In Exercises 57 and 58, verify the Divergence Theorem by evaluating SF N dSm as a

In Exercises 1 and 2, sketch several representative vectors in the vector field. Use a computer algebra system to verify your results.

In Exercises 1 and 2, sketch several representative vectors in the vector field. Use a computer algebra system to verify your results.

In Exercises 3 and 4, find the gradient vector field for the scalar function.

In Exercises 3 and 4, find the gradient vector field for the scalar function.

In Exercises 512, determine if the vector field is conservative. If it is, find a potential function for the vector field.

In Exercises 512, determine if the vector field is conservative. If it is, find a potential function for the vector field.
x, y
y x2 i 1 x j F

In Exercises 512, determine if the vector field is conservative. If it is, find a potential function for the vector field.

In Exercises 512, determine if the vector field is conservative. If it is, find a potential function for the vector field.F
x, y

2y3 sin 2xi 3y2
1 cos 2x j F
x,

In Exercises 512, determine if the vector field is conservative. If it is, find a potential function for the vector field.

In Exercises 512, determine if the vector field is conservative. If it is, find a potential function for the vector field.F
x, y, z
4xy z2i
2x2 6yzj 2xzk F
x, y,

In Exercises 512, determine if the vector field is conservative. If it is, find a potential function for the vector field.F
x, y, z yzi
xzj
xyk y2z2 F

In Exercises 512, determine if the vector field is conservative. If it is, find a potential function for the vector field.F
x, y, z sin z
y i x j k F
x,

In Exercises 1320, find (a) the divergence of the vector field F and (b) the curl of the vector field fF
x, y, z x2 i y2 j z2k F
x

In Exercises 1320, find (a) the divergence of the vector field F and (b) the curl of the vector field f

In Exercises 1320, find (a) the divergence of the vector field F and (b) the curl of the vector field fF
x, y, z
cos y y cos xi
sin x
x sin yj xyzk F
x, y

In Exercises 1320, find (a) the divergence of the vector field F and (b) the curl of the vector field f

In Exercises 1320, find (a) the divergence of the vector field F and (b) the curl of the vector field fF
x, y, z arcsin x i xy2 j yz2k F
x

In Exercises 1320, find (a) the divergence of the vector field F and (b) the curl of the vector field f

In Exercises 1320, find (a) the divergence of the vector field F and (b) the curl of the vector field fF
x, y, z ln
x2 y2i ln
x2 y2j zk F
x, y,

In Exercises 1320, find (a) the divergence of the vector field F and (b) the curl of the vector field fF
x, y, z z x i z y j z2k F
x

In Exercises 2126, evaluate the line integral along the given path(s). C
x2 y2 ds F

In Exercises 2126, evaluate the line integral along the given path(s). C xy ds

In Exercises 2126, evaluate the line integral along the given path(s). C
x2 y2 ds
0

In Exercises 2126, evaluate the line integral along the given path(s).

In Exercises 2126, evaluate the line integral along the given path(s). C
2x
y dx
x 3y dy C:

In Exercises 2126, evaluate the line integral along the given path(s). C
2x
y dx
x 3y dy y 3

In Exercises 27 and 28, use a computer algebra system to evaluate the line integral over the given path.

In Exercises 27 and 28, use a computer algebra system to evaluate the line integral over the given path.

Lateral Surface Area In Exercises 29 and 30, find the lateral surface area over the curve in the -plane and under the surface z
fx, y.

Lateral Surface Area In Exercises 29 and 30, find the lateral surface area over the curve in the -plane and under the surface z
fx, y.

In Exercises 3136, evaluate C F  dr. F
x, y xyi x2 j

In Exercises 3136, evaluate C F  dr.

In Exercises 3136, evaluate C F  dr. F
x, y, z x i yj zk C:

In Exercises 3136, evaluate C F  dr.
x, y, z
2y
zi
z
xj
x
yk C: r
t

In Exercises 3136, evaluate C F  dr.

In Exercises 3136, evaluate C F  dr. F
x, y, z
x2
zi
y2 zj x k

2, 2

In Exercises 37 and 38, use a computer algebra system to evaluate the line integral.In Exercises 37 and 38, use a computer algebra system to evaluate the line integral.

In Exercises 37 and 38, use a computer algebra system to evaluate the line integral. C F  dr

Work Find the work done by the force field along the path from to
4, 8. 3

Work A 20-ton aircraft climbs 2000 feet while making a turn in a circular arc of radius 10 miles. Find the work done by the engines.

In Exercises 41 and 42, evaluate the integral using the Fundamental Theorem of Line Integrals.

In Exercises 41 and 42, evaluate the integral using the Fundamental Theorem of Line Integrals.

Evaluate the line integral (a) (b) (c) Use the Fundamental Theorem of Line Integrals, where is a smooth curve from
1, 1
to 4,2

Area and Centroid Consider the region bounded by the axis and one arch of the cycloid with parametric equations and Use line integrals to find (a) the area of the region and (b) the centroid of the region.

In Exercises 45 50, use Greens Theorem to evaluate the line integral. C y dx 2x dy x

In Exercises 45 50, use Greens Theorem to evaluate the line integral.

In Exercises 45 50, use Greens Theorem to evaluate the line integral. C xy2 dx x2y dy

In Exercises 45 50, use Greens Theorem to evaluate the line integral. C
x2
y2 dx 2xy dy C:

In Exercises 45 50, use Greens Theorem to evaluate the line integral.

In Exercises 45 50, use Greens Theorem to evaluate the line integral. C y2 dx x43 dy y

In Exercises 51 and 52, use a computer algebra system to graph the surface represented by the vector-valued function.

In Exercises 51 and 52, use a computer algebra system to graph the surface represented by the vector-valued function.

Investigation Consider the surface represented by the vectorvalued function Use a computer algebra system to do the following. (a) Graph the surface for and (b) Graph the surface for and (c) Graph the surface for and (d) Graph and identify the space curve for and (e) Approximate the area of the surface graphed in part (b). (f) Approximate the area of the surface graphed in part (c).

Evaluate the surface integral over the surface s r
u, v
u vi
u
vj sin v k S: S where and 0 v .

Use a computer algebra system to graph the surface and approximate the surface integral where is the surface over and 0 0 u 2 v 2

Mass A cone-shaped surface lamina is given by At each point on the density is proportional to the distance between the point and the axis. (a) Sketch the cone-shaped surface. (b) Find the mass of the lamina.

In Exercises 57 and 58, verify the Divergence Theorem by evaluating S F  N dS m as a surface integral and as a triple integral

In Exercises 57 and 58, verify the Divergence Theorem by evaluating S F  N dS m as a surface integral and as a triple integral F
x, y, z x2 i xyj zk Solid region bounded by the coordinate planes and the plane 2x 3y 4z 12 Q:

In Exercises 59 and 60, verify Stokess Theorem by evaluating C F  dr as a line integral and as a double integral.

In Exercises 59 and 60, verify Stokess Theorem by evaluating C F  dr as a line integral and as a double integral.

Prove that it is not possible for a vector field with twicedifferentiable components to have a curl of xi yj zk.