Tools≻Tasks≻Browse:
Calculus - Vector≻Integration≻Flux≻3-D≻Through a Surface Defined over an Ellipse

Table 9.6.15(a)   Solution via task template

Initialize

Loading Student:-MultivariateCalculus

Define the vector field F

$⟨x\,y\,z⟩$$\stackrel{\text{assign to a name}}{\to }$$F$

Define the surface

$x^2\+y^2$$\stackrel{\text{assign to a name}}{\to }$$f$

Obtain N, a unit normal on the surface

$⟨-\frac{\partial }{\partial x} f\,-\frac{\partial }{\partial y} f\,1⟩$ = $\stackrel{\text{normalize}}{\to }$ $\stackrel{\text{assign to a name}}{\to }$$N$

Obtain $\mathrm{dsig}\=\sqrt{1\+f_x^2\+f_y^2}$ 

$\sqrt{1\+{\left(\frac{\partial }{\partial x} f\right)}^2\+{\left(\frac{\partial }{\partial y} f\right)}^2}$ = $\sqrt{4x^2\+4y^2\+1}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{dsig}$$$

Form $\mathbf{F}·\mathbf{N} \mathrm{dsig}$ and evaluate it on the surface

$\genfrac{}{}{0ex}{}{\mathbf{F}·\mathbf{N} \mathrm{dsig}}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{z\=f}$

$\left(-\frac{2x^2}{\sqrt{4x^2\+4y^2\+1}}-\frac{2y^2}{\sqrt{4x^2\+4y^2\+1}}\+\frac{x^2\+y^2}{\sqrt{4x^2\+4y^2\+1}}\right)\sqrt{4x^2\+4y^2\+1}$

$\stackrel{\text{simplify}}{\=}$

$-x^2-y^2$

Use polar coordinates to form and evaluate the flux integral

${\int }_0^{2 \mathrm{\π}}{\int }_0^{1\/\sqrt{4-3 {\cos }^2\left(\mathrm{\theta }\right)}}r \left(-r^2\right) \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $-\frac{5}{32}\mathrm{\π}$$$$$$$

Table 9.6.15(b)   Solution from first principles

Initialize

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{VectorCalculus}\right)\:$

$\mathbf{F}≔\mathrm{VectorField}\left(⟨x\,y\,z⟩\right)\:$

Form and evaluate the flux integral with the Flux command

$\mathrm{Flux}\left(\mathbf{F}\,\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Ellipse}\left(x^2\+4 y^2\=1\,\left[r\,\mathrm{\theta }\right]\right)\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^{\frac{1}{2}\sqrt{\frac{{\tan \left(\mathrm{\θ}\right)}^2\+1}{\frac{1}{4}\+{\tan \left(\mathrm{\θ}\right)}^2}}}\left(-r^2{\cos \left(\mathrm{\θ}\right)}^2-r^2{\sin \left(\mathrm{\θ}\right)}^2\right)r\ⅆr\ⅆ\mathrm{\θ}$

$\mathrm{Flux}\left(\mathbf{F}\,\mathrm{Surface}\left(⟨x\,y\,x^2\+y^2⟩\,\left[x\,y\right]\=\mathrm{Ellipse}\left(x^2\+4 y^2\=1\,\left[r\,\mathrm{\theta }\right]\right)\right)\right)$

$-\frac{5}{32}\mathrm{\π}$

Table 9.6.15(c)   Solution via the Flux command

Initialize

$\mathrm{BasisFormat}\left(\mathrm{false}\right)\:$

$f≔x^2\+y^2\:$

Obtain N, a unit normal on the surface

$\mathbf{N}≔\mathrm{Normalize}\left(⟨-\mathrm{diff}\left(f\,x\right)\,-\mathrm{diff}\left(f\,y\right)\,1⟩\right)$

Obtain $\mathrm{dsig}\=\sqrt{1\+f_x^2\+f_y^2}$ 

$\mathrm{dsig}≔\sqrt{1\+\mathrm{diff}{\left(f\,x\right)}^2\+\mathrm{diff}{\left(f\,y\right)}^2}$

$\sqrt{4x^2\+4y^2\+1}$

Obtain $\mathbf{F}·\mathbf{N} \mathrm{dsig}$ and evaluate it on the surface

$\mathrm{simplify}\left(\mathrm{eval}\left(\mathrm{DotProduct}\left(\mathbf{F}\,\mathbf{N}\right) \mathrm{dsig}\,z\=f\right)\right)$ = $-x^2-y^2$$$$$

Use the Int command and polar coordinates to form the flux integral and int to evaluate it.

$$\begin{gathered} \mathrm{Int}\left(r\left(- r^2\right)\,\left[ r\=0..1\/\sqrt{4-3 {\cos }^2\left(\mathrm{\theta }\right)}\,\mathrm{\θ}\=0..2 \mathrm{\π}\right]\right)\= \\ :-\mathrm{int}\left(r \left(-r^2\right)\,\left[r\=0..1\/\sqrt{4-3 {\cos }^2\left(\mathrm{\theta }\right)}\,\mathrm{\θ}\=0..2 \mathrm{\π}\right]\right) \end{gathered}$$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^{\frac{1}{\sqrt{4-3{\cos \left(\mathrm{\θ}\right)}^2}}}r\left(-r^2\right)\ⅆr\ⅆ\mathrm{\θ}\=-\frac{5}{32}\mathrm{\π}$

Table 9.6.15(d)   Solution from first principles