Chapter 9: Vector Calculus
Section 9.5: Line Integrals
Obtain the line integral of the scalar function fx,y,z=x y z, taken along the polygonal line connecting the points 1,2,3, 5,3,2, and 0,1,4, in that order.
The line integral of the scalar fx,y,z along a path described parametrically by x=xt,y=yt,z=zt, t∈a,b, is given by
∫abfxt,yt,zt dsdt dt
where s is arc length, so dsdt=x.2+y.2+z.2=ρ = R., with Rt=xt i+yt j+zt k being the vector form of the parametric representation of the path. However, there are two linear segments for the polygonal line.
A parametric representation of the first line segment is
R=x(t)y(t)z(t)=123+t (532−123) = 1+4 t2+ t3−t,0≤t≤1
ρ=dsdt=ddt1+4 t2+ddt2+ t2+ddt3−t2 = 42+12+12=32
and the line integral along this segment is given by
∫011+4 t 2+ t 3−t 3 2 dt=32 ∫01−4⁢t3+3⁢t2+25⁢t+6 dt=111/2
A parametric representation of the second line segment is
R=x(t)y(t)z(t)=532+t (014−532) = 5−5 t3−2 t2+2 t,0≤t≤1
ρ=dsdt=ddt5−5 t2+ddt3−2 t2+ddt2+2 t2 = 52+22+22=33
∫015−5 t 3−2 t 2+2 t 33 dt=33 ∫01102 t3−3 t2−2 t+3 dt=1533
Thus, the line integral along the polygonal path is the sum
1112+1533 ≐ 164.66
Tools≻Load Package: Student Vector Calculus
Access the PathInt command through the Context Panel
Write the scalar and press the Enter key.
Context Panel: Student Vector Calculus≻Line Integral (3D)
Complete the dialog as per Figure 9.5.4(a).
Context Panel: Evaluate (Integral)
x y z
Figure 9.5.4(a) Path Integral Domain dialog
Form and evaluate the line integral via the PathInt command
PathIntx y z,x,y,z=LineSegments1,2,3,5,3,2,0,1,4,output=integral
PathIntx y z,x,y,z=LineSegments1,2,3,5,3,2,0,1,4
Clearly, integrating along a polygonal line is equivalent to integrating along each segment of the path, and adding the resulting integrals.
PathIntx y z,x,y,z=Line1,2,3,5,3,2 = 1112⁢2
PathIntx y z,x,y,z=Line5,3,2,0,1,4 = 15⁢33
A solution from first principles is also possible, but the bulk of the work consists in obtaining parametric representations of each line segment.
Apply the BasisFormat command.
Let P be a list of the four nodes given as vectors.
Obtain parametric representations of each line segment, and calculate ρ for each segment
Write the equation of a line segment and press the Enter key.
Context Panel: Student Vector Calculus≻Differentiate≻With Respect To≻t
Context Panel: Student Vector Calculus≻Norm≻Euclidean
Now write and evaluate two integrals of the form ∫01fxt,yt,zt ρ dt.
∫011+4 t 2+ t 3−t 3 2 ⅆt = 1112⁢2
∫015−5 t 3−2 t 2+2 t 33 ⅆt = 15⁢33
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