Chapter 9: Vector Calculus
Section 9.8: Divergence Theorem
Example 9.8.3
Apply the Divergence theorem to the vector field F = x i + y j + z k and R , the rectangular box defined by x ∈ 1 , 2 , y ∈ 3 , 4 , z ∈ 5 , 6 .
Solution
Mathematical Solution
•
Figure 9.8.3(a) shows the box with outward normals on each face, and arrows of the field F .
∇ · F = ∂ x x + ∂ y y + ∂ z z = 1 + 1 + 1 = 3
•
The integral of ∇ · F over the interior of the box:
∫ 1 2 ∫ 3 4 ∫ 5 6 3 dz dy dx = 3
>
use Student:-VectorCalculus in module() local F,p; F:=VectorField(<x,y,z>); p:=Flux(F,Box(1..2,3..4,5..6),output=plot,caption="",axes=frame,labels=[x,y,z],tickmarks=[2,2,2],orientation=[-35,75,0],fieldoptions=[grid=[5,5,5]]); print(p); end module: end use:
Figure 9.8.3(a) Box and field F
The flux must be computed through each of the six faces of the box. To begin, note that
Face
dσ
N
F · N
x = 1
dy dz
− i
− 1
x = 2
dy dz
i
2
y = 3
dx dz
− j
− 3
y = 4
dx dz
j
4
z = 5
dx dy
− k
− 5
z = 6
dx dy
k
6
so that, by pairing opposing faces, the following three integrals represent the flux through the faces of the box.
∫ 3 4 ∫ 5 6 2 − 1 dz dy + ∫ 1 2 ∫ 5 6 4 − 3 dz dx + ∫ 1 2 ∫ 3 4 6 − 5 dy dx = 3
Maple Solution - Interactive
The Student VectorCalculus package is needed for calculating the divergence, but it then conflicts with any multidimensional integral set from the Calculus palette. Hence, the Student MultivariateCalculus package is installed to gain Context Panel access to the MultiInt command.
Initialize
•
Tools≻Load Package: Student Vector Calculus
Loading Student:-VectorCalculus
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Tools≻Tasks≻Browse: Calculus - Vector≻ Vector Algebra and Settings≻ Display Format for Vectors
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Press the Access Settings button and select "Display as Column Vector"
Display Format for Vectors
Define the vector field F
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Enter the components of F in a free vector. Context Panel: Evaluate and Display Inline
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Context Panel: Student Vector Calculus≻Conversions≻To Vector Field
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Context Panel: Assign to a Name≻ F
x , y , z =
→ to Vector Field → assign to a name F
Obtain ∇ · F , the divergence of F
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Common Symbols palette: Del and dot-product operators
•
Context Panel: Evaluate and Display Inline
∇ · F = 3
Integrate ∇ · F over the interior of the box
•
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
•
Context Panel: Student Multivariate Calculus≻Integrate≻Iterated Complete the dialogs as per the figures below.
•
Context Panel: Evaluate Integral
3 → MultiInt ∫ 1 2 ∫ 3 4 ∫ 5 6 3 ⅆ z ⅆ y ⅆ x = 3
Use a task template to compute the flux of F through the boundaries of R
Tools≻Tasks≻Browse: Calculus - Vector≻Integration≻Flux≻3-D≻Through a Box
Flux through a Box
For the Vector Field:
Select Coordinate System Cartesian [x,y,z] Cartesian - other cylindrical spherical bipolarcylindrical bispherical cardioidal cardioidcylindrical casscylindrical conical ellcylindrical hypercylindrical invcasscylindrical logcylindrical logcoshcylindrical oblatespheroidal paraboloidal paracylindrical prolatespheroidal rosecylindrical sixsphere tangentcylindrical tangentsphere toroidal
In the Divergence theorem, the volume integral on the left and the flux on the right both have the value 3.
Maple Solution - Coded
Initialize
•
Install the Student VectorCalculus package.
with Student :- VectorCalculus :
BasisFormat false :
F ≔ VectorField x , y , z :
Obtain ∇ · F , the divergence of F
Divergence F = 3
Use the int command to integrate the divergence of F over R
int 3 , x , y , z = Parallelepiped 1 .. 2 , 3 .. 4 , 5 .. 6 , output = integral
∫ 5 6 ∫ 3 4 ∫ 1 2 3 ⅆ x ⅆ y ⅆ z
int 3 , x , y , z = Parallelepiped 1 .. 2 , 3 .. 4 , 5 .. 6 = 3
Use the Flux command to obtain the flux of F through the boundaries of R
Flux F , Box 1 .. 2 , 3 .. 4 , 5 .. 6 , output = integral
∫ 5 6 ∫ 3 4 1 ⅆ s ⅆ t + ∫ 5 6 ∫ 1 2 1 ⅆ s ⅆ t + ∫ 3 4 ∫ 1 2 1 ⅆ s ⅆ t
Flux F , Box 1 .. 2 , 3 .. 4 , 5 .. 6 = 3
In the Divergence theorem, the volume integral on the left and the flux on the right both have the value 3.
Figure 9.8.3(a) is drawn with he Flux command by including the option "output = plot". The actual code for the figure is hidden behind the table cell in which the figure appears.
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