href="#ulink_3f1abf4d-261e-5c3e-88fb-61321dc23627">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
Figure 1.18 Illustration of Stokes' theorem.
Figure 1.19 Geometry of Example 1.5.
which leads to
The differential surface area is found from
Then
Now, we calculate the right‐hand side of the equation for each segment as
1 x = 0, z = 0, ,
2 x = 0, y = 1, ,
3 x = 0, z = 1, ,
4 x = 0, y = 0, ,
Hence
As a result, it is confirmed that
for Example 1.5.
1.4 Maxwell's Equations
1.4.1 Differential Forms of Maxwell's Equations
Maxwell's equations in differential forms in the presence of an impressed magnetic current density
where
The continuity
