MATH 251 Lecture 34

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Last Episode: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iint_\Omega f \,\mathrm{d}S}

This Week: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iint_\Omega (\vec{F}\cdot\vec{n}) \,\mathrm{d}S} called "flux"

Surface Area

${\mbox{S. A.}}=\iint _{\Omega }1\,\mathrm {d} S$

Example

Surface area of torus of mean radius $R$ and cross-sectional radius of $a$: position given by $\theta$ (polar angle) and $\phi$ (angle formed between vector from center of cross-sectional circle and point on surface and the radial vector)

Parameterization

$X(\phi ,\theta )={\begin{cases}x=(R+a\cos {\phi })\cos {\theta }\\y=(R+a\cos {\phi })\sin {\theta }\\z=a\sin {\phi }\end{cases}}$

Where $0\leq \phi \leq 2\pi$ and $0\leq \phi \leq 2\pi$.

Area Element

${\begin{aligned}{\vec {X}}_{\phi }&=\left\langle -a\,\sin {\phi }\,\cos {\theta },\,-a\,\sin {\phi }\,\sin {\theta },\,a\,\cos {\phi }\right\rangle \\{\vec {X}}_{\theta }&=\left\langle -(R+a\cos {\phi })\sin {\theta },(R+a\cos {\phi })\,\cos {\theta },0\right\rangle \\{\vec {X}}_{\phi }\times {\vec {X}}_{\theta }&=a(R+a\cos {\phi })\left\langle -\cos {\theta }\cos {\phi },-\sin {\theta }\cos {\phi },-\sin {\phi }\cos ^{2}{\theta }-\sin {\phi }sin^{2}{\theta }\right\rangle \\\left\|{\vec {X}}_{\phi }\times {\vec {X}}_{\theta }\right\|&=a(R+a\cos {\phi })\end{aligned}}$

Evaluate

$\int _{0}^{2\pi }\int _{0}^{2\pi }a(R+a\cos {\phi })\,\mathrm {d} \phi \,\mathrm {d} \theta =(2\pi a)(2\pi R)$

Another Example

$z=f(x,y)$, so $x(u,v)=u$, $y(u,v)=v$, and $z(u,v)=f(u,v)$.

${\begin{aligned}{\vec {X}}(u,v)&=\left\langle u,v,f(u,v)\right\rangle \\{\vec {X}}_{u}&=\left\langle 1,0,f_{u}(u,v)\right\rangle \\{\vec {X}}_{v}&=\left\langle 0,1,f_{v}(u,v)\right\rangle \\{\vec {X}}_{u}\times {\vec {X}}_{v}&=\left\langle -f_{u},-f_{v},1\right\rangle \\\left\|{\vec {X}}_{u}\times {\vec {X}}_{v}\right|&={\sqrt {{f_{u}}^{2}+{f_{v}}^{2}+1}}\end{aligned}}$

Flux

Vector field ${\vec {F}}$ flowing across a surface $\Omega$.

Assume $\Omega$ is orientable (has a well-defined normal vector field ${\hat {n}}$ (unit length that points outward))

Therefore ${\vec {F}}\cdot {\hat {n}}$ is the component of ${\vec {F}}$ that is perpendicular to $\Omega$.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iint_\Omega \vec{F} \cdot \hat{n} \,\mathrm{d}S}

Example

Calculate the flux of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{F} = \left\langle x-y, x^2, 1 \right\rangle} through a paraboloid Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = x^2 + y^2} over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [-1,1] \times [-1,1]} in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xy} plane.

Parameterize

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{X}(u,v) = \left\langle u, v, u^2+v^2 \right\rangle}

Find Normal Unit

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \vec{X}_u &= \left\langle 1,0,2u \right\rangle \\ \vec{X}_v &= \left\langle 0,1,2v \right\rangle \end{align}}

We want to have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{n}} pointing outward, so we take Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{n} = \vec{X}_v \times \vec{X}_u} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{n} = \frac{\vec{X}_v \times \vec{X}_u}{\left\| \vec{X}_v \times \vec{X}_u \right\|}}

Find Area Element

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{d}S = \left\| \vec{X}_u \times \vec{X}_v \right\|} This is true always.

Set up integral

Notice how Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{d}S} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{n}} can simplify to just Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{n}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-1}^1 \int_{-1}^1 \vec{F} \cdot \vec{X}_v \times \vec{X}_u \,\mathrm{d}u \,\mathrm{d}v = \int_{-1}^1 \int_{-1}^1 (2u^2-2uv+2vu^2-1) \,\mathrm{d}u\,\mathrm{d}v}