### Parametric Surfaces

(requires JavaScript)

1. Identify the surface with the vector equation

$\mathbf{r}\left(u,v\right)=2\mathrm{sin}\left(u\right)\mathbf{i}+3\mathrm{cos}\left(u\right)\mathbf{j}+v\mathbf{k}$

where $0\le v\le 2$.

2. Match the equations with the graphs, explain.

1. $\mathbf{r}\left(u,v\right)=⟨\mathrm{cos}\left(v\right),\mathrm{sin}\left(v\right),u⟩$
2. $\mathbf{r}\left(u,v\right)=⟨u\mathrm{cos}\left(v\right),u\mathrm{sin}\left(v\right),u⟩$
3. $\mathbf{r}\left(u,v\right)=⟨u\mathrm{cos}\left(v\right),u\mathrm{sin}\left(v\right),v⟩$
4. $x={u}^{3},\phantom{\rule{0.5em}{0ex}}y=u\mathrm{sin}\left(v\right),\phantom{\rule{0.5em}{0ex}}z=u\mathrm{cos}\left(v\right)$
5. $x=\left(u-\mathrm{sin}\left(u\right)\right)\mathrm{cos}\left(v\right)$,
$y=\left(1-\mathrm{cos}\left(u\right)\right)\mathrm{sin}\left(v\right)$,
$z=u$.
6. $x=\left(1-u\right)\left(3+\mathrm{cos}\left(v\right)\right)\mathrm{cos}\left(4\pi u\right)$,
$y=\left(1-u\right)\left(3+\mathrm{cos}\left(v\right)\right)\mathrm{sin}\left(4\pi u\right)$,
$z=3u+\left(1-u\right)\mathrm{sin}\left(v\right)$.
3. Find parametric equations for the surface obtained by rotating the curve $y={e}^{-x}$, $0\le x\le 3$, about the $x\text{-axis}$ and use them to graph the surface.
$x=x,\phantom{\rule{0.5em}{0ex}}y={e}^{-x}\mathrm{cos}\left(\theta \right),\phantom{\rule{0.5em}{0ex}}z={e}^{-x}\mathrm{sin}\left(\theta \right),\phantom{\rule{0.5em}{0ex}}0\le x\le 3$
4. Match the shown surfaces with their parametrizations. Justify your choices.

${\mathbf{r}}_{1}\left(u,v\right)=⟨\mathrm{sin}\left(u\right)\mathrm{cos}\left(v\right),\mathrm{sin}\left(u\right)\mathrm{sin}\left(v\right),\mathrm{cos}\left(u\right)⟩$

${\mathbf{r}}_{2}\left(u,v\right)=⟨u,v,{u}^{2}-{v}^{2}⟩$

${\mathbf{r}}_{3}\left(u,v\right)=⟨u\mathrm{cos}\left(v\right),u,u\mathrm{sin}\left(v\right)⟩$

${\mathbf{r}}_{4}\left(u,v\right)=⟨u,{e}^{u}\mathrm{cos}\left(v\right),{e}^{u}\mathrm{sin}\left(v\right)⟩$

${\mathbf{r}}_{5}\left(u,v\right)=⟨u,\sqrt{1+{u}^{2}}\mathrm{cos}\left(v\right),\sqrt{1+{u}^{2}}\mathrm{sin}\left(v\right)⟩$

${\mathbf{r}}_{6}\left(u,v\right)=⟨u,v,{u}^{3}+{v}^{3}⟩$

Surface 1.

Surface 2.

${\mathbf{r}}_{4}$ and ${\mathbf{r}}_{2}$.