Chapter 9: Vector Calculus
Section 9.5: Line Integrals
Example 9.5.19
Calculate the flux of the field through the circle whose center is and whose radius is 2. Take the outward normal along the circle.
Solution
Mathematical Solution
The circle with center and radius 2 is given parametrically by the position vector
=
The flux of the field F through this circle is given by the line integral
Maple Solution - Interactive
Obtain the flux via the following task template.
Tools≻Tasks≻Browse:
Calculus - Vector≻Integration≻Flux≻2-D≻Through a Circle
Flux through a Circle
Select Coordinate SystemCartesian [x,y]Cartesian - otherpolarbipolarcardioidcassinianelliptichyperbolicinvcassinianlogarithmiclogcoshparabolicrosetangent
Center :
Radius:
Obtain the flux via first principles.
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 free vector whose components are those of F. Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Conversions≻To Vector Field
Context Panel: Assign to a Name≻F
Define the path parametrically, as a position vector
Context Panel: Assign Name
Obtain , a tangent vector along the path
Calculus palette: Differentiation operator Context Panel: Evaluate and Display Inline
Obtain N, a unit normal vector along the path
Write the free vector whose components are those of , but switched, and the second one negated. Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Normalize≻Euclidean
Context Panel: Assign to a Name≻N
Obtain
Calculus palette: Differentiation operator
Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻rho
Obtain the integrand
Expression palette: Evaluation template Common Symbols palette: Dot product operator Press the Enter key.
Context Panel: Constructions≻Definite Integral≻ (Complete dialog as per figure at the right.)
Context Panel: Evaluate Integral
Maple Solution - Coded
Install the Student VectorCalculus package.
Set the display format for vectors via the BasisFormat command.
Define F via the VectorField command.
Apply the Flux command
Use the Flux command to generate a graph of F and the path of integration
It is also possible to obtain a solution from first principles.
Define the path parametrically, as the position vector R
Define R as the position vector parametrizing the line of integration with .
Apply the Norm command to the result obtained with the diff command.
Apply the Normalize command to the vector whose components are the interchanged components of , with the second one negated.
Form and evaluate the requisite line integral
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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