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Chapter 9: Vector Calculus
Section 9.6: Surface Integrals
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Example 9.6.10
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Obtain the flux of the field through the surface of the rectangular parallelepiped (box) whose faces lie in the planes .
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Solution
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Mathematical Solution
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Figure 9.6.10(a) shows the box, representative normal vectors on each face of the box, and field arrows representing the vector field.
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Table 9.6.10(a) summarizes over the six faces of the box.
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Face
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N
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-5
6
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Table 9.6.10(a)
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>
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use Student:-VectorCalculus in
module()
local F,p1;
F:=VectorField(<x,y,z>);
p1:=Flux(F,Box(1..2,3..4,5..6),output=plot,caption="",fieldoptions=[grid=[5,5,5]],tickmarks=[3,3,3],axes=frame,orientation=[-45,70,0]);
print(p1);
end module:
end use:
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Figure 9.6.10(a) Surface, normals, field
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Because opposite faces can be paired, there are only three double integrals to evaluate, namely
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Maple Solution - Interactive
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Table 9.6.10(b) provides a solution by task template.
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Tools≻Tasks≻Browse:
Calculus - Vector≻Integration≻Flux≻3-D≻Through a Box
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Flux through a Box
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For the Vector Field:
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Table 9.6.10(b) Solution via task template
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Table 9.6.10(c) provides a solution from first principles. The value(s) for on the faces of the box are taken from Table 9.6.10(a). On each face of the box, , where is the appropriate product of two of the differentials .
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Form and evaluate the flux integral
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Calculus palette:
Iterated double-integral template
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Context Panel: Evaluate and Display Inline
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=
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Table 9.6.10(c) Solution from first principles
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Maple Solution - Coded
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Table 9.6.10(d) provides a solution via the Flux command in the Student VectorCalculus package.
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Initialize
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Install the Student VectorCalculus package.
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Define F via the VectorField command.
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Invoke the Flux command with the Box option
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Table 9.6.10(d) Solution via the Flux command
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If the option "output = plot" is implemented in the Flux command, a version of Figure 9.6.10(a) will result. The additional graphing options in that figure can be seen in the version of the command used behind the table cell holding the figure.
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