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.

$∫Cx2dy+(2x−3xy)dx,∫Cx2dy+(2x−3xy)dx,$ along $C:y=12xC:y=12x$ from (0, 0) to (4, 2)

438.

$∫Cydx+xy2dy,∫Cydx+xy2dy,$ where $C:x=t,y=t−1,0≤t≤1C:x=t,y=t−1,0≤t≤1$

439.

$∬Sxy2dS,∬Sxy2dS,$ where S is surface $z=x2−y,0≤x≤1,0≤y≤4z=x2−y,0≤x≤1,0≤y≤4$

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.

$F(x,y,z)=yi−xj+zk,F(x,y,z)=yi−xj+zk,$ where $SS$ is the upper half of the unit sphere

445.

$F(x,y,z)=yi+xyzj−2zxk,F(x,y,z)=yi+xyzj−2zxk,$ 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.

$F(x,y,z)=(x3y)i+(3y−ex)j+(z+x)k,F(x,y,z)=(x3y)i+(3y−ex)j+(z+x)k,$ over cube $SS$ defined by $−1≤x≤1,−1≤x≤1,$ $0≤y≤2,0≤y≤2,$ $0≤z≤20≤z≤2$

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 $z≥0z≥0$ 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 $F(x,y,z)=3zi+2yj+2xk.F(x,y,z)=3zi+2yj+2xk.$