vector

436 Vectors
8. Evaluate F . nds ˆ. ,
s
and s is the region of the plane 2x + 2y + z = 6
where 2
F xyiˆ x ˆ j ( x z)
kˆ
in the first octant. (A.M.I.E.T.E., Summer 2004, Winter 2001) Ans. 27 4
9. Verify Green’s Theorem for
2 2
( xy y ) dx x dy
C
where C is the boundary by y = x and y = x 2 .
(AMIETE, June 2010)
5.38 STOKE’S THEOREM (Relation between Line Integral and Surface Integral)
(Uttarakhand, I Sem. 2008, U.P., Ist Semester, Dec. 2006)
Statement. Surface integral of the component of curl F along the normal to the surface S,
taken over the surface S bounded by curve C is equal to the line integral of the vector point function
F taken along the closed curve C.
Mathematically
F . d r = curl Fnds ˆ
S
where ˆn = cos î + cos ĵ + cos ˆk is a unit
external normal to any surface ds,
Proof. Let r = xiˆ
yj ˆ
zkˆ
dr = idx ˆ ˆj dy kdz ˆ
F = Fi ˆ
ˆ
1 F ˆ 2 j F3 k
On putting the values of F,
d r in the statement of the theorem
c
( Fi ˆ ˆ ˆ ˆ ˆ ˆ
1 F2 j Fk 3 ) ( idx j dy kdz)
=
i j k
S
x y z
( Fiˆ ˆ ˆ ˆ ˆ ˆ
1 F 2 j F3 k). (cos i cos j cos k)
ds
( F1 dx
F3 F2 1 3
2 1
F2 dy F3
dz)
=
ˆ F F
ˆ
F F
i j kˆ
.
S
y z z x x y
( iˆcos ˆj cos kˆ
cos )
ds
3 2 1 3
2 1
= F
F F F
F F
cos cos cos ds
S y z
z x
x y
...(1)
Let us first prove
F c
1 dx
1 1
= F
F
cos cos ds
S
z
y
...(2)
Let the equation of the surface S be z = g (x, y). The projection of the surface on x – y plane
is region R.
F 1 ( x , y , z ) dx
c
= F 1 [ x , y , g ( x , y )] dx
c
= F1 ( x, y, g)
dx dy [By Green’s Theorem]
R y
F1 F1
g
= dx dy
R
y z y
...(3)
The direction consines of the normal to the surface z = g(x, y) are given by
cos
= cos cos
g
g
1
x
y