vector

Vectors 421
5.33 LINE INTEGRAL
Let y,
F ( x, z)
be a vector function and a curve AB.
Line integral of a vector function F along the curve AB is defined as integral of the component
of F along the tangent to the curve AB.
Component of F along a tangent PT at P
= Dot product of F
and unit vector along PT
dr dr
= F is aunit vector along tangent PT
ds ds
dr
Line integral = F from A to B along the curve
ds
dr
Line integral = F
ds
c
ds
=
F
dr
c
Note (1) Work. If F represents the variable force acting on a particle along arc AB, then the
total work done =
B
F
A
dr
(2) Circulation. If V represents the velocity of a liquid then
V
c
dr is called the circulation
of V round the closed curve c.
If the circulation of V round every closed curve is zero then V is said to be irrotational there.
(3) When the path of integration is a closed curve then notation of integration is in place
of .
22.If 2
[(1 – x) (1 – 2x)] is equal to
(i) 2 (ii) 3 (iii) 4 (iv) 6 (A.M.I.E.T.E., Dec. 2009) Ans. (iii)
23.If R = xi + yj + zk and A
is a constant vector, curl ( A R ) is equal to
(i) R (ii) 2 R (iii) A (iv) 2 A (A.M.I.E.T.E., Dec. 2009) Ans. (iv)
1
24. If r is the distance of a point (x, y, z) from the origin, the value of the expression ˆj grad 2
equals
(i) 2 2 2
3
2 ˆ
( x y z ) ( ˆjz kx)
(ii)
(iii) zero (iv)
2
Example 65. If a force F 2x yiˆ 3xyj
ˆ displaces a particle in the xy-plane from (0, 0) to
(1, 4) along a curve y = 4 x 2 . Find the work done.
Solution. Work done = F
. dr
c r xiˆ yj ˆ
2
=
ˆ ˆ ˆ ˆ
(2 x yi 3 xyj).( dx i dyj)
dr dxiˆdyj
ˆ
=
c
c
2
(2 x y dx 3 xy dy)
3
2 2 2
( x y z ) 2 ( ˆjz
iz ˆ )
3
2 2 2 2 ˆ
( x y z ) ( ˆjy
kx)
(AMIETE, Dec. 2010) Ans. (ii)