vector

Vectors 437
And dx dy = projection of ds on the xy-plane = ds cos
Putting the values of ds in R.H.S. of (2)
F
z
F
y
1 1
cos cos ds
S
=
=
R
z
cos y
F1 cos F
1 dx dy
=
R
F1 F1
dx dy
cos cos
z
y
cos
F1 g
F1
dx dy
R
z y y
F1 F1
g
= dx dy
R
y z y
...(4)
From (3) and (4), we get
F c
1 = F1 F1
cos cos ds
S
z
y
...(5)
Similarly, F c
2 2 2
= F
F
cos cos ds
S
x
z
...(6)
and F c
3 3 3
= F
F
cos cos ds
S
y
x
...(7)
On adding (5), (6) and (7), we get
( F1 dx F2 dy F3
dz)
c
=
S
F1 F1 F2 F2
cos cos cos cos
z y x z
F3 F3
cos cos ds Proved.
y
x
5.39 ANOTHER METHOD OF PROVING STOKE’S THEOREM
The circulation of vector F around a closed curve C is equal to the flux of
the curve of the vector through the surface S bounded by the curve C.
F
d r
c = curl Fnds curl FdS
S S
Proof : The projection of any curved surface over xy-plane can be treated as kernal of the
surface integral over actual surface
Now,
S
( F)
kˆ
dS
= ( ) ( )
= [( iˆ)( F ˆj) ( ˆj)( Fiˆ)]
dx dy
F i j dx dy
[ kˆ iˆ
ˆj]
S
F F dx dy
= ( y) ( x)
S
S
x
y
Fx
dx Fy
dy [By Green’s theorem]
= [ ]
=
S
[ ˆ ˆ ] ( ˆ ˆ
iFx
jFy
idx j dy)
S
= F
. dr
c
curl F ndS ˆ = . d r .
S
c F
where, F = F iˆ F ˆj F kˆand
dr dx iˆ dyj ˆ dz kˆ
x y z
Example 85. Evaluate by Strokes theorem ( yz dx zx dy xy dz)
where C is the curve
C
x 2 + y 2 = 1, z = y 2 . (M.D.U., Dec 2009)
Solution. Here we have
yz dx zx dy xy dz
= ( yziˆ zxj ˆ xykˆ
).( idx ˆ ˆjdy kdz)