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Lecture XXVII
Stokes’s Theorem
In the previous lecture, we saw how Green’s theorem deals with integrals on
elementary regions in
E
2
and their boundaries. We also saw how the divergence
thoerem deals with elementary regions in
E
3
and their boundaries. We will now
look at Stokes’s theorem, which applies to integrals on elementary surfaces in
E
3
and their boundaries.
First let us extend the concept of rotational derivatives to
E
3
.
Defnition 1
Let
F
be a vector feld with a domain
D
in
E
3
, and let
P
be an
interior point oF
D
.
Given a unit vector
ˆ
u
in
E
3
let
M
be the plane that goes
through
P
and is normal to
ˆ
u
. Consider this plane directed, with the direction
given by
ˆ
u
.
±or
a >
0
, let
C
a
be the circle in
M
oF radius
a
and center
P
.
Consider the limit
1
d
lim
F
R
a
0
πa
2
C
a
·
→
IF the limit exists, then it is called the
rotational derivative of
F
in the direction
ˆ
F
ˆ
u
at
P
and it is denoted by
rot

u,P
.
F
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 Two '04
 Duorg
 Math, Integrals

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