Chapter 9: Vector Calculus
Section 9.10: Green's Theorem
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Example 9.10.6
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Apply the divergence-form of Green's theorem to , and , the region bounded by the parabola and the line .
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Solution
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Mathematical Solution
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Figure 9.10.6(a) shows the region , bounded by the curves and , curves that intersect at .
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Since , the left side of the divergence-form of Green's theorem becomes
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The flux through the boundary of is the sum of two separate flux integrals, namely,
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Figure 9.10.6(a) The region
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where the first integral is from left to right along ; and the second, from right to left, is along . The left-hand integral of evaluates to ; the right-hand integral, to . The sum is .
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Maple Solution - Interactive
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The divergence of F is integrated over the region in Table 9.10.6(a).
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 Cartesian vector field F
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Write the vector field as 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
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Compute the divergence of F
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Common Symbols palette:
Del and dot product operators
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Context Panel: Evaluate and Display Inline
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Obtain the -coordinates of the intersections of and
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Write the relevant equation.
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Context Panel: Solve≻Obtain Solutions for≻
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Context Panel: Assign to a Name≻
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Form, evaluate, and simplify the integral of the divergence
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Tools≻Load Package:
Student Multivariate Calculus
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Loading Student:-MultivariateCalculus
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Type and press the Enter key.
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Context Panel: Student Multivariate Calculus≻Integrate≻Iterated
Complete the two dialogs as per the figures below.
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Context Panel: Evaluate Integral
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Context Panel: Simplify≻Simplify
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Table 9.10.6(a) integrated over the region
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The flux of F through the boundary of is obtained in Table 9.10.6(b), where and . On the curve , , but on the curve , .
Compute , the flux of F through , the boundary of
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Calculus palette: Definite integral template
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Expression palette: Evaluation template
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Context Panel: Simplify≻Simplify
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Table 9.10.6(b) Flux of F through the boundary of
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The net flux is the sum .
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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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Use the BasisFormat command to set the display format for vectors.
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Obtain the -coordinates of the intersections of and by invoking the solve command.
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Use the int command to integrate over the region
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Use the Flux command to obtain the flux of F through
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Use the Flux command to obtain the flux of F through
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Obtain the total flux through the boundaries of the region
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