Chapter 9: Vector Calculus
Section 9.7: Conservative and Solenoidal Fields
Example 9.7.11
Show that F=x z i+x2−y z j+y2 k is solenoidal but not conservative, and find a vector potential both with Maple's VectorPotential command and by Recipe 2 in Table 9.7.4. If these differ, show that the difference is a gradient and find a scalar potential for this gradient.
Solution
Mathematical Solution
To show F is solenoidal, show that its divergence is zero. Hence, make the following calculation.
∇·F
=∂xx z+∂yx2−y z+∂zy2
=z−z+0
=0
To show F is not conservative, show that its curl is not the zero vector. Hence, make the following calculation.
∇×F=ijk∂x∂y∂zx zx2−y zy2 = 3 yx2 x≠0
Maple's VectorPotential command gives the vector
A=x2z−12yz2−13y3−12xz20
as a vector potential.
Recipe 2 in Table 9.7.4 gives
B=0∫axy2 dt−∫czat dt−∫axt2−yz dt = 0y2x−a−12a−c2+z213a3−13x3+yzx−a
The curl of the difference C=A−B=x2z−12yz2−13y3−12xz2−y2x−a+12a−c2+z2−13a3+13x3−yzx−a
is
∇×C=−zx−a−az+xz00=000
so that the difference C is a gradient. A scalar potential for this gradient is
ux,y,z=13x3z−12yz2x−13y3x+13y3a−12ac2y+12az2y−13a3z
Indeed, the gradient of u is C, as can be verified by the appropriate, but tedious, differentiations.
Maple Solution - Interactive
Table 9.7.11(a) shows that F is solenoidal because its divergence vanishes (∇·F=0) and that F is not conservative because its curl does not vanish (∇×F≠0). In addition, Table 9.7.11(a) provides Maple's vector 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
x z,x2−y z,y2 = →to Vector Field →assign to a nameF
Show ∇·F=0 and ∇×F≠0
Common Symbols palette: Del and dot-product or Del and cross-product operators Context Panel: Evaluate and Display Inline
∇·F = 0
∇×F =
Obtain Maple's vector potential for F
Write the name F. Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Vector Potential
Context Panel: Assign to a Name≻A
F = →vector potential →assign to a nameA
Table 9.7.11(a) F is solenoidal but not conservative, and A is Maple's vector potential for F
Table 9.7.11(b) applies Recipe 2 in Table 9.7.4 to obtain a vector potential for F. This result will differ from A, and the difference is shown to be a gradient vector.
Define the components of F as the functions f,g,h
Context Panel: Assign Function
fx,y,z=F1→assign as functionf
gx,y,z=F2→assign as functiong
hx,y,z=F3→assign as functionh
Apply Recipe 2, Table 9.7.4
Calculus palette: Definite-integral template Press the Enter key.
Context Panel: Assign to a Name≻B
0,∫axht,y,z ⅆt−∫czfa,y,t ⅆt,−∫axgt,y,z ⅆt
→to Vector Field
→assign to a name
B
Show ∇×A−B=0 so that C=A−B is a gradient vector
Common Symbols palette: Del and cross-product operators Context Panel: Evaluate and Display Inline
Context Panel: Simplify≻Simplify
∇×A−B = = simplify
Table 9.7.11(b) Recipe 2 in Table 9.7.4 and a vector potential for F
A scalar potential u for the difference C=A−B is found in Table 9.7.11(c) where it is also verified that ∇u=C.
Obtain u, a scalar potential for the difference A−B
Write the difference A−B and press the Enter key.
Context Panel: Student Vector Calculus≻Scalar Potential
Context Panel: Assign to a Name≻u
A−B
→scalar potential
13x3z−12yz2x−13y3x+13y3a−12ac2y+12az2y−13a3z
u
Verify that u is a scalar potential for C=A−B by showing ∇u=C
Calculus palette: Partial-derivative operator Context Panel: Evaluate and Display Inline
∂∂ x u,∂∂ y u,∂∂ z u =
Table 9.7.11(c) Scalar potential u for C=A−B and verification that ∇u=C
Maple Solution - Coded
Table 9.7.11(d) shows that F is solenoidal because its divergence vanishes (∇·F=0) and that F is not conservative because its curl does not vanish (∇×F≠0). In addition, Table 9.7.11(d) provides Maple's vector potential for F and applies Recipe 2 in Table 9.7.4, obtaining a different vector potential. The difference between the two vector potentials is shown to be a gradient vector.
Install the Student VectorCalculus package.
withStudent:-VectorCalculus:
Set display of vectors via BasisFormat.
BasisFormatfalse:
Define F via the VectorField command.
F≔VectorFieldx z,x2−y z,y2:
Apply the Divergence command.
DivergenceF = 0
Apply the Curl command.
CurlF =
Apply the VectorPotential command.
A≔VectorPotentialF
Implement Recipe 2 in Table 9.7.4
Use the unapply command to define the components of F as the functions f,g,h
f≔unapplyF1,x,y,z:
g≔unapplyF2,x,y,z:
h≔unapplyF3,x,y,z:
Use the VectorField and int commands to implement Recipe 2 in Table 9.7.4
B≔VectorField0,intht,y,z,t=a..x−intfa,y,t,t=c..z,−intgt,y,z,t=a..x
Apply the simplify command to the result produced by the Curl command.
simplifyCurlA−B =
Table 9.7.11(d) F is solenoidal but not conservative; a vector potential for F is unique up to an additive gradient
A scalar potential u for the difference C=A−B is found in Table 9.7.11(e) where it is also verified that ∇u=C.
Obtain a scalar potential for A−B by applying the ScalarPotential command.
u≔ScalarPotentialA−B
Use the Gradient command to obtain ∇u and compare to A−B Use the map command to apply the expand command to each component of A−B
Gradientu,x,y,z,mapexpand,A−B
Table 9.7.11(e) Scalar potential u for A−B and ∇u=A−B
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