Chapter 9: Vector Calculus
Section 9.9: Stokes' Theorem
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Example 9.9.1
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Apply Stokes' theorem to the vector field ; the curve , the unit circle with center at the origin; and the upper hemisphere as the capping surface .
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Solution
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Mathematical Solution
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Figure 9.9.1(a) shows arrows of the curl field of F, along with the upper hemisphere . The black arrow at the "north pole" is a representative normal taken on .
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A unit normal on can be obtained by normalizing , where R is a position-vector representation of .
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use Student:-VectorCalculus in
module()
local F,p;
F:=VectorField(<z,-x,-y>);
p:=Flux(Curl(F),Surface(<x,y,sqrt(-x^2-y^2+1)>,x=-1..1,y=-sqrt(-x^2+1)..sqrt(-x^2+1)),output=plot,fieldoptions=[grid=[3,3,3]],scaling=constrained,caption="",tickmarks=[3,3,3],axes=frame,orientation=[155,85,0]);
print(p);
end module:
end use:
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Figure 9.9.1(a) and the surface
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Let be a Cartesian representation of the upper hemisphere . It follows that
= = and
The element of surface area can be obtained from
=
or from . In either event,
so that can be implemented in Cartesian coordinates as
or in polar coordinates as
The line integral around , the unit circle centered at the origin, given by , can be evaluated if is parametrized by the position vector so that on
Consequently, = .
The parametrization chosen for induces a counterclockwise traverse of the circle, an orientation consistent with the choice of an outward normal on .
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Maple Solution - Interactive
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Initialize
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Tools≻Load Package: Student Vector Calculus
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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"
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Display Format for Vectors
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Define the vector field F
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Enter a free vector whose components are those of F.
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
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Obtain on the surface
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Common Symbols palette:
Del and cross-product operators
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Context Panel: Evaluate and Display Inline
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Context Panel: Assign to a Name≻curlF
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Table 9.9.1(a) contains a task template with which the flux of through is computed. Should the "Clear All and Reset" button in the Task Template be pressed, all the data that has been input to the template will be lost. In that event, the reader should simply re-launch the example to recover the appropriate inputs to the template.
Tools≻Tasks≻Browse:
Calculus - Vector≻Integration≻Flux≻3-D≻Through a Surface Defined over a Disk
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Flux through a Surface Defined over a Disk
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For the Vector Field:
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Table 9.9.1(a) Flux of through
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Table 9.9.1(b) contains the calculation of the line integral , where is the circle capped by .
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Write the name F.
Context Panel: Evaluate and Display Inline
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Context Panel: Student Vector Calculus≻Line Integral
(Complete the Line Integral Domain dialog as per Figure 9.9.1(b).)
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Context Panel: Evaluate Integral
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Figure 9.9.1(b) Line Integral Domain dialog
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Table 9.9.1(b) Line integral: tangential component of F along the circle
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The astute reader will realize that Maple has evaluated a line integral as an iterated double-integral by invoking Stokes' theorem! Consequently, a validation of Stokes' theorem demands that the line integral be evaluated from first principles. This is easily done if is parametrized by the position vector
so that on
and hence, = = .
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Maple Solution - Coded
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Initialize
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Install the Student VectorCalculus package.
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Define the upper hemisphere as .
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Use the Curl and Flux commands to obtain the flux of through
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Use the LineInt command to form and evaluate
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The astute reader will realize that Maple has evaluated a line integral as an iterated double-integral by invoking Stokes' theorem! Consequently, a validation of Stokes' theorem demands that the line integral be evaluated from first principles. This has already been done twice, in the previous two sections.
Figure 9.9.1(a) can be obtained with the Flux command, provided the integration is implemented in Cartesian coordinates with the following syntax. The actual options applied in the figure can be seen in the code hidden in the table cell containing the figure.
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