Example 9-8-5 - Maple Help

All Products Maple MapleSim

Home : Support : Online Help : Education : Study Guides : MultivariateCalculus : Chapter 9 : Examples : Section 9-8 : Example 9-8-5

Chapter 9: Vector Calculus

Section 9.8: Divergence Theorem

Example 9.8.5

Apply the Divergence theorem to the vector field $F = x y^{3} i + y z j + x^{2} z k$ and $R$ , the region bounded by two spheres centered at the origin, one with radius 1, the other with radius $1 / 2$ .

Solution

Mathematical Solution

Figure 9.8.5(a) contains a sketch of the region $R$ and representative outward normals on the two surfaces bounding $R$ .

use plots, VectorCalculus in
module()
local p1,p2,p3,p4,p5,p6,p7,N1,N2;
N1:=RootedVector(root=[.4,-.4,sqrt(.68)],<.4,-.4,sqrt(.68)>);
N2:=RootedVector(root=[.2,-.2,sqrt(.17)],<-.2,.2,-sqrt(.17)>);
p1:=sphereplot(1,theta=Pi/2..2*Pi, phi=0..Pi,color=red):
p2:=sphereplot(1/2,theta=Pi/2..2*Pi, phi=0..Pi,color=cyan):
p3:=PlotVector([N1,N2],color=black,width=.1);
p4:=plot3d([r,Pi/2,phi],r=1/2..1,phi=0..Pi,coords=spherical,color=black, style=hidden, grid=[5,25]):
p5:=plot3d([r,2*Pi,phi],r=1/2..1,phi=0..Pi,coords=spherical,color=black, style=hidden, grid=[5,25]):
p6:=display([p1,p2,p3,p4,p5], scaling=constrained, orientation=[45,70],tickmarks=[3,3,3],axes=frame,lightmodel=none):
print(p6);
end module:
end use:

Figure 9.8.5(a) Region $R$ and outward normals

The divergence of F:

$\nabla \cdot F = \partial_{x} (x y^{3}) + \partial_{y} (y z) + \partial_{z} (x^{2} z) = y^{3} + z + x^{2}$

Implement the integral of $\nabla \cdot F$ over the interior of $R$ in spherical coordinates:

$\int_{0}^{2 π} \int_{0}^{π} \int_{1 / 2}^{1} ({(ρ \sin (φ) \sin (θ))}^{3} + (ρ \cos (φ)) + {(ρ \sin (φ) \cos (θ))}^{2}) ρ^{2} \sin (φ) d ρ d φ d θ$ = $\frac{31}{120} π$

To compute the flux through $R$ , note that there are two boundaries, the outer and inner spheres. To compute the flux through the outer sphere, note that on that surface

$F \cdot N = [\begin{array}{c} x y^{3} \\ y z \\ x^{2} z \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}]$ = $x^{2} y^{3} + x^{2} z^{2} + y^{2} z$

Anticipating that this will be integrated over the sphere using spherical coordinates, write

$Q = {\sin (φ)}^{5} {\cos (θ)}^{2} {\sin (θ)}^{3} + {\sin (φ)}^{2} {\cos (φ)}^{2} {\cos (θ)}^{2} + {\sin (φ)}^{2} \cos (φ) {\sin (θ)}^{2}$

as the equivalent for $F \cdot N$ on the surface of the outer sphere. If this be integrated over the sphere in spherical coordinates, the result is

$\int_{0}^{2 π} \int_{0}^{π} Q \sin (φ) d φ d θ$ = $\frac{4}{15} π$

On the inner sphere, $N = - 2 (x i + y j + z k)$ , $dσ = {(1 / 2)}^{2} \sin (φ) d φ d θ$ , and $F \cdot N$ becomes

$q = - (\frac{1}{16} {\sin (φ)}^{5} {\cos (θ)}^{2} {\sin (θ)}^{3} + \frac{1}{8} {\sin (φ)}^{2} {\cos (φ)}^{2} {\cos (θ)}^{2} + \frac{1}{4} {\sin (φ)}^{2} \cos (φ) {\sin (θ)}^{2})$

The flux through the inner sphere is then $\int_{0}^{2 π} \int_{0}^{π} q \sin (φ) / 4 d φ d θ$ = $- \frac{1}{120} π$ , so the total flux through the boundaries of $R$ is then $(\frac{4}{15} - \frac{1}{120}) π = \frac{31}{120} π$ .

