9.5E: EXERCISES
( \newcommand{\kernel}{\mathrm{null}\,}\)
Exercise 9.5E.1: True or False
For the following exercises, determine whether the statement is true or false.
1. If the coordinate functions of ⇀F:R3→R3 have continuous second partial derivatives, then curl(div(F)) equals zero.
2. ∇⋅(xˆi+yˆj+zˆk)=1.
- Answer
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False
3. All vector fields of the form ⇀F(x,y,z)=f(x)ˆi+g(y)ˆj+h(z)ˆk are conservative.
4. If curl⇀F=0, then⇀F is conservative.
- Answer
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True
5. If ⇀F is a constant vector field then div(⇀F)=0.
6. If ⇀F is a constant vector field then curl(⇀F)=0.
- Answer
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True
Exercise 9.5E.2: Curl
For the following exercises, find the curl of ⇀F.
- ⇀F(x,y,z)=xy2z4ˆi+(2x2y+z)ˆj+y3z2ˆk
- ⇀F(x,y,z)=x2zˆi+y2xˆj+(y+2z)ˆk
Answer
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curl(⇀F)=ˆi+x2ˆj+y2ˆk
3. ⇀F(x,y,z)=3xyz2ˆi+y2sinzˆj+xe2zˆk
4. ⇀F(x,y,z)=x2yzˆi+xy2zˆj+xyz2ˆk
- Answer
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curl(⇀F)=(xz2−xy2)ˆi+(x2y−yz2)ˆj+(y2z−x2z)ˆk
5. ⇀F(x,y,z)=(xcosy)ˆi+xy2ˆj
6. ⇀F(x,y,z)=(x−y)ˆi+(y−z)ˆj+(z−x)ˆk
- Answer
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curl(⇀F)=ˆi+ˆj+ˆk
7. ⇀F(x,y,z)=xyzˆi+x2y2z2ˆj+y2z3ˆk
8. ⇀F(x,y,z)=xyˆi+yzˆj+xzˆk
- Answer
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curl(⇀F)=−yˆi−zˆj−xˆk
9. ⇀F(x,y,z)=x2ˆi+y2ˆj+z2ˆk
10. ⇀F(x,y,z)=axˆi+byˆj+cˆk for constants a, b, c
- Answer
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curl(⇀F)0
Exercise 9.5E.3: Divergence
For the following exercises, find the divergence of ⇀F.
1. ⇀F(x,y,z)=x2zˆi+y2xˆj+(y+2z)ˆk
2. ⇀F(x,y,z)=3xyz2ˆi+y2sinzˆj+xe2zˆk
- Answer
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div(⇀F)=3yz2+2ysinz+2xe2z
3. ⇀F(x,y)=(sinx)ˆi+(cosy)ˆj
4. ⇀F(x,y,z)=x2ˆi+y2ˆj+z2ˆk
- Answer
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div(⇀F)2(x+y+z)
5. ⇀F(x,y,z)=(x−y)ˆi+(y−z)ˆj+(z−x)ˆk
6. ⇀F(x,y)=x√x2+y2ˆi+y√x2+y2ˆj
- Answer
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div(⇀F)=1√x2+y2
8. ⇀F(x,y)=xˆi−yˆj
9. ⇀F(x,y,z)=axˆi+byˆj+cˆk for constants a, b, c
- Answer
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div(⇀F)=a+b
10. ⇀F(x,y,z)=xyzˆi+x2y2z2ˆj+y2z3ˆk
11. ⇀F(x,y,z)=xyˆi+yzˆj+xzˆk
- Answer
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div(⇀F)=x+y+z
Exercise 9.5E.4: Harmonic
For the following exercises, determine whether each of the given scalar functions is harmonic.
1. u(x,y,z)=e−x(cosy−siny)
2. w(x,y,z)=(x2+y2+z2)−1/2
- Answer
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Harmonic
Exercise 9.5E.5: CURL
1. If ⇀F(x,y,z)=2ˆi+2xˆj+3yˆk and ⇀G(x,y,z)=xˆi−yˆj+zˆk, find curl(F×G).
2. If ⇀F(x,y,z)=2ˆi+2xˆj+3yˆk and ⇀G(x,y,z)=xˆi−yˆj+zˆk, find div(F×G).
- Answer
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div(F×G)=2z+3x
3. Find divF, given that F=∇f, where f(x,y,z)=xy3z2.
4. Find the divergence of ⇀F for vector field ⇀F(x,y,z)=(y2+z2)(x+y)i+(z2+x2)(y+z)ˆj+(x2+y2)(z+x)ˆk.
- Answer
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div ⇀F= 2r^2\)
5. Find the divergence of ⇀F for vector field ⇀F(x,y,z)=f1(y,z)ˆi+f2(x,z)ˆj+f3(x,y)ˆk.
6. For the following exercises, use r=|r| and r=(x,y,z).
a) Find the curlr
- Answer
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curlr=0
b) Find the curlrr.
c) Find the curlrr3.
- Answer
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curlrr3=0
7. Let ⇀F(x,y)=−yˆi+xˆjx2+y2, where⇀Fis defined on {(x,y)∈R|(x,y)≠(0,0)}. Find curlF.
Exercise 9.5E.6: Curl
For the following exercises, use a computer algebra system to find the curl of the given vector fields.
