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Calculus - Vector≻Integration≻Surface Integration≻Over a parametrically Defined Surface

Table 9.6.3(a)   Solution by task template

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Loading Student:-MultivariateCalculus

Define the surface as the position vector R

$\mathbf{R}\=⟨u\+v\,u-v\,u v⟩$$\stackrel{\text{assign}}{\to }$

Obtain vectors ${\mathbf{R}}_u$ and ${\mathbf{R}}_v$ tangent to $u$- and $v$-coordinate curves, respectively

Obtain $\mathrm{d\σ}$ (sans differentials): $\mathrm{dsig}\=∥{\mathbf{R}}_u\times {\mathbf{R}}_v∥$ 

$\mathbf{Ru}\times \mathbf{Rv}$ = $\stackrel{\text{Euclidean-norm}}{\to }$$\sqrt{2u^2\+2v^2\+4}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{dsig}$$$

Write and evaluate the relevant surface integral

${\int }_0^1{\int }_0^u\left(u\+v\right) \left(u-v\right) u v \mathrm{dsig} \mathit{\ⅆ}v \mathit{\ⅆ}u$ = $-\frac{16}{105}-\frac{3}{5}\sqrt{3}\sqrt{2}\+\frac{128}{105}\sqrt{2}$$$

Table 9.6.3(b)   Solution from first principles

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

$$\begin{gathered} \mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨u\+v\,u-v\,u v⟩\,u\=0..1\,v\=0..u\right)\,\mathrm{output}\=\mathrm{integral}\right)\= \\ \mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨u\+v\,u-v\,u v⟩\,u\=0..1\,v\=0..u\right)\right) \end{gathered}$$

${\∫}_0^1{\∫}_0^u\left(u\+v\right)\left(u-v\right)uv\sqrt{4\+{\left(u\+v\right)}^2\+{\left(-u\+v\right)}^2}\ⅆv\ⅆu\=-\frac{16}{105}-\frac{3}{5}\sqrt{3}\sqrt{2}\+\frac{128}{105}\sqrt{2}$

Table 9.6.3(c)   Solution via the Surface option in the SurfaceInt command

Initialize

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

$f≔\left(x\,y\,z\right)\to x y z\:$

$\mathbf{R}≔⟨u\+v\,u-v\,u v⟩\:$

$\mathbf{Ru}\,\mathbf{Rv}≔\mathrm{diff}\left(\mathbf{R}\,u\right)\,\mathrm{diff}\left(\mathbf{R}\,v\right)\:$

$\mathrm{dsig}≔\mathrm{Norm}\left(\mathrm{CrossProduct}\left(\mathrm{Ru}\,\mathrm{Rv}\right)\right)$

$\sqrt{2u^2\+2v^2\+4}$

Use the top-level Int and int commands to write and evaluate the surface integral

$$\begin{gathered} \mathrm{Int}\left(f\left(u\+v\,u-v\,u v\right) \mathrm{dsig}\,\left[ v\=0..u\,u\=0..1\right]\right)\= \\ :-\mathrm{int}\left(f\left(u\+v\,u-v\,u v\right) \mathrm{dsig}\,\left[ v\=0..u\,u\=0..1\right]\right) \end{gathered}$$

${\∫}_0^1{\∫}_0^u\left(u\+v\right)\left(u-v\right)uv\sqrt{2u^2\+2v^2\+4}\ⅆv\ⅆu\=-\frac{16}{105}\+\frac{128}{105}\sqrt{2}-\frac{3}{5}\sqrt{2}\sqrt{3}$

Table 9.6.3(d)   Solution from first principles