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

•	Tools≻Tasks≻Browse: Calculus - Vector≻ Vector Algebra and Settings≻ Display Format for Vectors

•	Press the Access Settings button and select "Display as Column Vector"

Display Format for Vectors

Define the vector field F

•	Enter the components of F in a free vector. Context Panel: Evaluate and Display Inline

•	Context Panel: Student Vector Calculus≻Conversions≻To Vector Field

•	Context Panel: Assign to a Name≻F

$⟨x y^{3}, y z, x^{2} z⟩$ = $\overset{to Vector Field}{\to}$ $\overset{assign to a name}{\to}$ $F$

Obtain $\nabla \cdot F$ , the divergence of F, and represent it parametrically

•	Context Panel: Assign name

$X = ρ \sin (φ) \cos (θ)$ $\overset{assign}{\to}$

$Y = ρ \sin (φ) \sin (θ)$ $\overset{assign}{\to}$

$Z = ρ \cos (φ)$ $\overset{assign}{\to}$

•	Expression palette: Evaluation template

•	Common Symbols palette: Del and dot-product operators

•	Context Panel: Evaluate and Display Inline

•	Context Panel: Assign to a Name≻ $divF$

$\binom{\nabla \cdot F}{} | \binom{}{x = X, y = Y, z = Z}$ = $ρ^{3} {\sin (φ)}^{3} {\sin (θ)}^{3} + ρ^{2} {\sin (φ)}^{2} {\cos (θ)}^{2} + ρ \cos (φ)$ $\overset{assign to a name}{\to}$ $divF$

Use spherical coordinates to obtain the volume integral of the divergence of F

•	Tools≻Load Package: Student Multivariate Calculus

Loading Student:-MultivariateCalculus

•	Write the name given to the divergence and press the Enter key.

•	Context Panel: Student Multivariate Calculus≻Integrate≻Iterated Complete the dialogs as per the figures below.

•	Context Panel: Evaluate Integral

$divF$

$ρ^{3} {\sin (φ)}^{3} {\sin (θ)}^{3} + ρ^{2} {\sin (φ)}^{2} {\cos (θ)}^{2} + ρ \cos (φ)$

$\overset{MultiInt}{\to}$

$\int_{0}^{2 π} \int_{0}^{π} \int_{\frac{1}{2}}^{1} (ρ^{3} {\sin (φ)}^{3} {\sin (θ)}^{3} + ρ^{2} {\sin (φ)}^{2} {\cos (θ)}^{2} + ρ \cos (φ)) ρ^{2} \sin (φ) &DifferentialD; ρ &DifferentialD; φ &DifferentialD; θ$

$=$

$\frac{31}{120} π$

There are two parts to the boundary of $R$ , the surface of the outer sphere, and the surface of the inner sphere. For the flux through the outer surface surface, use a task template.

Tools≻Tasks≻Browse:
Calculus - Vector≻Integration≻Flux≻3-D≻Through a Sphere

Flux through a Sphere

For the Vector Field:

On the inner sphere, the normal that is outward on $R$ is inward for this smaller sphere. The same task template can be used to obtain the flux through this smaller sphere.

Tools≻Tasks≻Browse:
Calculus - Vector≻Integration≻Flux≻3-D≻Through a Sphere

Flux through a Sphere

For the Vector Field:

The total flux through $R$ is then $(\frac{4}{15} - \frac{1}{120}) π = \frac{31}{120} π$ , the same value that is obtained for the volume integral of the divergence in $R$ .

Maple Solution - Coded

Initialize

•	Install the Student VectorCalculus package.

$with (Student :- VectorCalculus) &colon;$

•	Set display of vectors via BasisFormat.

$BasisFormat (false) &colon;$

•	Define F via the VectorField command.

$F ≔ VectorField (⟨x y^{3}, y z, x^{2} z⟩) &colon;$

Obtain $\nabla \cdot F$ , the divergence of F

•	Invoke the Divergence command.

$divF ≔ Divergence (F)$

$y^{3} + x^{2} + z$

Use the int command to integrate the divergence of F over $R$

$int (divF, [x, y, z] = Sphere (⟨0, 0, 0⟩, 1), output = integral) - int (divF, [x, y, z] = Sphere (⟨0, 0, 0⟩, 1 / 2), output = integral)$

