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After looking at your pdf files I see that neither theorem is appropriate for proving that the shape of the surface enscribed by a contour (a loop) is immaterial. I have to apologize to you for going in that direction for so long but i did need to see those pdf pages.Pushoam said:
In fact, either theorem requires the inclusion of a maxwell relation. And here's the problem with that: when dealing with moving media such as the loop of fig. 7.13 the maxwell relations are often irrelevant! The author himself points that out (p. 298 lines 8 and 9).
So, bottom line, I conclude that neither the Stokes nor the Divergence theorem is apposite to proving what he seems to be referring to. Referring again to fig. 7.13, the emf is generated differentially for every segment of the loop dl, so the attached surface is immaterial. The loop of fig. 7.13 is an example of where what I call the "Blv law" is the correct law to invoke, not any of the four maxwell relations: d(emf) = B⋅(dl x v) = (v x B)⋅dl. And so the total emf around the loop is just ∫(v x B)⋅dl. The shape of the surface has nothing to do with this integral!
As an example of where you luck out with maxwell is fig. 7.16. In this case emf = - dΦ/dt (based on maxwell's ∇ x E = - ∂B/∂t plus Stokes) happens to be correct but safer is to use the BLv law: emf = Blv based on the Lorentz law F = qv x B.
