Parametric Surfaces

$\newenvironment {prompt}{}{} \newcommand {\DeclareMathOperator }[2]{\@OldDeclareMathOperator {##1}{##2}\immediate \write \myfile {\unexpanded {\DeclareMathOperator }{\unexpanded {##1}}{\unexpanded {##2}}}} \newcommand {\reserved@a }[2]{} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument } \newcommand {\reserved@a }[1]{} \newcommand {\reserved@a }[2]{} \newcommand {\AppendGraphicsExtensions }[0]{\@ifundefined {Gin@extensions}{\let \Gin@extensions \@empty }{}\@ifstar {\grfext@Append \grfext@Check }{\grfext@Append \grfext@@Add }} \newcommand {\PrependGraphicsExtensions }[0]{\@ifundefined {Gin@extensions}{\let \Gin@extensions \@empty }{}\@ifstar {\grfext@Prepend \grfext@Check }{\grfext@Prepend \grfext@@Add }} \newcommand {\RemoveGraphicsExtensions }[1]{\@ifundefined {Gin@extensions}{\def \Gin@extensions {}}{\edef \grfext@tmp {\zap@space ##1 \@empty }\@for \grfext@ext :=\grfext@tmp \do {\def \grfext@next {\let \grfext@tmp \Gin@extensions \@expandtwoargs \@removeelement \grfext@ext \Gin@extensions \Gin@extensions \ifx \grfext@tmp \Gin@extensions \let \grfext@next \relax \fi \grfext@next }\grfext@next }}\grfext@Print \RemoveGraphicsExtensions } \newcommand {\epstopdfsetup }[0]{\setkeys {ETE}} \newcommand {\epstopdfcall }[1]{\ifETE@InsideSetfile \expandafter \@firstoftwo \else \expandafter \@secondoftwo \fi {‘##1}{\Gin@base \Gin@ext }} \newcommand {\epstopdfDeclareGraphicsRule }[4]{\ifx \\##4\\\@PackageError {epstopdf-base}{Conversion command is missing}\@ehc \else \begingroup \@ifundefined {Gin@rule@##1}{}{\@PackageInfo {epstopdf-base}{Redefining graphics rule for ‘##1’}}\endgroup \@namedef {Gin@rule@##1}####1{{##2}{##3}{\epstopdfcall {##4}}}\fi } \newcommand {\GetTitleStringSetup }[0]{\setkeys {gettitlestring}} \newcommand {\GetTitleString }[0]{\ifGTS@expand \expandafter \GetTitleStringExpand \else \expandafter \GetTitleStringNonExpand \fi } \newcommand {\GetTitleStringExpand }[1]{\def \GetTitleStringResult {##1}\begingroup \GTS@DisablePredefinedCmds \GTS@DisableHook \edef \x {\endgroup \noexpand \def \noexpand \GetTitleStringResult {\GetTitleStringResult }}\x } \newcommand {\GetTitleStringNonExpand }[1]{\def \GetTitleStringResult {##1}\global \let \GTS@GlobalString \GetTitleStringResult \begingroup \GTS@RemoveLeft \GTS@RemoveRight \endgroup \let \GetTitleStringResult \GTS@GlobalString } \newcommand {\GTS@DisableHook }[0]{} \newcommand {\label@hook }[0]{} \newcommand {\@currentlabelname }[0]{} \newcommand {\reserved@a }[0]{} \newcommand {\@safe@activestrue }[0]{} \newcommand {\@safe@activesfalse }[0]{} \newcommand {\nameref }[0]{\@ifstar \T@nameref \T@nameref } \newcommand {\Sectionformat }[2]{##1} \newcommand {\vnameref }[1]{\unskip ~\nameref {##1}\@vpageref [\unskip ]{##1}} \newcommand {\ref }[0]{\@ifstar \@refstar \T@ref } \newcommand {\pageref }[0]{\@ifstar \@pagerefstar \T@pageref } \newcommand {\nameref }[0]{\@ifstar \@namerefstar \T@nameref } \newcommand {\reserved@a }[2]{} \newcommand {\reserved@a }[0]{\AtBeginDocument } \newcommand {\reserved@a }[1]{} \newcommand {\reserved@a }[2]{}$

Objectives:

1.: Know what it means to parametrize a surface.
2.: Understand why surface parametrizations always have two variables.
3.: Be able to parametrize various surfaces.
4.: Understand how to find the tangent plane to a parametrized surface.
5.: Know how to calculate the surface area of a parametrized surface.

Recap Video

Here is a video highlights the main points of the section.

