Chapter 9: Vector Calculus
Section 9.7: Conservative and Solenoidal Fields
Example 9.7.7
Prove that F=2xz+y2 i+2 x y−z3 j+x2−3yz2 k is conservative by showing it is curl free.
Solution
Mathematical Solution
∇×F=ijk∂x∂y∂z2 x z+y22 x y−z3x2−3 y z2 = −3 z2+3 z2−(2 x−2 x)2 y−2 y = 000
Maple Solution - Interactive
Table 9.7.7(a) provides a solution in which the curl of F is calculated in two different syntax-free ways.
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
Show that the curl of F is the zero vector
Common Symbols palette: Del operator
Common Symbols palette: Cross-product operator
Context Panel: Evaluate and Display Inline
∇×F =
Alternate access to the curl operator
Write the name F. Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Curl
F = →curl
Table 9.7.7(a) Computation of the curl of F
Maple Solution - Coded
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 Curl command to show that the curl of F is the zero vector
CurlF =
Table 9.7.7(b) Curl obtained by the Curl command
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