Chapter 9: Vector Calculus
Section 9.7: Conservative and Solenoidal Fields
Example 9.7.9
Show that F=2xz+y2 i+2 x y−z3 j+x2−3yz2 k is not solenoidal.
Solution
Mathematical Solution
A vector field F is solenoidal if its divergence vanishes identically, that is, if ∇·F=0. If the divergence does not vanish identically, then the field is not solenoidal. The divergence of the given vector field is
∇·F
=∂x2xz+y2+∂y2 x y−z3+∂zx2−3yz2
=2 z+2 x−6 y z
≠0
Maple Solution - Interactive
Table 9.7.9(a) provides the calculation whereby F is shown not to be solenoidal. In short, the calculation shows that the divergence of F does not vanish identically.
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 z+y2,2 x y−z3,x2−3 y z2 = →to Vector Field →assign to a nameF
Calculate the divergence of F
Common Symbols palette: Del and dot-product operators
Context Panel: Evaluate and Display Inline
∇·F = −6yz+2x+2z
Alternate calculation of the divergence of F
Write the name F. Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Divergence
F = →divergence−6yz+2x+2z
Table 9.7.9(a) F is not solenoidal because ∇·F≠0
Maple Solution - Coded
Table 9.7.9(b) provides the calculation whereby F is shown not to be solenoidal. In short, the calculation shows that the divergence of F does not vanish identically.
Install the Student VectorCalculus package.
withStudent:-VectorCalculus:
Set display of vectors via BasisFormat.
BasisFormatfalse:
Define F via the VectorField command.
F≔VectorField2 x z+y2,2 x y−z3,x2−3 y z2:
Use the Divergence command to obtain a scalar potential for F
DivergenceF = −6yz+2x+2z
Table 9.7.9(b) F is not solenoidal because ∇·F≠0
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