THE THEOREMS
37
2.5.3
Stokes’s
Theorem
Stokes’s Theorem derives from Equation
2.74,
(2.85)
where the path
C
encloses the differential surface
dZ
in the right-hand sense.
The development of Stokes’s theorem follows steps similar to those for Gauss’s
theorem. Equation
2.85
is
applied to two adjacent differential surfaces that have a
common border, as shown in Figure 2.13. The result is
(2.86)
where the path
CI+~
is the closed path surrounding both
dal
and
daz.
The line
integrals along the common edge
of
C1
and C2 cancel exactly. Any number of these
differential areas can be added up to give
an
arbitrary surface
S
and the closed contour
C
which surrounds
S.
The result
is
Stokes’s
Theorem:
(2.87)
There
is
an important consequence of Stokes’s Theorem for vector fields that have
-
zero curl. Such a field can always be derived from a scalar potential. That is to say, if
V
X
x
=
0
everywhere, then there exists a scalar function
a@)
such that
A
=
-v@.
To see this, consider the two points
1
and 2 and two arbitrary paths between them,
Path
A
and Path
B,
as shown in Figure 2.14.
A
closed line integral can be formed
by combining Path
A
and the
reversal
of
path
B.
If
X
A
=
0
everywhere, then
Equation 2.87 lets
us
write
(2.88)
Figure
2.13
The
Sum
of
Two
Differential
Surfaces
