Chapter 9: Vector Calculus
Section 9.5: Line Integrals
Calculate the work that the field F=x y i+x+y j does on a unit positive charge as the charge moves along the polygonal line connecting the points 1,2, 3,5, 4,0, 2,7, in that order.
The line segment from 1,2 to 3,5 is given parametrically by the position vector
R=xy = 12+t (35−12) = 1+2 t2+3 t
The line segment from 3,5 to 4,0 is given parametrically by the position vector
R=xy = 35+t (40−35) = 3+ t5−5 t
The line segment from 4,0 to 2,7 is given parametrically by the position vector
R=xy = 40+t (27−40) = 4− 2 t7 t
The work done by the field F is given by the sum of three line integrals of the form
=∫abf dx+g dy
=∫01x yx=a|f(x)x=xt,y=yt⋅ddtxt+x+yx=a|f(x)x=xt,y=yt⋅ddtyt dt
=∫011+2 t2+3 t⋅2+1+2 t+2+3 t⋅3 dt
From Example 9.5.10, the work done over the first line segment is
∫011+2 t2+3 t⋅2+1+2 t+2+3 t⋅3 dt
Similarly, the work done on the second segment is
∫013+t5−5 t⋅1+3+t+5−5 t⋅−5 dt
=−5∫01t2−2 t+5 dt
Finally, the work done on the third segment is
∫014−2 t7 t⋅−2+4−2 t+7 t⋅7 dt
=7∫014 t2−3 t+4 dt
The work done over the polygonal path is then 632−653+1616=1103.
Maple Solution - Interactive
Tools≻Load Package: Student Vector Calculus
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 free vector whose components are those of F.
Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Conversions≻To Vector Field
Context Panel: Assign to a Name≻F
x y,x+y = x⁢yx+y→to Vector Fieldx⁢yx+y→assign to a nameF
Obtain the line integral via the Context Panel
Write the name F and press the Enter key.
Context Panel: Student Vector Calculus≻Line Integral
(Complete the dialog as per Figure 9.5.11(a).)
Context Panel: Evaluate Integral
Figure 9.5.11(a) Line Integral Domain dialog
In the "Line Integral Domain" dialog, select "Segment or Polygonal Line" and enter the nodes of the line (or line segment) as a sequence of vectors. At the bottom of the dialog, select "integral" as the return option. The alternatives are "value" or "plot". The graph returned by the "plot" option can be seen in the next section (Maple Solution - Coded).
A solution from first principles would require implementing the calculations in Example 9.5.10 three times, and adding the results. Such efforts are left to the interested reader.
Maple Solution - Coded
Install the Student VectorCalculus package.
Set the display format for vectors via the BasisFormat command.
Define F via the VectorField command.
Apply the LineInt command
Use the LineInt command to generate a graph of F and the path of integration
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