- The document proves Stokes' theorem, which states that the circulation of a vector field F around a closed curve C is equal to the flux of the curl of F through any surface S bounded by C. - It considers a vector function F over a rectangular surface S bounded by the closed curve C. - Using properties of the dot product and differential operators, it evaluates the left and right hand sides of Stokes' theorem for this example, showing they are equal, thereby proving the theorem.

2.  Statement :If S be an open surface bounded by a
closed curve C and F be continuous and
differentiable vector function, then

 


S
C
dS
n
r
d
F

).
F
(
)
.
(

4. )
2
...(
..........
ˆ
.
ˆ
ˆ
.
ˆ
ˆ
.
ˆ
ˆ
ˆ
ˆ
.
ˆ
ˆ
ˆ
ˆ
ˆ
).
ˆ
(
,
)
1
...(
ˆ
).
ˆ
(
ˆ
).
ˆ
(
ˆ
).
ˆ
(
ˆ
)
ˆ
ˆ
ˆ
(
ˆ
.
F
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
F
Let
1
1
1
1
1
1
1
3
2
1
3
2
1
3
2
1
dS
n
k
y
F
n
j
z
F
dS
n
i
F
z
F
j
y
F
k
dS
n
i
F
z
k
y
j
x
i
dS
n
i
F
Consider
dS
n
k
F
dS
n
j
F
dS
n
i
F
dS
n
k
F
j
F
i
F
dS
n
k
z
j
y
i
x
r
k
F
j
F
i
F
S
S
S
S
S
S
S
S






























































































5. k
j
y
i
x
k
z
j
y
i
x
r
ˆ
y)
f(x,
ˆ
ˆ
ˆ
ˆ
ˆ
y),
f(x,
z
be
S
surface
the
of
equation
Let







k
f
j ˆ
y
ˆ
y
r
y,
w.r.t.
ting
Diffrentia






.....(3)
n̂
.
ˆ
y
n̂
.
ˆ
n̂
.
y
r
,
n̂
th
product wi
dot
Taking
k
f
j







6. 0
n̂
.
y
r
S.
surface
the
to
normal
is
n̂
and
l
tangentia
is
y
r





 
)
4
.....(
n̂
.
ˆ
y
y
.
n̂
.
ˆ
y
n̂
.
ˆ
y
dS
n̂
.
ˆ
F
eq.(2),
in
ing
Substituit
y)]
f(x,
z
....[
n̂
.
ˆ
y
n̂
.
ˆ
y
n̂
.
ˆ
n̂
.
ˆ
y
n̂
.
ˆ
0
eq.(3),
in
ing
Substituit
1
1
1
1
S
1
dS
k
F
z
z
F
dS
k
F
k
z
z
F
i
k
z
k
f
j
k
f
j
S
S




























































7.  
y
z
y
y
y
x
G
y
x
f
y
x
z
y
x













G
.
F
F
G
y,
w.r.t
partially
ting
Diffrentia
say
)
,
(
)
,
(
,
,
F
)
,
,
(
F
y)
f(x,
z
is
surface
the
of
Equation
1
1
1
1
dxdy
dS
n
k
dS
n
k
y
dS
n
i
F
S
S






 

ˆ
.
ˆ
then
plane,
xy
on the
dS
of
projection
the
is
dxdy
and
plane
-
on xy
S
of
projction
the
is
R
Let
ˆ
.
ˆ
G
ˆ
).
ˆ
(
(4)
eq
in
ing
Substituit
1

8. theorm)
s
en'
......(Gre
ˆ
).
ˆ
(
Thus,
1
1










C
R
S
Gdx
dxdy
y
G
dS
n
i
F

 
 
 
 















C
S
C
S
C
S
C
S
C
r
d
F
dz
F
dy
F
dx
F
dS
n
i
dz
F
dS
n
k
dy
F
dS
n
j
dx
F
dS
n
i
F
)
.
(
)
(
).
ˆ
F
(
)
(c
in
)
(
and
)
(
),
(
Equation
ing
Substituit
......(
).
ˆ
F
(
......(
).
ˆ
F
(
planes,
zx
and
yz
on
S
surface
the
projecting
by
Similarly,
......(
).
ˆ
F
(
get
we
C,
and
C
curves
both the
for
same
axis
and
C
z)of
y,
point(x,
each
at
of
value
the
as
same
is
C
of
y)
(x,
point
each
at
G
of
value
the
Since
3
2
1
1
3
3
2
2
1
1
1
1
1





9. Verifying R.H.S.

 


S
C
dS
n
r
d
F

).
F
(
)
.
(
By stokes’ theorem,
k
y
y
y
k
j
i
xy
y
x
z
y
x
k
j
i
F
ˆ
4
)
2
2
(
ˆ
)
0
(
ˆ
)
0
(
ˆ
0
2
ˆ
ˆ
ˆ
.
1
2
2
















10. D(-a,b)
A(-a,0)
C’(a,b)
B(a,0)
x=-a x=a
y=0
y=b
y
x

11. )
1
..(
.
4
2
2
4
4
ˆ
.
ˆ
4
dS
n̂
.
F
a.
to
a
-
from
x varies
R,
region
in the
b
to
0
from
y varies
PQ,
strip
vertical
the
Along
plane.
-
in xy
ABCD
rectancle
by the
bounded
region
the
be
R
Let
.
3
.
ˆ
.
n̂
dy
dx
dS
and
ˆ
n̂
plane
-
in xy
ABCD
rectangle
is
S
Surface
.
2
2
2
0
2
0
ab
x
b
dx
y
ydydx
dxdy
k
k
y
dy
dx
k
k
a
a
b
a
a
x
a
a
x
b
y
R
S














 

 



12. ....(2)
.
.
.
.
)
.
(
'
'


 
 




DA
D
C
AB BC
C
r
d
F
r
d
F
r
d
F
r
d
F
r
d
F
0)
to
b
from
x varies
,
const.
x
(keeping
ab
.
a)
-
to
a
from
x varies
,
const.
y
(keeping
2
3
2
.
b)
to
0
from
y varies
,
const.
x
(keeping
ab
.
a)
to
a
-
from
x varies
,
const.
y
(keeping
3
2a
.
2
2
3
'
2
'
3










DA
D
C
BC
AB
r
d
F
ab
a
r
d
F
r
d
F
r
d
F

13. (3),
&
(1)
eqn.
From
)
........(3
4ab
ab
2
3
2
ab
3
2a
.
(2),
.
eq
value
abv
the
ing
Substituit
2
2
2
3
2
3
n







 ab
a
r
d
F
C
verified.
is
theorem
s
strokes'
Hence,
).
F
(
)
.
( 
 


S
C
dS
n
r
d
F
