9E: Chapter Exercises
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Exercise 9E.1: True or False?
Justify your answer with a proof or a counterexample.
1. Vector field ⇀F(x,y)=x2yˆi+y2xˆj is conservative.
- Answer
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False
2. For vector field ⇀F(x,y)=P(x,y)ˆi+Q(x,y)ˆj, if Py(x,y)=Qz(x,y) in open region D, then ∫∂DPdx+Qdy=0.
3. The divergence of a vector field is a vector field.
- Answer
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False
4. If curl⇀F=0, then ⇀F is a conservative vector field.
Exercise 9E.2
Draw the following vector fields.
1. ⇀F(x,y)=12ˆi+2xˆj
- Answer
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2. ⇀F(x,y)=yˆi+3xˆjx2+y2
Exercise 9E.3
Are the following the vector fields conservative? If so, find the potential function f such that ⇀F=∇f.
1. ⇀F(x,y)=yˆi+(x−2ey)ˆj
- Answer
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Conservative, f(x,y)=xy−2ey
2. ⇀F(x,y)=(6xy)ˆi+(3x2−yey)ˆj
3. ⇀F(x,y)=(2xy+z2)ˆi+(x2+2yz)ˆj+(2xz+y2)ˆk
- Answer
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Conservative, f(x,y,z)=x2y+y2z+z2x
4. ⇀F(x,y,z)=(exy)ˆi+(ex+z)ˆj+(ex+y2)ˆk
Exercise 9E.4
Evaluate the following integrals.
1. ∫Cx2dy+(2x−3xy)dx, along C:y=12x from (0, 0) to (4, 2)
- Answer
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−163
2. ∫Cydx+xy2dy, where C:x=√t,y=t−1,0≤t≤1
3. ∬Sxy2dS, where S is surface z=x2−y,0≤x≤1,0≤y≤4
- Answer
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A32√29(3√3−1)
Exercise 9E.5
Find the divergence and curl for the following vector fields.
1. ⇀F(x,y,z)=3xyzˆi+xyexˆj−3xyˆk
2. ⇀F(x,y,z)=exˆi+exyˆj−exyzˆk
- Answer
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Divergence: ex+xexy+xyexyz, curl: xzexyzˆi−yzexyzˆj+yexyˆk
Exercise 9E.6
Use Green’s theorem to evaluate the following integrals.
1. ∫C3xydx+2xy2dy, where C is a square with vertices (0,0),(0,2),(2,2) and (2,0).
2. ∮C3ydx+(x+ey)dy, where C is a circle centered at the origin with radius 3.
- Answer
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−2π
Exercise 9E.7
Use Stokes’ theorem to evaluate ∬Scurl⇀F⋅⇀dS.
1. ⇀F(x,y,z)=yˆi−xˆj+zˆk, where S is the upper half of the unit sphere
2. ⇀F(x,y,z)=yˆi+xyzˆj−2zxˆk, where S is the upward-facing paraboloid z=x2+y2 lying in cylinder x2+y2=1
- Answer
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−π
Exercise 9E.8
Use the divergence theorem to evaluate ∬S⇀F⋅⇀dS.
1. ⇀F(x,y,z)=(x3y)ˆi+(3y−ex)ˆj+(z+x)ˆk, over cube S defined by −1≤x≤1,0≤y≤2,0≤z≤2
2. ⇀F(x,y,z)=(2xy)ˆi+(−y2)ˆj+(2z3)ˆk, where S is bounded by paraboloid z=x2+y2 and plane z=2
- Answer
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31π/2
Exercise 9E.9
1. Find the amount of work performed by a 50-kg woman ascending a helical staircase with radius 2 m and height 100 m. The woman completes five revolutions during the climb.
2. Find the total mass of a thin wire in the shape of a semicircle with radius √2, and a density function of ρ(x,y)=y+x2.
- Answer
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√2(2+π)
3. Find the total mass of a thin sheet in the shape of a hemisphere with radius 2 for z≥0 with a density function ρ(x,y,z)=x+y+z.
4. Use the divergence theorem to compute the value of the flux integral over the unit sphere with ⇀F(x,y,z)=3zˆi+2yˆj+2xˆk.
- Answer
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2π/3