Divergence, Curl, Line Integrals ((c) of M55 Upd)

Mathematics Second Long Exam 55

Exercises

I. Divergence and Curl

1. Compute for the curl and divergence of the following vector fields.

(a) F(x,y,z) = (b) G(x,y,z)=(x - y)

ˆ i + (x + z)

ˆ j + (y - z)

k ˆ

(c) H(x,y,z) = (d) F(x,y,z) = (siny + 3z)

k ˆ

(e) G(x,y,z) =

2. Using the fact that we can write F(x, y) = as F(x,y,z) = , compute for the

divergence and curl of the following:

(a) F(x, y) =

ˆ i + (ex - y + 2z)

ˆ j + (3x + 2y + z)

〈

-

〉

(b) G(x, y)=2x

x y

,

x 1

ˆ i + 3y

ˆ j (c) H(x, y) = <3x2y,-2xy3>

II. Conservative Vector Fields

1. Determine if the following vector fields are conservative. If yes, find a potential function for

them.

(a) F(x, y) = <2x + y,x + 2y - 2> (b) F(x, y) = (c) F(x,y,z) = <6xy sinz + 2x,3x2 sinz + 5,3x2y cosz> (d) F(x,y,z) = (e) F(x,y,z) =

III. Triple Integrals

1. Evaluate

∫

1

0

∫ √

1−x2

0

∫

1−x2−y2

0

(1 + x2 + y2) dx dy dz using cylindrical coordinates.

2. Let ∫∫∫

S

S ((x2 be + the y2) solid dV .

bounded by the paraboloid z = 1 - x2 - y2 and the xy-plane. Evaluate

3. (a) Write the equation of the cone z =

√

3(x2 + y2) spherical coordinates. (b) Set-up the integral in spherical coordinates that gives the volume of the region inside the

sphere x2 + y2 + z2 = 1 and above the cone in the previous item.

4. Find the volume of the solid bounded by the following surfaces: S

1

: z2 - x2 - y2 = 0, S

2

: z2 - x2 - y2 = 1, and S

3

: z = 5

5. Set-up a triple integral in spherical coordinates xy-plane that lies within the sphere x2 + y2 + z2 that = 4 and gives below the volume the cone of z the =

√

solid 3x2 + above 3y2.

the

6. Evaluate using cylindrical coordinates

∫

2

−2

∫ √

4−z2

∫

5−y

0

x2+y2

dz dy dx

7. Evaluate

∫

2

∫

/

4−y2

0

0

∫

/

/

x2+y2

8−x2−y2

x2 + y2 1

+ z2

dz dx dy using spherical coordinates.

8. Set-up the iterated triple integral in cylindrical coordinates that gives the volume of the solid

in the first octant bounded by the cylinder x2 + y2 = 1 and the plane x = z.

9. Let coordinates S be the to region evaluate

bounded ∫∫∫

S

√

by x2 the + y2 spheres + z2 ρ = dV .

2, ρ = 4, and the cone φ = π 4

. Use spherical

10. Set-up paraboloid the x2 integral +y2 +z in = cartesian 6 and the form cone that z =

will √

x2 give + y2. the The mass density of the function solid S of bounded S at any by point the

is ρ(x,y,z) = kz.

11. Let Q be the solid bounded by the planes y = 0, x = y, 2x+3z = 6, and the xy-plane. Set-up the iterated triple integral that will calculate the mass of Q if the density at any point (x,y,z) on Q is directly proportional to its distance in the xy-plane.

IV. Vector Fields, Line Integrals, and Green’s Theorem

1. Given F(x, y) =

(

3x2y +

)

ˆ i +

(

x3 -

)

(a) Show that the vector field F is conservative. (b) Evaluate

1 xey

lnx ey

+ 2y

∫

C

F · dr where C is any path from (e,0) to (1,1).

2. Let F(x, C y) be = y

the ˆ

i + part x

ˆ j. Evaluate

of the parabola ∫

C

F y = · T dr.

x2 from the point (0,0) to the point (1,1), and let

3. Evaluate the line integral given below where C is the circle centered at (2,3) of radius 2.

∮

(6y + x) dx + (y + 2x) dy

4. Given F(x, y)=(ex lny + cosxcosy)

(

ex y

)

ˆ j.

(a) Show that F is a conservative vector field. (b) Evaluate

ˆ i +

- sinxsiny

∫

C

F · dr, where C is a path from (π 2

,1) to (0,e).

5. Use a line integral to find the area of the region enclosed by the ellipse

x2 9

+

y2 4

= 1

6. Given F(x, y) =

(

lny -

cosx y

)

ˆ i +

(

x y

+

sinx y2

) + 2y

ˆ j.

(a) Show that F is conservative.

∫ (b) Evaluate

C F · 7. Use the Green’s Theorem dr along any path from (0,2) to (π,1).

to evaluate

∮

(ey -3y2) dx+(xey +6xy) dy, where C consists of the line segment from (-1,-1) to (2,2), and the portion of y2 - 2 from (2,2) to (-1,-1).