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Mathematics LibreTexts

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

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

False

4. If curlF=0, then F is a conservative vector field.

Exercise 9E.2

Draw the following vector fields.

1. F(x,y)=12ˆi+2xˆj

Answer

A vector field in two dimensions. All quadrants are shown. The arrows are larger the further from the y axis they become. They point up and to the right for positive x values and down and to the right for negative x values. The further from the y axis they are, the steeper the slope they have.

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+(x2ey)ˆj

Answer

Conservative, f(x,y)=xy2ey

2. F(x,y)=(6xy)ˆi+(3x2yey)ˆj

3. F(x,y)=(2xy+z2)ˆi+(x2+2yz)ˆj+(2xz+y2)ˆk

Answer

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+(2x3xy)dx, along C:y=12x from (0, 0) to (4, 2)

Answer

163

2. Cydx+xy2dy, where C:x=t,y=t1,0t1

3. Sxy2dS, where S is surface z=x2y,0x1,0y4

Answer

A3229(331)

Exercise 9E.5

Find the divergence and curl for the following vector fields.

1. F(x,y,z)=3xyzˆi+xyexˆj3xyˆk

2. F(x,y,z)=exˆi+exyˆjexyzˆk

Answer

Divergence: ex+xexy+xyexyz, curl: xzexyzˆiyzexyzˆj+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

2π

Exercise 9E.7

Use Stokes’ theorem to evaluate ScurlFdS.

1. F(x,y,z)=yˆixˆj+zˆk, where S is the upper half of the unit sphere

2. F(x,y,z)=yˆi+xyzˆj2zxˆk, where S is the upward-facing paraboloid z=x2+y2 lying in cylinder x2+y2=1

Answer

π

Exercise 9E.8

Use the divergence theorem to evaluate SFdS.

1. F(x,y,z)=(x3y)ˆi+(3yex)ˆj+(z+x)ˆk, over cube S defined by 1x1,0y2,0z2

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

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

2(2+π)

3. Find the total mass of a thin sheet in the shape of a hemisphere with radius 2 for z0 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

2π/3


This page titled 9E: Chapter Exercises is shared under a not declared license and was authored, remixed, and/or curated by Pamini Thangarajah.

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