag{(2)} \end{equation}% (2) n\'{a}m ukazuje, \v{z}e dvojn\'{y} integr\'{a}l mo\v{z}no po\v{c}\'{\i}ta% \v{t} ako iterovan\'{y} dvojn\'{a}sobn\'{y} integr\'{a}l. Niekedy sa zvykne p% \'{\i}sa\v{t} \[ \int_{c}^{d}dy\int_{a}^{b}f\left( x,y\right) dx\text{ \ namiesto \ }% \int_{c}^{d}\left( \int_{a}^{b}f\left( x,y\right) dx\right) dy. \]% Iterovan\'{e} integr\'{a}ly m\^{o}\v{z}eme vypo\v{c}\'{\i}ta\v{t} pomocou element\'{a}rneho integr\'{a}lneho po\v{c}tu funkci\'{\i} jednej premennej. Ale ak $\int_{E}f$ existuje, potom e\v{s}te (2) nemus\'{\i} plati\v{t}. Skuto% \v{c}ne, najjednoduch\v{s}\'{\i} pr\'{\i}klad je, ke\v{d} $f$ je spojit\'{a} s v\'{y}nimkou \v{c}iary $y=c,a\leq x\leq b.$ T\'{a}to mno\v{z}ina m\'{a} dvojdimenzion\'{a}lnu mieru nula, teda $\int_{E}f$ existuje, ale $% \int_{a}^{b}f\left( x,y\right) dx$ neexistuje, preto\v{z}e funkcia $f(.,c)$ je nespojit\'{a} na mno\v{z}ine (jednodimenzion\'{a}lnej) kladnej miery. Samozrejme, \v{z}e tak\'{e} nie\v{c}o sa nem\^{o}\v{z}e sta\v{t}, ak je funkcia $f$ spojit\'{a} na $E$. Dok\'{a}za\v{t} mo\v{z}no ale viac: \begin{theorem} \label{3}(Fubiniho veta pre $n$-kv\'{a}dre) Nech $f$ je riemannovsky integrovate\v{l}n\'{a} na $E=\left\langle a,b\right\rangle \times \left\langle c,d\right\rangle $ a predpokladajme, \v{z}e pre ka\v{z}d\'{e} pevn\'{e} $x\in \left\langle a,b\right\rangle $ je funkcia $f(x,.)$ riemannovsky integrovate\v{l}n\'{a} na $\left\langle c,d\right\rangle .$ Potom \[ \int_{E}f=\int_{a}^{b}F\left( x\right) dx,\text{ \ kde \ }% F(x)=\int_{c}^{d}f\left( x,y\right) dy, \]% podobne ak $f$ a $f(.,y)$ s\'{u} integrovate\v{l}n\'{e}, tak \[ \int_{E}f=\int_{c}^{d}G(y)dy,\text{ \ kde \ }G(y)=\int_{a}^{b}f\left( x,y\right) dx. \] \end{theorem} \begin{tabular}{|c|} \hline {\small %TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO531.tex}}% %BeginExpansion \msihyperref{D\^{o}kaz}{}{}{DO531.tex}% %EndExpansion } \\ \hline \end{tabular}% \label{1} D\'{a} sa overi\v{t}, \v{z}e (2) plat\'{\i}, ak $f$ m\'{a} najviac kone\v{c}% ne mnoho bodov nespojitosti v $E$. Av\v{s}ak (2) plat\'{\i} ak mno\v{z}ina bodov nespojitosti funkcie $f$ a mno\v{z}ina bodov nespojitosti $f(x,.)$ (a $% f(.,y)$) m\'{a} jednodimenzion\'{a}lnu mieru nula. \begin{theorem} Nech $f$ je riemannovsky integrovate\v{l}n\'{a} na $E=\left\langle a_{1},b_{1}\right\rangle \times \left\langle a_{2},b_{2}\right\rangle \times \left\langle a_{3},b_{3}\right\rangle $ a predpokladajme, \v{z}e pre ka\v{z}d% \'{e} pevn\'{e} $z$ je funkcia $f(.,.,z)$ riemannovsky integrovate\v{l}n\'{a} na $E_{12}=\left\langle a_{1},b_{1}\right\rangle \times \left\langle a_{2},b_{2}\right\rangle .$Potom \begin{equation} \int_{E}f=\int_{a_{3}}^{b_{3}}dz\int_{E_{12}}f(x,y,z)dxdy, \tag{(3)} \end{equation}% podobne ak $\forall (x,y)$ je funkcia $f(x,y,.)$ integrovate\v{l}n\'{a}, tak \begin{equation} \int_{E}f=\int_{E_{12}}dxdy\int_{a_{3}}^{b_{3}}f(x,y,z)dz. \tag{(4)} \end{equation} \end{theorem} \begin{tabular}{|c|} \hline {\small %TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO532.tex}}% %BeginExpansion \msihyperref{D\^{o}kaz}{}{}{DO532.tex}% %EndExpansion } \\ \hline \end{tabular}% \label{2} Kombin\'{a}ciou (3) a (4) dostaneme \[ \int \int \int_{E\left( \mathbf{a},\mathbf{b}\right) }f=\int_{a_{3}}^{b_{3}}dz\int_{a_{2}}^{b_{2}}dy% \int_{a_{1}}^{b_{1}}f(x,y,z)dx \]% a tieto integr\'{a}ly sa m\^{o}\v{z}u bra\v{t} aj v inom porad\'{\i}, v z% \'{a}vislosti od toho, \v{c}i existuj\'{u}. \begin{example} Vypo\v{c}\'{\i}tajte $\int \int_{E}x\sin ydxdy,$ ak $E=\left\langle 0,1\right\rangle \times \left\langle 0,\frac{\pi }{2}\right\rangle .$ \end{example} \begin{solution} \[ \int \int_{E}x\sin ydxdy=\int_{0}^{1}xdx\int_{0}^{\frac{\pi }{2}}\sin ydy=% \frac{1}{2}.\,\square \] \end{solution} \begin{center} \begin{tabular}{|c|c|c|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maiindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maiindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{Ma5.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{Ma5.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{Ma52.tex}}% %BeginExpansion \msihyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{Ma52.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma54.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma54.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O5.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O5.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C5.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C5.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{Integr\'{a}lny po\v{c}et} \end{document}