In (1.81), the first three terms are zero, hence (1.81) reduces to
or
From (1.76)
(1.83)
Hence, (1.82) can be written as
(1.84) leads to
(1.85)
The left‐hand side of the equation now can be found as
(1.86)
1.3.6 Stokes' Theorem
Stokes' theorem states that the line integral of a vector around a closed contour is equal to the surface integral of the curl of a vector over an open surface. It is expressed by
This can be illustrated in Figure 1.18. The small interior contours have adjacent sides that are in opposite directions yielding no net line integral contribution, as shown in Figure 1.18 [1]. The net nonzero contribution occurs due to contours with having a side on the open boundary L. Then, the total result of adding the contributions for all the contours can be represented as given in (1.87) by Stokes' theorem.
Example 1.5 Stokes' Theorem
Please verify Stokes' theorem for the geometry given in Figure 1.19 for the vector
Solution
We first calculate the left‐hand side of Eq. (1.87) as