Stokes’ theorem relates the integral over an open surface
A to the line integral around the surface’s bounding curve
C. Here, unlike Gauss’ theorem, the inside and outside of
A are not well defined so an arbitrary choice must be made for the direction of the outward normal
n (here it always originates on the outside of
A). Once this choice is made, the unit tangent vector to
C,
t, points in the counterclockwise direction when looking at the outside of
A. The final unit vector,
p, is perpendicular the curve
C and tangent to the surface, so it is perpendicular to
n and
t. Together the three unit vectors form a right-handed system:
t ×
n =
p (see
Figure 2.10). For this geometry, Stokes’ theorem states:
where
s is the arc length of the closed curve
C. This theorem signifies that the surface integral of the curl of a vector field
u is equal to the line integral of
u along the bounding curve of the surface. In fluid mechanics, the right side of
(2.34) is called the
circulation of
u about
C. In addition,
(2.34) can be used to define the curl of a vector through the limit of the circulation about an infinitesimal surface as:
The advantage of integral definitions of field derivatives is that such definitions do not depend on the coordinate system.