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Chapter 14: Q 30. (page 1107)

Evaluate each of the vector field line integrals over the indicated curves.

Short Answer

Expert verified

The required integral is∫CF·dr=-8π

Step by step solution

01

Given Information

The vector field is F(x,y)=yi-xj.

Cis circle of radius 2centered at origin and traversed clockwise.

02

Solving parametric equations

The parametric equations are

x(t)=2cost,y(t)=2sint,0≤t≤2π

⇒x'(t)=-2sint,y'(t)=2cost

Hence, C:r(t)=⟨2cost,2sint⟩

Also

F(x(t),y(t))=2sinti-2costj

and

r'(t)=x'(t),y'(t)

=⟨-2sint,2cost⟩

=(-2sint)i+(2cost)j

03

Finding line integral

The line integral is

∫F·dr=∫F(x(t),y(t))·r'(t)dt

=∫02π[2sinti-2costj]·[(-2sint)i+(2cost)j]dt

=∫02π-4sin2t-4cos2tdt

=∫02π(-4)dt

=-4[2π]

=-8π

Hence,∫CF·dr=-8π

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Most popular questions from this chapter

F(x,y,z)=zcosyzj+zsinyzk, where S is the portion of the plane with equation 2x−8y−10z=42 that lies on the positive side of the rectangle with corners0,-π,0(0,π,0),(0,π,π),(0,−π,π)in theyz-plane.

ComputethedivergenceofthevectorfieldsinExercises17–22.F(x,y,z)=xeyzi+yexzj+zexyk

Q. Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A smooth surface with a smooth boundary.

(b) A surface that is not smooth, but that has a smooth boundary.

(c) A surface that is smooth, but does not have a smooth boundary

Integrate the given function over the accompanying surface in Exercises 27–34.

f(x,z)=e-(x2+z2), where S is the unit disk centered at the point (0, 2, 0)and in the plane y = 2.

Make a chart of all the new notation, definitions, and theorems in this section, including what each new thing means in terms you already understand.
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