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PostPosted: Fri Aug 31, 2018 4:23 pm 
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Location: Ioannina, Greece
Let $E$ be the surface with parametric representation
\begin{align*}
\overline{R}:[0,6)&\times[0,2\pi]\longrightarrow{\mathbb{R}}^3\,; \quad
\overline{R}(r,\theta)=\left({\begin{array}{c}
\frac{r^2}{\sqrt{36-r^2}}\,\cos{\theta}\\
\frac{r^2}{\sqrt{36-r^2}}\,\sin{\theta}\\
r
\end{array}}\right)\,,
\end{align*} and $S_{+}$ the upper hemisphere of the sphere $x^2+y^2+z^2=36$.

  1. Find the volume of the solid $\Sigma$ which is enclosed by the surfaces $E,\, S_{+}$ and the plane $z=0$.

  2. Find the area of the surface $(S_{+})\cap \Sigma$.

  3. Let the function $f:{\mathbb{R}}^3\longrightarrow{\mathbb{R}}\,; \; f(x,y,z)=x\,z\,(36-z^2)^2$ and $c$ the curve which is the section of the surface $\partial\Sigma$ and the half-plane $\Pi: \big\{(x,0,z)\in{\mathbb{R}}^3\;|\; x\geqslant0 \big\}$. Find the line integral $\oint_{c}f\,ds$.

  4. Let the vector field $\overline{F}:{\mathbb{R}}^3\longrightarrow{\mathbb{R}}^3\,;\quad\overline{F}(x,y,z)=\left({y-x\,,\,y^3z\,,\,z^2}\right)\,.$ Find the line integral $\oint_{c}\overline{F}\cdot d\overline{r}$.

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Grigorios Kostakos


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