To summarize:

A parametrization of a surface $S$ consists of a description of the points of the surface as a function of two variables, say $\textbf {r}(u,v)$ , where $(u,v)$ live in some domain $D$ .
If a surface $S$ is parametrized as $\textbf {r}(u,v)$ , then the partials $\textbf {r}_u$ and $\textbf {r}_v$ are computed componentwise and give tangent vectors to $S$ , and their cross product gives the normal vector to the tangent plane.
The surface area of a surface $S$ given by $\textbf {r}(u,v)$ , where $(u,v)$ are in some domain $D$ , is given by $\iint _D \| \textbf {r}_u \times \textbf {r}_v \| du dv.$

Problems

Parametrizing Surfaces

The function $\textbf {r}(u,v) = \left \langle u+v,3u-v,2u+1 \right \rangle$ parametrizes a plane. The equation of the plane (in the form $ax+by+c=d$ ) is $\answer {x+y-2z}=-2$ .

Find a relation among $x$ , $y$ , and $z$ that eliminates $u$ and $v$ .

Notice that $x+y-2z$ is a constant.

Use the spherical coordinate method demonstrated in class to parametrize the portion of the sphere $x^2+y^2+z^2=9$ which lies in the first octant. (INPUT NOTE: In your answer, write p instead of phi. For example, write $\sin (p)$ instead of $\sin (\phi )$ .)

The parametrization is $\textbf {r}(\theta ,\phi ) = \left \langle \answer {3\cos (\theta )\sin (p)},\answer {3\sin (\theta )\sin (p)},\answer {3\cos (p)} \right \rangle$ and the bounds are $0\leq \theta \leq \answer {\pi /2},\quad 0 \leq \phi \leq \answer {\pi /2}.$

The spherical coordinate parametrization of a sphere of radius $R$ centered at the origin is $(R\cos (\theta )\sin (\phi ),R\sin (\theta )\sin (\phi ),R\cos (\phi )).$

The bounds are $0$ to $\pi /2$ for both because $\theta$ controls rotation around the $z$ -axis and $\phi$ controls movement from the north pole to the south pole.

The surface with parametrization $G(x,\theta ) = \left \langle x,\sin (x)\cos (\theta ),\sin (x)\sin (\theta ) \right \rangle$ for $0 \leq x \leq \pi$ and $0 \leq \theta \leq 2\pi$ can be gotten by revolving the graph of $y = \answer {\sin (x)}$ for $0 \leq x \leq \pi$ around the -axis.

The $x$ in the first coordinate says that the $x$ -axis is the axis of rotation, and the $\sin (x)$ in the second and third components tell us that that is the graph.

Let $S$ be the portion of the graph of $z = 16-x^2-y^2$ where $z \geq 0$ .

A parametrization for $S$ in Cartesian coordinates could be $\textbf {r}(x,y) = \left \langle x,y,\answer {16-x^2-y^2} \right \rangle$ for $x^2+y^2 \leq 16$ .
A parametrization for $S$ using polar coordinates in $x$ and $y$ would be $\textbf {r}(r,\theta ) = \left \langle r\cos (\theta ),r\sin (\theta ),\answer {16-r^2} \right \rangle$ for $0 \leq r \leq 4$ and $0 \leq \theta \leq 2\pi$ .

Which of the following parametrizes the portion of $z = \sqrt {x^2+y^2}$ with $0 \leq z \leq 4$ . Select all that apply.

Tangent Planes

Consider a surface given by $\textbf {r}(u,v) = \left \langle u,v,uv \right \rangle$ . Find the tangent plane to the surface at the point $(u,v) = (1,2)$ .

The partials are $\textbf {r}_u = \left \langle \answer {1},\answer {0},\answer {v} \right \rangle ,$ $\textbf {r}_v = \left \langle \answer {0},\answer {1},\answer {u} \right \rangle .$ At the point $(u,v) = (1,2)$ , we have $\textbf {r}_u(1,2) = \left \langle \answer {1},\answer {0},\answer {2} \right \rangle ,$ $\textbf {r}_v(1,2) = \left \langle \answer {0},\answer {1},\answer {1} \right \rangle .$ The cross product is $\textbf {r}_u(1,2) \times \textbf {r}_v(1,2) = \left \langle \answer {-2},\answer {-1},\answer {1} \right \rangle .$ The point $(x,y,z)$ corresponding to $u=1$ and $v=2$ is $(x,y,z) = (\answer {1},\answer {2},\answer {2}).$ Therefore, the equation of the plane in the form $ax+by+cz=d$ is $-2x+\answer {-1}y+\answer {1}z = \answer {-2}.$

