National Institute of Technology Karnataka, Surathkal
Department of Mathematical and Computational Sciences
MA110 - Engineering Mathematics - I
Problem Sheet - 8 (with solutions)
Line Integrals & Vector Fields, Work, Circulation, and Flux & Path Independence, Potential
Functions and Conservative Fields
1. Evaluate
(ANS:
Z
13
)
2
C
( xy + y + z) ds along the curve r(t) = 2ti + tj + (2 − 2t)k, 0 6 t 6 1.
2. Find the integral of f ( x, y, z) = 3x2 + (2xz − y) + z over the stright-line segment from
(0, 0, 0) to (2, 1, 3).
√
(ANS:3 14)
3. Find the work done by force F from (0, 0, 0) to (1, 1, 1) over each of the following paths:
a. The stright-line path C1 : r(t) = ti + tj + tk, 0 6 t 6 1
b. The curve path C2 : r(t) = ti + t2 j + t4 k, 0 6 t 6 1
c. The path C3 ∪ C4 consisting of the line segment from (0, 0, 0) to (1, 1, 0) followed by
the segment from (1, 1, 0) to (1, 1, 1)
(i). F = 3yi + 2xj + 4zk
(ii). F = xyi + yzj + xzk
13
9
9
, c. ,
(ANS:[(i).] a. , b.
2
3
2
17
5
[(ii ).] a.1, b. , c. )
18
6
4. Find the work done by F = 2yi + 3xj + ( x + y)k over the curve r(t) = (cos t)i + (sin t)j +
(t/6)k, 0 6 t 6 2π in the direction of increasing t.
(ANS:π)
5. Evaluate
(ANS:0)
6. Evaluate
Z
C
Z
C
x dx + y dy across the ellipse x2 + 4y2 = 4.
( x − y) dx + ( x + y) dy counterclockwise around the triangle with vertices
(0, 0), (1, 0) and (0, 1).
(ANS:1)
1
7. Find the circulation/flow of the velocity field F = ( x + y)i − ( x2 + y2 )j along each of the
following paths from (1, 0) to (−1, 0) in the xy-plane.
(a) The upper half of the circle x2 + y2 = 1
(b) The line segment from (1, 0) to (−1, 0)
(c) The line segment from (1, 0) to (0, −1) followed by the line segment from (0, −1) to
(−1, 0)
(ANS:( a) −
π
, (b) 0, (c)1.)
2
8. Find the flux of the field F = ( x + y)i − ( x2 + y2 )j outward across the triangle with vertices (1, 0), (0, 1), (−1, 0).
1
(ANS:From (1, 0) to (0, 1) :
3
2
From (0, 1) to (−1, 0) : −
3
1
From (−1, 0) to (1, 0) :
3
)
9. Which fields in the following are conservative, and which are not?
(a) F = yzi + xzj + xyk
(b) F = (y sin z)i + ( x sin z)j + ( xy cos z)k
(c) F = (e x cos y)i − (e x sin y)j + zk
(ANS:(a) Conservative, (b) Conservative, (c) Conservative.)
10. Find a potential function f for the following fields.
(a) F = 2xi + 3yj + 4zk.
(b) F = (2xy − z2 )i + ( x2 + 2yz)j + (y2 − 2zx )k
3y2
(ANS:(a) f ( x, y, z) = x2 +
+ 2z2 + C
2
(b) f ( x, y, z) = (y + z) x + zy + C )
11. Show that the differential forms in the integrals are exact. Then evaluate the integrals.
(a)
(b)
Z (2,3,−6)
(0,0,0)
Z (1,2,3)
(0,0,0)
2x dx + 2y dy + 2z dz
2xy dx + ( x2 − z2 ) dy − 2yz dz
(ANS:( a)49, (b) − 16)
2
x
12. Find a potential function for F = (e x ln y)i + ( ey + sin z)j + (y cos z)k
(ANS:F = ∇(e x ln y + y sin z))
13. (a) How are the constants a, b, and c related if the following differential form is exact?
( ay2 + 2czx ) dx + y(bx + cz) dy + ( ay2 + cx2 ) dz
(b) For what values of b and c will F = (y2 + 2czx )i + y(bx + cz)j + (y2 + cx2 )k be a
gradient field?
(ANS:( a) b = 2a and c = 2a
(b) b = 2 and c = 2)
14. Evaluate
Z
F · dr, where F = ( x2 + y2 )i − 2xyj and the curve C is the rectangle in the
C
xy-plane bounded by y = 0, x = a, y = b, x = 0.
(ANS:−2ab2 )
15. Find the work done when a force F = ( x2 − y2 + x )i − (2xy + y)j moves a particle in the
xy plane from (0, 0) to (1, 1) along the parabola y2 = x.
2
(ANS:− )
3
16. Evaluate
Z
F · dr, where F = xyi + ( x2 + y2 )j and the curve C is the arc of y = x2 − 4
C
from (2, 0) to (4, 12) in the xy-plane.
(ANS:732)
17. If F = yzi + zxj − xyk find
to Q(2, 4, 8).
(ANS:0)
18. Prove that
Z
C
F · dr = −
Z
C
Z
−C
F · dr, where C is given by x = t, y = t2 , z = t3 from P(0, 0, 0)
F · dr.
3