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\QTR{small}{Matematick\U{e1} anal\U{fd}za II online - Krivkov\U{e9} integr\U{e1}ly - Cesty a krivky\dotfill \thepage }}
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\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}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\textbf{%
%TCIMACRO{\hyperref{Obsah}{}{}{maiindex.tex}}%
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%TCIMACRO{\hyperref{Index}{}{}{Glos.tex}}%
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} \\ \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 vysvetli\v{t} pojem cesty a krivky v $\mathbf{R}^{n},$
\item rozozna\v{t} r\^{o}zne typy kriviek pod\v{l}a ich vlastnost\'{\i},
\item vypo\v{c}\'{\i}ta\v{t} d\'{l}\v{z}ku krivky,
\item definova\v{t} krivkov\'{y} integr\'{a}l zo skal\'{a}rnej funkcie,
\item definova\v{t} krivkov\'{y} integr\'{a}l z vektorovej funkcie,
\item aplikova\v{t} tieto znalosti pri v\'{y}po\v{c}te krivkov\'{y}ch integr%
\'{a}lov,
\item ur\v{c}i\v{t}, \v{c}i je krivkov\'{y} integr\'{a}l z vektorovej
funkcie nez\'{a}visl\'{y} od cesty.
\end{itemize}
\paragraph{Po\v{z}adovan\'{e} vedomosti:}
\begin{itemize}
\item znalos\v{t} spojit\'{y}ch a diferencovate\v{l}n\'{y}ch funkci\'{\i},
\item znalos\v{t} met\'{o}d v\'{y}po\v{c}tu jednorozmern\'{y}ch integr\'{a}%
lov.
\end{itemize}
\subsection{Cesty a krivky.}
Nech $\mathbf{c}:\left\langle a,b\right\rangle \longrightarrow \mathbf{R}%
^{n} $ je spojit\'{a} funkcia. Pre ka\v{z}d\'{e} $t\in \left\langle
a,b\right\rangle $ existuje $\mathbf{c}(t)\in \mathbf{R}^{n}.$ M\^{o}\v{z}me
predpoklada\v{t}, \v{z}e $t$ reprezentuje \v{c}as a $\mathbf{c}(t)$ je
polohu pohybuj\'{u}ceho sa bodu v \v{c}ase $t$. Ako sa men\'{\i} $t\in
\left\langle a,b\right\rangle $, tak pohybuj\'{u}ci bod vytv\'{a}ra krivku.
\begin{definition}
Spojit\'{u} funkciu $\mathbf{c}:\left\langle a,b\right\rangle
\longrightarrow \mathbf{R}^{n}$ naz\'{y}vame \label{4}\emph{cestou, }$\,$jej
obraz - mno\v{z}inu $C=\mathbf{c}\left( \left\langle a,b\right\rangle
\right) $ naz\'{y}vame \label{5}\emph{krivkou.}
Ak je funkcia $\mathbf{c}$ injekt\'{\i}vna na $\left\langle a,b\right\rangle
,\,$cestu naz\'{y}vame \label{6}\emph{jednoduchou.}
Ak je $\mathbf{c}$ injekt\'{\i}vna na $\left\langle a,b\right) $ a $\mathbf{c%
}(a)=\mathbf{c}(b),$ potom $\mathbf{c}$ naz\'{y}vame \label{7}\emph{jednoduch%
\'{a} uzavret\'{a} cesta.}
Ak $\mathbf{c}\in C^{1},$ potom $\mathbf{c}$ naz\'{y}vame \label{8}$C^{1}$%
\emph{\ cestou.}
Ak $\mathbf{c}\in C^{1}$\ a $\mathbf{c\,\,}^{\prime }(t)\neq \mathbf{0}%
,\,\forall t\in \left\langle a,b\right\rangle $ potom $\mathbf{c}$\ naz\'{y}%
vame \label{9}\emph{hladkou cestou.}
Funkciu $\mathbf{c}(t)$ sa naz\'{y}va \label{10}\emph{parametriz\'{a}cia, }%
alebo \emph{parametrick\'{e} rovnice krivky \ }$C.$ Zvykneme p\'{\i}sa\v{t}
aj $C=\mathbf{c}\left( \left\langle a,b\right\rangle \right) =\left[ \mathbf{%
c}\right] .$
\end{definition}
Cesty maj\'{u} \emph{orient\'{a}ciu} - krivka sa vykres\v{l}uje v smere rast%
\'{u}ceho $t.$
V\v{s}imnime si, \v{z}e krivka \emph{nie je grafom} funkcie $\mathbf{c,}$
ale jej obrazom. Injekt\'{\i}vnos\v{t} funkcie $\mathbf{c}$ znamen\'{a},
\v{z}e krivka je jednoduch\'{a} t.j. tak\'{a}, ktor\'{a} sama seba nepret%
\'{\i}na, t.j. nem\'{a} vlastn\'{e} priese\v{c}n\'{\i}ky.
\begin{example}
Nech $\mathbf{c}:\left\langle 0,\frac{\pi }{2}\right\rangle \longrightarrow
\mathbf{R}^{2},\,$ $\mathbf{c}\left( t\right) =\left( \cos t,\sin t\right) .$
Zistite, \v{c}i $\mathbf{c}$ je cesta ak \'{a}no na\v{c}rtnite jej obraz.
\end{example}
\begin{solution}
Funkcia $\mathbf{c}$ je spojit\'{a}, $\mathbf{c}\in C^{1},$ je to $C^{1}$%
cesta, jej obrazom je krivka $C=\mathbf{c}\left( \left\langle 0,\frac{\pi }{2%
}\right\rangle \right) ,\,$t.j. \v{s}tvr\v{t}kru\v{z}nica v prvom kvadrante
v $\mathbf{R}^{2}.$ Krivka $C$ sa vykres\v{l}uje od bodu $\left[ 1,0\right] $
po bod $\left[ 0,1\right] .$\FRAME{dtbpFX}{4.4996in}{3in}{0pt}{}{}{Plot}{%
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\end{solution}
D\^{o}le\v{z}it\'{y}m pojmom je d\'{l}\v{z}ka cesty.
\begin{definition}
Nech $\mathbf{c}:\left\langle a,b\right\rangle \longrightarrow \mathbf{R}%
^{n} $ je cesta. Pre \v{l}ubovo\v{l}n\'{e} delenie intervalu $\left\langle
a,b\right\rangle $
\[
\mathcal{P}=\left\{ a=t_{0}