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%
\begin{tabular}
[c]{ll}%
$\text{UFPB/CCEN/CAMPUS I}$ & $\text{PER\'{I}ODO 09.2}$\\
$\text{P\'{O}S-GRADUA\c{C}\~{A}O EM MATEM\'{A}TICA}$ & $\text{DATA:22.08.2009
}$\\
$\text{DISCIPLINA: }$An\'{a}lise Real I & $\text{TURMA: 01}$\\
$\text{PROFESSOR: Milton }$ & $\text{TURNO: Tarde}$\\
{\ ALUNO:}\underline{\hspace{5cm}}\quad MATRICULA:\underline{\hspace{2cm}} &
\\
&
\end{tabular}


\bigskip

\begin{center}
$6^{a}$ Lista de Exerc\'{\i}cios

\bigskip
\end{center}

\begin{enumerate}
\item Sejam $T:\mathbb{R} \longrightarrow\mathbb{R}$ definida por $T(x)=
\alpha x + \beta$ com $\alpha\neq0$ e $I\subset\mathbb{R},$ compacto. Dada
$f:J \longrightarrow\mathbb{R}$ limitada em $J=T(I)$ tem-se
\[
\overline{\int}_{J} f(y)dy = |\alpha| \overline{\int}_{I} f(\alpha x +
\beta)dx
\]


\item Sejam $Y \subset\mathbb{R}$, $[a,b]$ contendo Y e sua imagem $T(Y),$
$(onde \ \ T(x)= \alpha x + \beta)$ e $f:[a,b] \longrightarrow\mathbb{R}$ uma
fun\c{c}\~{a}o limitada, que se anula fora de $T(Y).$ Ent\~{a}o%

\[
\bar\int_{a}^{b} f(y)dy =\bar\int_{a}^{b} f(\alpha x + \beta)|\alpha|dx
\]


\item Toda transforma\c{c}\~ao linear invert\'{\i}vel $T:\mathbb{R}^{m}
\rightarrow\mathbb{R}^{m}$ pode ser escrita como produto de
transforma\c{c}\~oes lineares elementares(necessariamente invert\'{\i}veis)
$E_{1}\cdot E_{2} \cdots E_{k}$ dos tipos listados abaixo:

\begin{enumerate}
\item $x=(x_{1},\cdots,x_{n})\rightarrowtail E\cdot x=(\phi(x), \cdots,
x_{k})$, onde $\phi(x)=\sum_{i=1}^{m} \alpha_{i} x_{i}.$

\item $x=(x_{1},\cdots, x_{i},\cdots, x_{j},\cdots,x_{m})\rightarrowtail
E\cdot x=(x_{1},\cdots,x_{j},\cdots, x_{i},\cdots,x_{m}).$
\end{enumerate}

\item Se $h:U \longrightarrow V$ \'{e} um difeomorfismo de classe $C^{1}$
entre abertos de $\mathbb{R}^{m}$ ent\~{a}o (por $h$ ser um homeomorfismo)
tem-se $h(\partial X )= \partial h(X)$ para todo conjunto $X$ tal que $\Bar X
\subset U.$

\item Se $X \subset\mathbb{R}^{m} $ tem medida nula e $f:X \longrightarrow
\mathbb{R}^{m}$ \'{e} localmente Lipschitziana, ent\~{a}o$f(X)$ tem medida
nula em $\mathbb{R}^{m}.$

\item Se $X$ \'{e} J-mensur\'{a}vel e $\bar X \subset U,$ sua imagem $h(X)$
\'{e} J-mensur\'{a}vel.

\item Se $f:h(X)\longrightarrow\mathbb{R}$ ent\~{a}o o conjunto dos pontos de
descontinidade de $f$ e $f \circ h:X\longrightarrow\mathbb{R}$ est\~{a}o
relacionadas pela igualdade%

\[
D(f \circ h)= h^{-1}(D(f))
\]


\item $h^{-1}$ localmente Lipschitziana, temos $med[D(f \circ h)]=0
\Longleftrightarrow med[D(f)]=0,$ isto \'{e} $f$ \'{e} integr\'{a}vel se, e
somente se $f \circ h$ \'{e} integr\'{a}vel.

