Chapter 9: Vector Calculus
Section 9.7: Conservative and Solenoidal Fields
Example 9.7.5
For ; , the line from the origin to the point ; and , the polygonal path from the origin to to , to , show that the line integral of F along and have the same value.
Solution
Mathematical Solution
If is parametrized by , , then the line integral along is
If the three segments of are parametrized by the position vectors
, ,
then on each segment becomes , respectively. Hence the line integral along is
Maple Solution - Interactive
Table 9.7.5(a) provides a solution in which the requisite line integrals are formed via the LineInt command accessed through the Context Panel.
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
=
Form and evaluate the line integral of F along
Write the name F. Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Line Integral Complete the dialog as per the figure at the right.
Context Panel: Evaluate Integral
Table 9.7.5(a) Solution via the LineInt command accessed through the Context Panel
To emphasize how much tedious work the LineInt command actually saves in the line integration over , Table 9.7.5(b) containing a solution from first principles, is provided. Three separate line segments must be parametrized, and must be evaluated on each.
Form a list of nodes for the polygonal line
Write a list of nodes. Context Panel: Assign to a Name≻
Form a position-vector representation of each segment of
Write a vector representation of each segment. Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name: r[k],
Obtain a parametric representation of each line segment in
Form a sequence of the general position vector and the position vector of a segment of . Context Panel: Evaluate and Display Inline
Context Panel: Equate
Context Panel: Assign to a Name≻s[k],
Evaluate on each segment of
Expression palette: Evaluation template
Common Symbols palette: Dot-product operator
Calculus palette: Differentiation operator
Context Panel: Evaluate and Display Inline
Form and evaluate line integrals on each segment of
Calculus palette: definite-integral operator
Table 9.7.5(b) Line integral on from first principles
Maple Solution - Coded
Table 9.7.5(c) provides a solution in which the requisite line integrals are formed and evaluated with the LineInt command in the Student VectorCalculus package.
Install the Student VectorCalculus package.
Set display of vectors via BasisFormat.
Define F via the VectorField command.
Table 9.7.5(c) Solution via the LineInt command
In Table 9.7.5(d), the line integral of F along is obtained from first principles.
Define the nodes on
Use the Equate command to obtain a parametric representation of each line segment in
Obtain dr with the diff command.
Obtain with the DotProduct command.
Evaluate on each segment of with the eval command.
Use the Int and top-level int commands to form and evaluate line integrals on each segment of
Table 9.7.5(d) Line integral along implemented from first principles
With the Student VectorCalculus package installed, access to the top-level int command requires the "colon dash" prefix.
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