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Calculus Volume 3

Review Exercises

Calculus Volume 3Review Exercises

Review Exercises

True or False? Justify your answer with a proof or a counterexample.

427.

Vector field F(x,y)=x2yi+y2xjF(x,y)=x2yi+y2xj is conservative.

428.

For vector field F(x,y)=P(x,y)i+Q(x,y)j,F(x,y)=P(x,y)i+Q(x,y)j, if Py(x,y)=Qx(x,y)Py(x,y)=Qx(x,y) in open region D,D, then DPdx+Qdy=0.DPdx+Qdy=0.

429.

The divergence of a vector field is a vector field.

430.

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

Draw the following vector fields.

431.

F ( x , y ) = 1 2 i + 2 x j F ( x , y ) = 1 2 i + 2 x j

432.

F ( x , y ) = y i + 3 x j x 2 + y 2 F ( x , y ) = y i + 3 x j x 2 + y 2

Are the following the vector fields conservative? If so, find the potential function ff such that F=f.F=f.

433.

F ( x , y ) = y i + ( x 2 e y ) j F ( x , y ) = y i + ( x 2 e y ) j

434.

F ( x , y ) = ( 6 x y ) i + ( 3 x 2 y e y ) j F ( x , y ) = ( 6 x y ) i + ( 3 x 2 y e y ) j

435.

F ( x , y , z ) = ( 2 x y + z 2 ) i + ( x 2 + 2 y z ) j + ( 2 x z + y 2 ) k F ( x , y , z ) = ( 2 x y + z 2 ) i + ( x 2 + 2 y z ) j + ( 2 x z + y 2 ) k

436.

F ( x , y , z ) = ( e x y ) i + ( e x + z ) j + ( e x + y 2 ) k F ( x , y , z ) = ( e x y ) i + ( e x + z ) j + ( e x + y 2 ) k

Evaluate the following integrals.

437.

Cx2dy+(2x3xy)dx,Cx2dy+(2x3xy)dx, along C:y=12xC:y=12x from (0, 0) to (4, 2)

438.

Cydx+xy2dy,Cydx+xy2dy, where C:x=t,y=t1,0t1C:x=t,y=t1,0t1

439.

Sxy2dS,Sxy2dS, where S is surface z=x2y,0x1,0y4z=x2y,0x1,0y4

Find the divergence and curl for the following vector fields.

440.

F ( x , y , z ) = 3 x y z i + x y e z j 3 x y k F ( x , y , z ) = 3 x y z i + x y e z j 3 x y k

441.

F ( x , y , z ) = e x i + e x y j + e x y z k F ( x , y , z ) = e x i + e x y j + e x y z k

Use Green’s theorem to evaluate the following integrals.

442.

C3xydx+2xy2dy,C3xydx+2xy2dy, where C is a square with vertices (0, 0), (0, 2), (2, 2) and (2, 0) oriented counterclockwise.

443.

C3ydx+(x+ey)dy,C3ydx+(x+ey)dy, where C is a circle centered at the origin with radius 3

Use Stokes’ theorem to evaluate ScurlF·dS.ScurlF·dS.

444.

F(x,y,z)=yixj+zk,F(x,y,z)=yixj+zk, where SS is the upper half of the unit sphere

445.

F(x,y,z)=yi+xyzj2zxk,F(x,y,z)=yi+xyzj2zxk, where SS is the upward-facing paraboloid z=x2+y2z=x2+y2 lying in cylinder x2+y2=1x2+y2=1

Use the divergence theorem to evaluate SF·dS.SF·dS.

446.

F(x,y,z)=(x3y)i+(3yex)j+(z+x)k,F(x,y,z)=(x3y)i+(3yex)j+(z+x)k, over cube SS defined by −1x1,−1x1, 0y2,0y2, 0z20z2

447.

F(x,y,z)=(2xy)i+(y2)j+(2z3)k,F(x,y,z)=(2xy)i+(y2)j+(2z3)k, where SS is bounded by paraboloid z=x2+y2z=x2+y2 and plane z=2z=2

448.

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.

449.

Find the total mass of a thin wire in the shape of an upper semicircle with radius 2,2, and a density function of ρ(x,y)=y+x2.ρ(x,y)=y+x2.

450.

Find the total mass of a thin sheet in the shape of a hemisphere with radius 2 for z0z0 with a density function ρ(x,y,z)=x+y+z.ρ(x,y,z)=x+y+z.

451.

Use the divergence theorem to compute the value of the flux integral over the unit sphere with F(x,y,z)=3zi+2yj+2xk.F(x,y,z)=3zi+2yj+2xk.

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