\item O que \'e uma decomposi\c{c}\~{a}o de $X \subset\mathbb{R}^{m}$
J-mensur\'{a}vel? O que \'e uma decomposi\c{c}\~ao pontilhada?Defina a soma de
Riemann de $f: X \longrightarrow\mathbb{R}$ (limitada) relativamente a
decomposi\c{c}\~{a}o pontilhada $D^{*}=(D,(\xi_{i}))$ de $X$.

\item Sejam $T:\mathbb{R}^{m} \longrightarrow\mathbb{R}^{m} $ uma
transforma\c{c}\~{a}o linear invert\'{\i}vel, $X \subset\mathbb{R}^{m} $
J-mensur\'{a}vel e $f:T(X)\longrightarrow\mathbb{R}$ integr\'{a}vel.
Ent\~{a}o
\[
\int_{T(X)} f(y)dy = \int_{X} f(T.x)\mid\det T \mid dx .
\]
Este \'{e} o caso linear do Teorema de mudan\c{c}a de vari\'aveis.

\item Se $X \subset\mathbb{R}^{m}$ um conjunto J-mensur\'{a}vel. Para toda
transforma\c{c}\~{a}o linear $T:\mathbb{R}^{m} \longrightarrow\mathbb{R}^{m}
,$ tem-se%
\[
vol\, T(X)= \mid\det T \mid. vol \,X
\]


\item Sejam $X$ compacto, $U$ aberto, $X \subset U \subset\mathbb{R}^{m},$ e
$\varphi: U\times U \longrightarrow\mathbb{R}$ cont\'{\i}nua, com
$\varphi(x,x)=1 \forall\ x \in X.$ Dado $\epsilon>0,$ pode-se obter $\delta>0$
tal que $\mid\varphi(x,y)-1 \mid< \epsilon$ quaisquer que sejam $x,y \in X $
com $\mid y -x \mid<\delta.$

\item Sejam $U, V \subset\mathbb{R}^{m} $ abertos, $h:U \longrightarrow V$ um
difeomorfismo $C^{1}$, $X \subset U$ compacto J-mensur\'{a}vel e $N=N(h,X)=
\sup\{\mid h^{\prime}(x)\mid; x \in X\}.$ Ent\~{a}o $h(X)$ \'{e}
J-mensur\'{a}vel e $vol h(X) \leq N^{m}.vol X$

\item Lema de Duhamel: Seja $f:X \longrightarrow\mathbb{R}$ integr\'{a}vel no
conjunto J-mensur\'{a}vel $X \subset\mathbb{R}^{m}.$ Para cada
decomposi\c{c}\~{a}o $D={X_{1},...,X_{k}}$ de $X,$ suponhamos dados os
n\'{u}meros $\eta_{1}=\eta_{1}(D),...,\eta_{k}=\eta_{k}(D)$ tais que
$\displaystyle\lim_{\mid D\mid\longrightarrow0} \eta_{i} = 0. $ Nestas
condi\c{c}\~{o}es, tem-se
\[
\displaystyle\lim_{\mid D\mid\longrightarrow0} \sum[f(\xi_{i})+\eta
_{i}]volX_{i}=\int_{X} f(x)dx
\]


\item Junte todos os itens anteriores e prove o Teorema de Mudan\c{c}a de
Vari\'{a}veis: Sejam $h:U\longrightarrow V$ um difeomorfismo de classe $C^{1}$
entre abertos, $U,V\subset\mathbb{R}^{m},\ X\subset U$ compacto
J-mensur\'{a}vel e $f:h(X)\longrightarrow\mathbb{R}$ integr\'{a}vel. Ent\~{a}o
$f\circ h:X\longrightarrow\mathbb{R}$ \'{e} integr\'{a}vel e
\[
\int_{h(X)}f(y)dy=\int_{X}f(h(x)).\det\mid h^{\prime}(x)\mid dx
\]

\end{enumerate}


\end{document}
