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Chapter 9: Vector Calculus
Section 9.6: Surface Integrals
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Example 9.6.11
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Working in spherical coordinates, obtain the flux of the field through the surface of the unit sphere centered at the origin. Use a parametric representation of the surface.
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Solution
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Mathematical Solution
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Figure 9.6.11(a) shows the sphere, a representative normal on its surface, and field arrows.
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The arrows suggest the field is radially outward, so in spherical coordinates it would be , and on the sphere it is just . The outward normal on the sphere is also just , so . Thus, the integral of over the surface of the sphere is
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use Student:-VectorCalculus in
module()
local F,p1;
F:=VectorField(<x,y,z>);
p1:=Flux(F,Sphere(<0,0,0>,1),output=plot,caption="",tickmarks=[3,3,3],axes=frame,orientation=[-55,65,0],fieldoptions=[grid=[5,5,5]]);
print(p1);
end module:
end use:
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Figure 9.6.11(a) Sphere, field, normal
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Maple Solution - Interactive
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Table 9.6.11(a) provides a solution via task template. The vector field F is given in Cartesian coordinates, but could also be expressed in spherical coordinates. Table 9.6.11(b) shows how the same task template can be used for the alternate calculation.
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Tools≻Tasks≻Browse:
Calculus - Vector≻Integration≻Flux≻3-D≻Through a Sphere
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Flux through a Sphere
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For the Vector Field:
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Table 9.6.11(a) Solution by task template
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Table 9.6.11(b) uses the same task template, but expresses the vector field F in spherical coordinates.
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Tools≻Tasks≻Browse:
Calculus - Vector≻Integration≻Flux≻3-D≻Through a Sphere
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Flux through a Sphere
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For the Vector Field:
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Table 9.6.11(b) Task-template solution for field in spherical coordinates
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Maple Solution - Coded
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Table 9.6.11(c) provides a solution in which the vector field is given in Cartesian coordinates, but in which the Flux command in the Student VectorCalculus package writes the integral in spherical coordinates.
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Initialize
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Install the Student VectorCalculus package.
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Use the Flux command with the Sphere option to compute the flux
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Table 9.6.11(c) Solution via the Flux command
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Figure 9.6.11(a) can be obtained by changing the option "output = integral" to "output = plot" in Table 9.6.11(c). The version of this command used for this figure is hidden behind the cell containing the graph.
Table 9.6.11(d) provides a solution from first principles. The surface is represented by a position vector that parametrizes the surface via spherical coordinates. However, the position vector is essentially a Cartesian vector, so the vector field F is also given in Cartesian coordinates. In the expression , the Cartesian variables are simply replaced with their equivalents on the surface of the sphere.
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Initialize
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Set the display of vectors with the BasisFormat command.
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Define the sphere as the position vector R.
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Obtain
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In the Student VectorCalculus package, diff maps onto the components of vectors.
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Obtain a unit outward normal on the surface of the sphere
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In the Student VectorCalculus package, diff maps onto the components of vectors.
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Evaluate on the surface of the sphere
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Use the eval command to make the substitutions for as returned by the Equate command.
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Form and evaluate the flux integral
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Use the Int and int commands, accessing the top-level versions via the prefix operator
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Table 9.6.11(d) Solution from first principles with F given in Cartesian coordinates
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