1. [T] ⇀Fx,y,z)=arctan(xy)ˆi+ln√x2+y2ˆj+ˆk
- Answer
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curl(⇀F)=2xx2+y2ˆk
2. [T] ⇀F(x,y,z)=sin(x−y)ˆi+sin(y−z)ˆj+sin(z−x)ˆk
Exercise 9.5E.7: Divergence
For the following exercises, find the divergence of⇀Fat the given point.
1. ⇀F(x,y,z)=ˆi+ˆj+ˆk at (2,−1,3)
- Answer
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div(⇀F)=0
2. ⇀F(x,y,z)=xyzˆi+yˆj+zˆk at (1,2,3)
3. ⇀F(x,y,z)=e−xyˆi+exzˆj+eyzˆk at (3,2,0)
- Answer
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div⇀F=2−2e−6
4. ⇀F(x,y,z)=xyzˆi+yˆj+zˆk at (1,2,1)
5. ⇀F(x,y,z)=exsinyˆi−excosyˆj at (0,0,3)
- Answer
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div(⇀F)=0
Exercise 9.5E.8: CURL
For the following exercises, find the curl of⇀Fat the given point.
1. ⇀F(x,y,z)=ˆi+ˆj+ˆk at (2,−1,3)
2. ⇀F(x,y,z)=xyzˆi+yˆj+zˆk at (1,2,3)
- Answer
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curl(⇀F)=2ˆj−3ˆk
3. ⇀F(x,y,z)=e−xyˆi+exzˆj+eyzˆk at (3,2,0)
4. ⇀F(x,y,z)=xyzˆi+yˆj+zˆk at (1,2,1)
- Answer
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curl(⇀F)=2ˆj−ˆk
5. ⇀F(x,y,z)=exsinyˆi−excosyˆj at (0,0,3)
Exercise 9.5E.9
Let ⇀F(x,y,z)=(3x2y+az)ˆi+x3ˆj+(3x+3z2)ˆk. For what value of a is⇀Fconservative?
- Answer
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a=3
Exercise 9.5E.10
1. Given vector field ⇀F(x,y)=1x2+y2(−y,x) on domain D=R2{(0,0)}={(x,y)∈R2|(x,y)≠(0,0)}, is⇀Fconservative?
2. Given vector field ⇀F(x,y)=1x2+y2(x,y) on domain D=R2{(0,0)}, is⇀Fconservative?
- Answer
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⇀F is conservative.
3. Find the work done by force field ⇀F(x,y)=e−yˆi−xe−yˆj in moving an object from P(0, 1) to Q(2, 0). Is the force field conservative?
Exercise 9.5E.11
1. Compute divergence ⇀F=(sinhx)ˆi+(coshy)ˆj−xyzˆk.
- Answer
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div(⇀F)=coshx+sinhy−xy
2. Compute curl(⇀F)=(sinhx)ˆi+(coshy)ˆj−xyzˆk.
- Answer
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TBA
Exercise 9.5E.12
For the following exercises, consider a rigid body that is rotating about the x-axis counterclockwise with constant angular velocity ω=⟨a,b,c⟩. If P is a point in the body located at ⇀r=xˆi+yˆj+zˆk, the velocity at P is given by vector field ⇀F=ω×⇀r.
a) Express⇀F in terms of ˆi,ˆj, and ˆk vectors.
- Answer
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(bz−cy)ˆi+(cx−az)ˆj+(ay−bx)ˆk
b) Find div⇀F.
c) Find curl⇀F
- Answer
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curl(⇀F)=2ω
Exercise 9.5E.13
In the following exercises, suppose that ∇⋅⇀F=0 and ∇⋅⇀G=0.
a) Does ⇀F+⇀G necessarily have zero divergence?
b) Does ⇀F×⇀G necessarily have zero divergence?
- Answer
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⇀F×⇀G does not have zero divergence.
Exercise 9.5E.14
In the following exercises, suppose a solid object in R3 has a temperature distribution given by T(x,y,z). The heat flow vector field in the object is ⇀F=−k∇T, where k>0 is a property of the material. The heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is ∇⋅⇀F=−k∇⋅∇T=−k∇2T.
a) Compute the heat flow vector field.
b) Compute the divergence.
- Answer
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∇⋅⇀F=−200k[1+2(x2+y2+z2)]e−x2+y2+z2
Exercise 9.5E.15
[T] Consider rotational velocity field ⇀v=⟨0,10z,−10y⟩. If a paddlewheel is placed in plane x+y+z=1 with its axis normal to this plane, using a computer algebra system, calculate how fast the paddlewheel spins in revolutions per unit time.
Glossary
curl
the curl of vector field F=⟨P,Q,R⟩, denoted ∇×F, is the “determinant” of the matrix |iˆjˆk∂∂x∂∂y∂∂zPQR| and is given by the expression (Ry−Qz)ˆi+(Pz−Rx)ˆj+(Qx−Py)ˆk; it measures the tendency of particles at a point to rotate about the axis that points in the direction of the curl at the point
divergence
the divergence of a vector field F=⟨P,Q,R⟩, denoted ∇×F, is Px+Qy+Rz; it measures the “outflowing-ness” of a vector field