Begin2.DVI

6-54. Evaluate the line integral I=

∫

C

F
·d
r, where F
= 3(x+y)ˆe 1 + 5xy ˆe 2 and Cis

the curve y=x^2 between the points (0 ,0) and (2,4).

6-55. For F
=xˆe 1 + 2xy ˆe 2 +xy ˆe 3 evaluate the line integral

∫

CF
·d
r, where Cis the

curve consisting of the straight-line segments OA +AB, where O is the origin and

A, B are respectively the points (1 , 1 ,0), (1 , 1 ,2).

6-56. For P 1 = (1 , 1 ,1) and P 2 = (2, 3 ,5),evaluate the line integral

I=

∫P 2

P 1

A
·d
r, where A
=yz ˆe 1 +xz ˆe 2 +xy eˆ 3

and the integration is

(a) Along the straight-line joining P 1 and P 2.

(b) Along any other path joining P 1 to P 2.

6-57. Evaluate the line integral

∫

CF
·d
r, where F
=yz ˆe^1 + 2xˆe^2 +yˆe^3 and Cis the

unit circle x^2 +y^2 = 1 lying in the plane z= 2.

6-58. Find the work done in moving a particle in the force field F
=xˆe 1 −zˆe 2 +2 yˆe 3

along the parabola y=x^2 , z = 2 between the points (0 , 0 ,2) and (1 , 2 ,2).

6-59. Find the work done in moving a particle in the force field F
=yˆe 1 −xˆe 2 +zˆe 3

along the straight-line path joining the points (1 , 1 ,1) and (2 , 3 ,5).

6-60. Sketch some level curves φ(x, y ) = kfor the values of kindicated.

(a) φ=4x− 2 y, k =− 2 ,− 1 , 0 , 1 , 2

(b) φ=xy, k =− 2 ,− 1 , 0 , 1 , 2

(c) φ=x^2 +y^2 , k = 0, 1 , 9 , 25

(d) φ=9x^2 + 4y^2 , k = 16, 36 , 64

Give a physical interpretation to your results.

6-61. Sketch the two-dimensional vector fields or their associated field lines.

(a) F
=xˆe 1 −yeˆ 2 (b) F
= 2xeˆ 1 + 2yˆe 2 (c) 
 2 yˆe 1 + 2xˆe 2

6-62.

(a) Show line through (x 0 , y 0 , z 0 )and parallel to vector A
is (
r −
r 0 )×A
=
0

(b) Show line through (x 0 , y 0 , z 0 )and (x 1 , y 1 , z 1 )is given by (
r −
r 0 )×(
r −
r 1 ) =
0

(c) Show line through (x 0 , y 0 , z 0 )and perpendicular to the vectors A
and B
is given

by (
r −
r 0 )×(A
×B
) =
0

(d) Find equation of line through (x 0 , y 0 , z 0 )and perpendicular to plane through the

noncolinear points P 1 ,P 2 and P 3.