Chapter 9: Vector Calculus
Section 9.5: Line Integrals
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Example 9.5.20
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Calculate the flux of the field through the ellipse whose center is and whose semi-major and semi-minor axes, parallel to the coordinate axes, are 2 and 1, respectively. Take the outward normal along the ellipse.
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Solution
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Mathematical Solution
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The ellipse with center , semi-major axis 2, and semi-minor axis 1, is given parametrically by the position vector
=
The flux of the field F through this ellipse is given by the line integral
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Maple Solution - Interactive
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Obtain the flux via the following task template.
Tools≻Tasks≻Browse:
Calculus - Vector≻Integration≻Flux≻2-D≻Through an Ellipse
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Flux through an Ellipse
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Obtain the flux via first principles.
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 the 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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=
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Define the path parametrically, as a position vector
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Context Panel: Assign Name
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Obtain , a tangent vector along the path
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Calculus palette: Differentiation operator
Context Panel: Evaluate and Display Inline
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Obtain N, a unit normal vector along the path
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Write the free vector whose components are those of , but switched, and the second one negated.
Context Panel: Evaluate and Display Inline
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Context Panel: Student Vector Calculus≻Normalize≻Euclidean
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Context Panel: Assign to a Name≻N
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Obtain
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Calculus palette: Differentiation operator
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Context Panel: Evaluate and Display Inline
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Context Panel: Assign to a Name≻rho
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Obtain the integrand
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Expression palette: Evaluation template
Common Symbols palette: Dot product operator
Press the Enter key.
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Context Panel: Constructions≻Definite Integral≻
(Complete dialog as per figure at the right.)
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Context Panel: Evaluate Integral
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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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Set the display format for vectors via the BasisFormat command.
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Apply the Flux command
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Use the Flux command to generate a graph of F and the path of integration
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It is also possible to obtain a solution from first principles.
Define the path parametrically, as the position vector R
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Define R as the position vector parametrizing the line of integration with .
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Obtain
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Apply the Norm command to the result obtained with the diff command.
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Obtain N, a unit normal vector along the path
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Apply the Normalize command to the vector whose components are the interchanged components of , with the second one negated.
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Form and evaluate the requisite line integral
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Compute with the DotProduct command, and use the eval command to evaluate this expression along the path. Then use the Int and int commands.
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