vector

432 Vectors
Thus,
dx
c
=
Similarly, it can be shown that
dx dy
...(1)
R y
dy
c
= dx dy ...(2)
x
On adding (1) and (2), we get
( dx
dy)
=
dx dy
R
Proved.
x
y
Note. Green’s Theorem in vector form
where,
Fdr
= ( F ) ˆ
c
R
k dR
F iˆ ˆj, r xiˆ yj ˆ,
kˆ
is a unit vector along z-axis and dR = dx dy.
Example 79. A vector field F is given by F sin yiˆ x (1 cos y) ˆj.
Evaluate the line integral Fdr
where C is the circular path given by x 2 + y 2 = a 2 .
C
Solution. F sin yiˆ x(1 cos y)
ˆj
Fdr
C
= [sin ˆ (1 cos ) ˆ] ( ˆ ˆ
yi x y j idx jdy)
C
= sin ydx x (1 cos y)
dy
C
On applying Green’s Theorem, we have
( dx dy)
c
=
dx dy
s
x
y
= [(1
cos y) cos y]
dx dy
s
where s is the circular plane surface of radius a.
= dx dy
s
= Area of circle = a 2 . Ans.
2 2
Example 80. Using Green’s Theorem, evaluate ( x ydx x dy),
where c is the boundary
described counter clockwise of the triangle with vertices (0, 0), (1, 0), (1, 1).
(U.P., I Semester, Winter 2003)
Solution. By Green’s Theorem, we have
Y
A
(1, 1)
=
( dx dy)
c
= dx dy
R
x
y
2 2
( x y dx
2
x dy)
= (2 x x ) dxdy
c
1 2
(2 x x ) dx dy =
0 0
x
=
R
1 2
0
(2 x x ) dx[ y]x
1 2
(2 x x )( x ) dx =
0
2 1
=
3 4 = 5
12
Example 81. State and verify Green’s Theorem in the plane for
c
0
1 2 3
(2 x x ) dx =
0
2 2
(0, 0)
(1, 0)
3 4
2 x x
3 4
1
0
Ans.
(3 x –8 y ) dx (4 y – 6 xy)
dy where C is the boundary of the region bounded by x 0, y 0 and 2x – 3y = 6.
(Uttarakhand, I Semester, Dec. 2006)
O
y = x
X