$\int_{0}^{1} \int_{0}^{π} \int_{0}^{2 π} r^{3} (\sin (φ) \sin (θ) {\cos (θ)}^{2} {\cos (φ)}^{2} r^{2} - \sin (φ) \sin (θ) {\cos (θ)}^{2} r^{2} - \sin (φ) \sin (θ) {\cos (φ)}^{2} r^{2} - {\cos (θ)}^{2} {\cos (φ)}^{2} r + \sin (φ) \sin (θ) r^{2} + {\cos (θ)}^{2} r + \cos (φ)) \sin (φ) &DifferentialD; θ &DifferentialD; φ &DifferentialD; r - (\int_{0}^{\frac{1}{2}} \int_{0}^{π} \int_{0}^{2 π} r^{3} (\sin (φ) \sin (θ) {\cos (θ)}^{2} {\cos (φ)}^{2} r^{2} - \sin (φ) \sin (θ) {\cos (θ)}^{2} r^{2} - \sin (φ) \sin (θ) {\cos (φ)}^{2} r^{2} - {\cos (θ)}^{2} {\cos (φ)}^{2} r + \sin (φ) \sin (θ) r^{2} + {\cos (θ)}^{2} r + \cos (φ)) \sin (φ) &DifferentialD; θ &DifferentialD; φ &DifferentialD; r)$

$int (divF, [x, y, z] = Sphere (⟨0, 0, 0⟩, 1)) - int (divF, [x, y, z] = Sphere (⟨0, 0, 0⟩, 1 / 2))$

$\frac{31}{120} π$

Use the Flux command to obtain the flux of F through the outer sphere

$Flux (F, Sphere (⟨0, 0, 0⟩, 1), output = integral)$

$\int_{0}^{2 π} \int_{0}^{π} {\sin (φ)}^{3} (\sin (φ) {\cos (θ)}^{4} \sin (θ) {\cos (φ)}^{2} - \sin (φ) {\cos (θ)}^{4} \sin (θ) - \sin (φ) {\cos (θ)}^{2} \sin (θ) {\cos (φ)}^{2} + \sin (φ) {\cos (θ)}^{2} \sin (θ) + {\cos (θ)}^{2} {\cos (φ)}^{2} - {\cos (θ)}^{2} \cos (φ) + \cos (φ)) &DifferentialD; φ &DifferentialD; θ$

$Flux (F, Sphere (⟨0, 0, 0⟩, 1))$ = $\frac{4}{15} π$

Use the Flux command to obtain the flux of F through the inner sphere

$Flux (F, Sphere (⟨0, 0, 0⟩, 1 / 2, inward), output = integral)$

$- (\int_{0}^{2 π} \int_{0}^{π} \frac{1}{64} {\sin (φ)}^{3} (\sin (φ) {\cos (θ)}^{4} \sin (θ) {\cos (φ)}^{2} - \sin (φ) {\cos (θ)}^{4} \sin (θ) - \sin (φ) {\cos (θ)}^{2} \sin (θ) {\cos (φ)}^{2} + \sin (φ) {\cos (θ)}^{2} \sin (θ) + 2 {\cos (θ)}^{2} {\cos (φ)}^{2} - 4 {\cos (θ)}^{2} \cos (φ) + 4 \cos (φ)) &DifferentialD; φ &DifferentialD; θ)$

$Flux (F, Sphere (⟨0, 0, 0⟩, 1 / 2, inward))$ = $- \frac{1}{120} π$

The total flux through $R$ is then $(\frac{4}{15} - \frac{1}{120}) π = \frac{31}{120} π$ , the same value that is obtained for the volume integral of the divergence in $R$ .

<< Previous Example Section 9.8 Next Example >>

© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.

For more information on Maplesoft products and services, visit www.maplesoft.com

Download Help Document

Maple

Maple Add-Ons

MapleSim

MapleSim Add-Ons

Systems Engineering

Consulting Services

Maple T.A. and Möbius

Education

Industries

Automotive and Aerospace

Robotics

Machine Design & Industrial Automation

Other

Application Areas

Product Pricing

Purchasing

Institutional Student Licensing

Maplesoft Elite Maintenance (EMP)

Support

Product Training

Online Product Help

Webinars & Events

Publications

Content Hubs

Examples & Applications

Community

About Maplesoft

Media Center

User Community

Contact

Online Help

All Products Maple MapleSim

Maple

Powerful math software that is easy to use

Maple Add-Ons

MapleSim

Advanced System Level Modeling

MapleSim Add-Ons

Systems Engineering

Consulting Services

Maple T.A. and Möbius

Education

Industries

Automotive and Aerospace

Robotics

Machine Design & Industrial Automation

Other

Application Areas

Product Pricing

Purchasing

Institutional Student Licensing

Maplesoft Elite Maintenance (EMP)

Support

Product Training

Online Product Help

Webinars & Events

Publications

Content Hubs

Examples & Applications

Community

About Maplesoft

Media Center

User Community

Contact

Online Help

All Products Maple MapleSim