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Mathematics LibreTexts

9.5E: EXERCISES

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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:R3R3 have continuous second partial derivatives, then curl(div(F)) equals zero.

2. (xˆi+yˆj+zˆk)=1.

Answer

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 curlF=0, thenF is conservative.

Answer

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

True

Exercise 9.5E.2: Curl

For the following exercises, find the curl of F.

  1. F(x,y,z)=xy2z4ˆi+(2x2y+z)ˆj+y3z2ˆk
  2. F(x,y,z)=x2zˆi+y2xˆj+(y+2z)ˆk

Answer

curl(F)=ˆi+x2ˆj+y2ˆk

3. F(x,y,z)=3xyz2ˆi+y2sinzˆj+xe2zˆk

4. F(x,y,z)=x2yzˆi+xy2zˆj+xyz2ˆk

Answer

curl(F)=(xz2xy2)ˆi+(x2yyz2)ˆj+(y2zx2z)ˆk

5. F(x,y,z)=(xcosy)ˆi+xy2ˆj

6. F(x,y,z)=(xy)ˆi+(yz)ˆj+(zx)ˆk

Answer

curl(F)=ˆi+ˆj+ˆk

7. F(x,y,z)=xyzˆi+x2y2z2ˆj+y2z3ˆk

8. F(x,y,z)=xyˆi+yzˆj+xzˆk

Answer

curl(F)=yˆizˆjxˆk

9. F(x,y,z)=x2ˆi+y2ˆj+z2ˆk

10. F(x,y,z)=axˆi+byˆj+cˆk for constants a, b, c

Answer

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

div(F)=3yz2+2ysinz+2xe2z

3. F(x,y)=(sinx)ˆi+(cosy)ˆj

4. F(x,y,z)=x2ˆi+y2ˆj+z2ˆk

Answer

div(F)2(x+y+z)

5. F(x,y,z)=(xy)ˆi+(yz)ˆj+(zx)ˆk

6. F(x,y)=xx2+y2ˆi+yx2+y2ˆj

Answer

div(F)=1x2+y2

8. F(x,y)=xˆiyˆj

9. F(x,y,z)=axˆi+byˆj+cˆk for constants a, b, c

Answer

div(F)=a+b

10. F(x,y,z)=xyzˆi+x2y2z2ˆj+y2z3ˆk

11. F(x,y,z)=xyˆi+yzˆj+xzˆk

Answer

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)=ex(cosysiny)

2. w(x,y,z)=(x2+y2+z2)1/2

Answer

Harmonic

Exercise 9.5E.5: CURL

1. If F(x,y,z)=2ˆi+2xˆj+3yˆk and G(x,y,z)=xˆiyˆj+zˆk, find curl(F×G).

2. If F(x,y,z)=2ˆi+2xˆj+3yˆk and G(x,y,z)=xˆiyˆj+zˆk, find div(F×G).

Answer

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

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

curlr=0

b) Find the curlrr.

c) Find the curlrr3.

Answer

curlrr3=0

7. Let F(x,y)=yˆi+xˆjx2+y2, whereFis 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+lnx2+y2ˆj+ˆk

Answer

curl(F)=2xx2+y2ˆk

2. [T] F(x,y,z)=sin(xy)ˆi+sin(yz)ˆj+sin(zx)ˆk

Exercise 9.5E.7: Divergence

For the following exercises, find the divergence ofFat the given point.

1. F(x,y,z)=ˆi+ˆj+ˆk at (2,1,3)

Answer

div(F)=0

2. F(x,y,z)=xyzˆi+yˆj+zˆk at (1,2,3)

3. F(x,y,z)=exyˆi+exzˆj+eyzˆk at (3,2,0)

Answer

divF=22e6

4. F(x,y,z)=xyzˆi+yˆj+zˆk at (1,2,1)

5. F(x,y,z)=exsinyˆiexcosyˆj at (0,0,3)

Answer

div(F)=0

Exercise 9.5E.8: CURL

For the following exercises, find the curl ofFat 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

curl(F)=2ˆj3ˆk

3. F(x,y,z)=exyˆi+exzˆj+eyzˆk at (3,2,0)

4. F(x,y,z)=xyzˆi+yˆj+zˆk at (1,2,1)

Answer

curl(F)=2ˆjˆk

5. F(x,y,z)=exsinyˆiexcosyˆj 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 isFconservative?

Answer

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)}, isFconservative?

2. Given vector field F(x,y)=1x2+y2(x,y) on domain D=R2{(0,0)}, isFconservative?

Answer

F is conservative.

3. Find the work done by force field F(x,y)=eyˆixeyˆj 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)ˆjxyzˆk.

Answer

div(F)=coshx+sinhyxy

2. Compute curl(F)=(sinhx)ˆi+(coshy)ˆjxyzˆk.

Answer

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 three dimensional diagram of an object rotating about the x axis in a counterclockwise manner with constant angular velocity w = <a,b,c>. The object is roughly a sphere with pointed ends on the x axis, which cuts it in half. An arrow r is drawn from (0,0,0) to P(x,y,z) and down from P(x,y,z) to the x axis.

a) ExpressF in terms of ˆi,ˆj, and ˆk vectors.

Answer

(bzcy)ˆi+(cxaz)ˆj+(aybx)ˆk

b) Find divF.

c) Find curlF

Answer

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

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=kT, 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=kT=k2T.

a) Compute the heat flow vector field.

b) Compute the divergence.

Answer

F=200k[1+2(x2+y2+z2)]ex2+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.

A three dimensional diagram of a rotational velocity field. The arrows are showing a rotation in a clockwise manner. A paddlewheel is shown in plan x + y + z = 1 with n extended out perpendicular to the plane.

Glossary

curl

the curl of vector field F=P,Q,R, denoted ×F, is the “determinant” of the matrix |iˆjˆkxyzPQR| and is given by the expression (RyQz)ˆi+(PzRx)ˆj+(QxPy)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


This page titled 9.5E: EXERCISES is shared under a not declared license and was authored, remixed, and/or curated by Pamini Thangarajah.

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