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\title[{\sf{Análise Matemática III -- Teste 1 -- 28/Outubro/2006}}]
{\sf{Análise Matemática III\\
\vspace*{2ex}
todos os cursos excepto LEIC, LCI, LEA, LEBM, LEFT e LMAC\\
\vspace*{2ex}
Teste 1 -- 28 de Outubro de 2006}}
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{\sf{\textbf{Instituto Superior Técnico\\
Departamento de Matemática\\
Secção de Álgebra e Análise}}}\end{center}}

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\vspace*{.5ex}

\centerline{\textbf{apresente e justifique todos os cálculos}}

\vspace*{.7ex}

\centerline{duração: hora e meia (11:00-12:30)}

\vspace*{1ex}

\begin{enumerate}

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\item
% parte I -- integrais múltiplos
% teorema de Fubini
Considere o sólido
\[
   S = \{ (x,y,z) \in {\mathbb R}^3 \, :\, 0<x<1\,;\, x<y<1\,;\, 0<z<\sqrt{2-x^2-y^2}\} \ .
\]

\begin{enumerate}
\item Calcule \marginpar{\ \ \ \ \ \ \ \ \ \ \ \ \ (3.5 val.)}
o integral $\displaystyle{\iiint _S z}$.

\item

 Escreva uma expressão para o volume  de $S$ em termos de integrais \marginpar{\ \ \ \ \ \ \ \ \ \ \ \ \ (3.5 val.)} iterados da forma
\[
   \int_{\ldots}^{\ldots} \left( \int_{\ldots}^{\ldots} \left( \int_{\ldots}^{\ldots}
  \, dx \right) \, dy \right) \, dz \ .
\]
\end{enumerate}


\vspace*{2ex}

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\item
% parte I -- mudanças de coordenadas
Calcule o volume do sólido \marginpar{\ \ \ \ \ \ \ \ \ \ \ \ \ (3 val.)}
 \[
    B = \{ (x,y,z) \in {\mathbb R}^3 \, :  1+\frac 12 \left(x²+y²\right)< z <\frac 32 \sqrt{x²+y²} , \ x>0, \ y>0 \}.
\]

\vspace*{2ex}

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% parte II --  integrais de linha

\item Seja $\Gamma \subset \mathbb{R}^3$ a curva definida por
$$
\Gamma=\{ (x,y,z)\in \mathbb{R}^3:\,\, z=2-y,\,\, x^2+y^2+(z-2)^2=1\}.
$$
Calcule a massa de $\Gamma$ sabendo \marginpar{\ \ \ \ \ \ \ \ \ \ \ \ \ (2 val.)}  que a densidade de massa é dada por $\sigma=z$.

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\vspace*{2ex}          
\item
Considere os campos vectoriais 
\begin{eqnarray*} \nonumber
\mathbf{F}(x,y,z) & = &(\sen(xz)z,\, y,\, \sen(xz)x)\quad \quad \text{e}\quad \\ \nonumber \mathbf{G}(x,y,z) & = & (-\frac{2y}{x^2+y^2},\,\frac{2x}{x^2+y^2},\, 0).
\end{eqnarray*}
\begin{enumerate}
\item[a)] Mostre que\marginpar{\ \ \ \ \ \ \ \ \ \ \ \ \ (2.5 val.)} $\mathbf{F}$ é um campo fechado e indique, justificando, se $\mathbf{F}$ é um gradiente no seu domínio. Em caso afirmativo determine um potencial escalar.
\vspace*{.3ex}
\item[b)]  Calcule $\int_{\Gamma}(\mathbf{F}+ \mathbf{G}) \cdot\,  d\mathbf{g}$ \marginpar{\ \ \ \ \ \ \ \ \ \ \ \ \  (1.5 val.)} onde $\Gamma$ é a curva definida por $z=2-y$ e $x^2+y^2+(z-2)^2=1$, percorrida no sentido horário quando observada do ponto $(0,0,10)$.
 \vspace*{1ex}
\item[c)] Obtenha uma equação que descreva  \marginpar{\ \ \ \ \ \ \ \ \ \ \ \ \ (1 val.)} todos os pontos $P$ tais que, se $C$ é uma curva regular com pontos inicial e final $(0,0,1)$ e $P$, então $\int_{C} \mathbf{F}\cdot \, d\mathbf{g} =1$.
\end{enumerate}

\vspace*{2ex}
\item
% Seja $\bf F$ um campo vectorial contínuo \marginpar{\ \ \ \ \ \ \ \ \ \ \ \ \ (2 val.)}  num domínio $S$ de $\mathbb{R}^n$, seja
% $C\subset S$ uma curva dada por uma 
% parametrização $ {\bf g}:[a,b]\rightarrow \mathbb{R}^n$ de classe $C^1$, e seja $k:[c,d]\rightarrow [a,b]$ uma função de
% classe $C^1$ tal que $k(c)=a,\, k(d)=b$. Considerando a parametrização  
% $
% {\bf h}={\bf g}\circ k:[c,d]\rightarrow \mathbb{R}^n,
% $ 
% mostre que
% $
% \int_{C}{\bf F.}\, d{\bf h}= \int_C {\bf F.} \, d{\bf g}.
% $
Seja  $D=\left\{(x,y) \in \mathbb{R}² \ : \ x²+y²\leq R²\right\}$ e $f:\mathbb{R}^2 \to \mathbb{R}$ \marginpar{\ \ \ \ \ \ \ \ \ \ \ \ \ (3 val.)} uma função
de classe $C^2$ tal que $f(x,y)=0$ na fronteira de $D$.
Mostre que
$
\iint_D \ f \Delta f \ \leq 0
$,
onde $\Delta f = \frac{\partial^2 f}{\partial x²} +  \frac{\partial^2 f}{\partial y²}$.




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\end{enumerate}

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