Chapter 9: Vector Calculus
Section 9.6: Surface Integrals
Example 9.6.10
Obtain the flux of the field F=x i+y j+z k through the surface of the rectangular parallelepiped (box) whose faces lie in the planes x=1,x=2,y=3,y=4,z=5,z=6.
Solution
Mathematical Solution
Figure 9.6.10(a) shows the box, representative normal vectors on each face of the box, and field arrows representing the vector field.
Table 9.6.10(a) summarizes F·N over the six faces of the box.
Face
N
F·N
x=1
x=2
−i
i
−1
2
y=3
y=4
−j
j
−3
4
z=5
z=6
−k
k
-5
6
Table 9.6.10(a) F·N
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:
Figure 9.6.10(a) Surface, normals, field
Because opposite faces can be paired, there are only three double integrals to evaluate, namely
∫56∫341 dy dz+∫12∫561 dz dx+∫12∫341 dy dx=1+1+1=3
Maple Solution - Interactive
Table 9.6.10(b) provides a solution by task template.
Tools≻Tasks≻Browse: Calculus - Vector≻Integration≻Flux≻3-D≻Through a Box
Flux through a Box
For the Vector Field:
Select Coordinate SystemCartesian [x,y,z]Cartesian - othercylindricalsphericalbipolarcylindricalbisphericalcardioidalcardioidcylindricalcasscylindricalconicalellcylindricalhypercylindricalinvcasscylindricallogcylindricallogcoshcylindricaloblatespheroidalparaboloidalparacylindricalprolatespheroidalrosecylindricalsixspheretangentcylindricaltangentspheretoroidal
Table 9.6.10(b) Solution via task template
Table 9.6.10(c) provides a solution from first principles. The value(s) for F·N on the faces of the box are taken from Table 9.6.10(a). On each face of the box, dσ=1 dA, where dA is the appropriate product of two of the differentials dx, dy, dz.
Form and evaluate the flux integral
Calculus palette: Iterated double-integral template
Context Panel: Evaluate and Display Inline
∫56∫341 ⅆy ⅆz+∫12∫561 ⅆz ⅆx+∫12∫341 ⅆy ⅆx = 3
Table 9.6.10(c) Solution from first principles
Maple Solution - Coded
Table 9.6.10(d) provides a solution via the Flux command in the Student VectorCalculus package.
Initialize
Install the Student VectorCalculus package.
withStudent:-VectorCalculus:
Define F via the VectorField command.
F≔VectorFieldx,y,z:
Invoke the Flux command with the Box option
FluxF,Box1..2,3..4,5..6,output=integral
∫56∫341ⅆsⅆt+∫56∫121ⅆsⅆt+∫34∫121ⅆsⅆt
FluxF,Box1..2,3..4,5..6 = 3
Table 9.6.10(d) Solution via the Flux command
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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