Chapter 9: Vector Calculus
Section 9.7: Conservative and Solenoidal Fields
Example 9.7.4
If is a scalar potential for , show that , where is that part of the parabola between P and Q, the points , and , respectively.
Solution
Mathematical Solution
Parametrize as , and let the components of F be and so that becomes .
From Example 9.7.3, a scalar potential for F is , so that becomes
=
Maple Solution - Interactive
Table 9.7.4(a) provides a Context Panel construction of , a scalar potential for F.
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
Obtain Maple's scalar potential
Write the name F. Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Scalar Potential
Context Panel: Assign to a Name≻
Make a function of and
Context Panel: Assign Function
Table 9.7.4(a) Context Panel construction of the scalar potential
Table 9.7.4(b) demonstrates that holds.
Evaluate
Context Panel: Student Vector Calculus≻Line Integral Complete the Line Integral Domain dialog as per Figure 9.7.4(a).
Figure 9.7.4(a) Line Integral Domain dialog
Evaluate the difference
Context Panel: Evaluate and Display Inline
Table 9.7.4(b) Verification that
Maple Solution - Coded
Table 9.7.4(c) contains the calculations that verify .
Install the Student VectorCalculus package.
Set display of vectors via BasisFormat.
Define F via the VectorField command.
Obtain , a scalar potential for F
Invoke the ScalarPotential command.
Use the unapply command to create a function.
Calculate and evaluate the difference
Invoke the LineInt command.
Table 9.7.4(c) Verification that
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