Consider the surface $S$ given by $\textbf {r}(\theta ,z) = \left \langle 2\cos (\theta ),2\sin (\theta ),z \right \rangle .$ The equation of the tangent plane to $S$ at the point $\textbf {r}(\pi /4,5)$ in the form $ax+by+cz=d$ is $\answer {x+y} = 2\sqrt {2}.$

Surface Area

Consider the surface $S$ gotten by revolving the portion of $y = x^3$ where $0 \leq x \leq 2$ around the $x$ -axis. This surface can be parametrized as $\textbf {r}(x,\theta ) = \left \langle x,x^3\cos (\theta ),x^3\sin (\theta ) \right \rangle .$ Find the surface area of $S$ .

We compute the partials: $\textbf {r}_x(x,\theta ) = \left \langle \answer {1},\answer {3x^2\cos (\theta )},\answer {3x^2\sin (\theta )} \right \rangle$ $\textbf {r}_{\theta }(x,\theta ) = \left \langle \answer {0},\answer {-x^3\sin (\theta )},\answer {x^3\cos (\theta )} \right \rangle .$ The cross product is $\textbf {r}_x\times \textbf {r}_{\theta } = \left \langle \answer {3x^5},\answer {-x^3\cos (\theta )},\answer {-x^3\sin (\theta )} \right \rangle$ which has magnitude $\textbf {r}_x \times \textbf {r}_{\theta } = \sqrt {\answer {9x^{10}+x^6}}.$ The bounds for $x$ and $\theta$ are $\answer {0} \leq x \leq \answer {2},\quad 0 \leq \theta \leq \answer {2\pi }.$ The surface area is therefore $\int _0^{2\pi } \int _0^2 \answer {\sqrt {9x^{10}+x^6}}dx d\theta = \int _0^{2\pi } \int _0^2 x^3\sqrt {9x^4+1}dxd\theta .$ The integral evaluates to $\answer {\frac {\pi }{27}(145\sqrt {145}-1)}$ .

The surface area of the portion of $y = 9-z^2$ where $0 \leq x \leq z \leq 3$ is equal to $\answer {\frac {1}{12}(37\sqrt {37}-1)}$ .

Parametrize the surface as $\textbf {r}(x,z) =(x,9-z^2,z)$ . The partials are $\textbf {r}_x = \left \langle 1,0,0 \right \rangle ,\quad \textbf {r}_z = \left \langle 0,-2z,1 \right \rangle .$ The cross product is $\left \langle 0,-1,-2z \right \rangle$ which has magnitude $\sqrt {1+4z^2}$ .

Press...	...to do
left/right arrows	Move cursor
shift+left/right arrows	Select region
ctrl+a	Select all
ctrl+x/c/v	Cut/copy/paste
ctrl+z/y	Undo/redo
ctrl+left/right	Add entry to list or column to matrix
shift+ctrl+left/right	Add copy of current entry/column to to list/matrix
ctrl+up/down	Add row to matrix
shift+ctrl+up/down	Add copy of current row to matrix
ctrl+backspace	Delete current entry in list or column in matrix
ctrl+shift+backspace	Delete current row in matrix

Type...	...to get
norm	$\|\|\blue{[?]}\|\|$
text	$\text{\blue{[?]}}$
sym_name	$\backslash\texttt{\blue{[?]}}$
abs	$\left\|\blue{[?]}\right\|$
sqrt	$\sqrt{\blue{[?]}}$
paren	$\left(\blue{[?]}\right)$
floor	$\lfloor \blue{[?]} \rfloor$
factorial	$\blue{[?]}!$
exp	${\blue{[?]}}^{\blue{[?]}}$
sub	${\blue{[?]}}_{\blue{[?]}}$
frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
cot	$\cot\left(\blue{[?]}\right)$
log	$\log\left(\blue{[?]}\right)$
ln	$\ln\left(\blue{[?]}\right)$
alpha	$\alpha$
beta	$\beta$
gamma	$\gamma$
delta	$\delta$
epsilon	$\epsilon$
zeta	$\zeta$
eta	$\eta$
theta	$\theta$
iota	$\iota$
kappa	$\kappa$
lambda	$\lambda$
mu	$\mu$
nu	$\nu$
xi	$\xi$
omicron	$\omicron$
pi	$\pi$
rho	$\rho$
sigma	$\sigma$
tau	$\tau$
upsilon	$\upsilon$
phi	$\phi$
chi	$\chi$
psi	$\psi$
omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

Recap Video

Problems

Parametrizing Surfaces

Tangent Planes

Surface Area

Controls

Symbols

Settings