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Calculus - Vector≻Integration≻Surface Integration≻Over a Sphere

Table 9.6.2(a)   Solution via task template

$x\=A\+a \sin \left(\mathrm{\φ}\right)\cos \left(\mathrm{\θ}\right)$ 

$y\=B\+a \sin \left(\mathrm{\φ}\right)\sin \left(\mathrm{\θ}\right)$ 

$z\=C\+a \cos \left(\mathrm{\φ}\right)$ 

Tools≻Tasks≻Browse
Calculus - Vector≻Integration≻Surface Integration≻Over a Parametrically Defined Surface

Table 9.6.2(b)   Alternate solution by task template

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

$$\begin{gathered} \mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Sphere}\left(⟨A\,B\,C⟩\,a\right)\,\mathrm{output}\=\mathrm{integral}\right)\= \\ \mathrm{SurfaceInt}\left(x y z\,\left[x\,y\,z\right]\=\mathrm{Sphere}\left(⟨A\,B\,C⟩\,a\right)\right) \end{gathered}$$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}{a}^2\sin \left(\mathrm{\φ}\right)\left(-\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\θ}\right){\cos \left(\mathrm{\φ}\right)}^3{a}^3-C\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\θ}\right){\cos \left(\mathrm{\φ}\right)}^2{a}^2\+A\sin \left(\mathrm{\φ}\right)\sin \left(\mathrm{\θ}\right)\cos \left(\mathrm{\φ}\right)a^2\+B\sin \left(\mathrm{\φ}\right)\cos \left(\mathrm{\θ}\right)\cos \left(\mathrm{\φ}\right)a^2\+\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\θ}\right)\cos \left(\mathrm{\φ}\right)a^3\+AC\sin \left(\mathrm{\φ}\right)\sin \left(\mathrm{\θ}\right)a\+BC\sin \left(\mathrm{\φ}\right)\cos \left(\mathrm{\θ}\right)a\+C\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\θ}\right)a^2\+AB\cos \left(\mathrm{\φ}\right)a\+ABC\right)\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}\=4ABC\mathrm{\π}{a}^2$

Table 9.6.2(c)   Solution via the Sphere option in the SurfaceInt command

Initialize

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

Write parametric equations in which $u\to \mathrm{\φ}$ and $v\to \mathrm{\θ}$

$$\begin{gathered} X≔A\+a \sin \left(u\right)\cos \left(v\right)\: \\ Y≔B\+a \sin \left(u\right)\sin \left(v\right)\: \\ Z≔C\+a \cos \left(u\right)\: \end{gathered}$$

Express the surface of the sphere via the position vector R

$\mathbf{R}≔⟨X\,Y\,Z⟩\:$

Obtain the vectors ${\mathbf{R}}_u\={\partial }_u\mathbf{R}$ and ${\mathbf{R}}_v\={\partial }_v\mathbf{R}$ tangent to coordinate curves

Apply the diff command, which automatically maps onto the components of vectors.

Set the names ${\mathbf{R}}_u$ and ${\mathbf{R}}_v$ as Atomic Variables, something best done in typeset math.

$\mathrm{R\_\_u}≔\mathrm{diff}\left(\mathbf{R}\,u\right)$

$\mathrm{R\_\_v}≔\mathrm{diff}\left(\mathbf{R}\,v\right)$

Except for the differentials, $\mathrm{d\σ}\=∥{\mathbf{R}}_u\times {\mathbf{R}}_v∥$

$\mathrm{dsig}≔\mathrm{simplify}\left(\mathrm{Norm}\left(\mathrm{CrossProduct}\left(\mathrm{R\_\_u}\,\mathrm{R\_\_v}\right)\right)\right) assuming a\>0\,u\colon\colon \mathrm{RealRange}\left(0\,\mathrm{\π}\right)$

$\sin \left(u\right)a^2$

Form and evaluate the appropriate surface integral

$\mathrm{Int}\left(X Y Z \mathrm{dsig}\,\left[u\=0..\mathrm{\π}\,v\=0..2 \mathrm{\π}\right]\right)\= :-\mathrm{int}\left(X Y Z \mathrm{dsig}\,\left[u\=0..\mathrm{\pi }\,v\=0..2 \mathrm{\pi }\right]\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}\left(A\+a\sin \left(u\right)\cos \left(v\right)\right)\left(B\+a\sin \left(u\right)\sin \left(v\right)\right)\left(C\+a\cos \left(u\right)\right)\sin \left(u\right)a^2\ⅆu\ⅆv\=4ABC\mathrm{\π}{a}^2$