Chapter 9: Vector Calculus
Section 9.7: Conservative and Solenoidal Fields
Example 9.7.4
If ux,y is a scalar potential for F=2 x y−y3 i+x2−3 x y2 j, show that ∫CF·dr=uQ−uP, where C is that part of the parabola y=x2 between P and Q, the points −2,4, and 2,4, respectively.
Solution
Mathematical Solution
Parametrize C as xt=t,yt=t2,t∈−2,2, and let the components of F be f and g so that ∫CF·dr becomes ∫−22ft,t2+gt,t2⋅2 t dt= −256.
From Example 9.7.3, a scalar potential for F is ux,y=x yx−y2, so that uQ−uP becomes
2×4×2−16−−2×4×−2−16 = −112−144=−256
Maple Solution - Interactive
Table 9.7.4(a) provides a Context Panel construction of ux,y, 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
2 x y−y3,x2−3 x y2 = →to Vector Field →assign to a nameF
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≻U
F = →scalar potential−xy3+x2y→assign to a nameU
Make U a function of x and y
Context Panel: Assign Function
ux,y=U→assign as functionu
Table 9.7.4(a) Context Panel construction of the scalar potential
Table 9.7.4(b) demonstrates that ∫CF·dr=uQ−uP holds.
Evaluate ∫CF·dr
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
F = →line integral−256
Evaluate the difference uQ−uP
Context Panel: Evaluate and Display Inline
u2,4−u−2,4 = −256
Table 9.7.4(b) Verification that ∫CF·dr=uQ−uP
Maple Solution - Coded
Table 9.7.4(c) contains the calculations that verify ∫CF·dr=uQ−uP.
Install the Student VectorCalculus package.
withStudent:-VectorCalculus:
Set display of vectors via BasisFormat.
BasisFormatfalse:
Define F via the VectorField command.
F≔VectorField2 x y−y3,x2−3 x y2:
Obtain ux,y, a scalar potential for F
Invoke the ScalarPotential command.
Use the unapply command to create a function.
U≔ScalarPotentialF:u≔unapplyU,x,y:
Calculate ∫CF·dr and evaluate the difference uQ−uP
Invoke the LineInt command.
LineIntF,Pathx,x2,x=−2..2 = −256
Table 9.7.4(c) Verification that ∫CF·dr=uQ−uP
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