Integration theorems over curves

A line integral around a curve C can be related to a double Riemaim integral over the enclosed area S by Green s theorem ... 

Integrate this identity over the region Q of the valley enclosed by the curve L along the top of the ridge, using Gauss theorem in two dimensions,... 

Barrett [50] has interestingly reviewed and compared the properties of the Abelian and non-Abelian Stokes theorems, a review and comparison that makes it clear that the Abelian and non-Abelian Stokes theorems must not be confused [83,95]. The Abelian, or original, Stokes theorem states that if A(x) is a vector field, S is an open, orientable surface, C is the closed curve bounding S, dl is a line element of C, n is the normal to S, and C is traversed in a right-handed (positive direction) relative to n, then the line integral of A is equal to the surface integral over 5 of V x A-n ... 

According to (9.3.17) we then require either that all f be zero or that all Fijn vanish. The first alternative cannot be correct since all the functions except fk may be chosen arbitrarily and fk is absent from the summation over i. This leaves only the alternative that all Fijk = 0. From the earlier discussion involving Eq. (9.2.20) it follows that the Pfaffian dL = dxi is integrable. We have thereby established the necessary condition for the Caratheodory theorem of Section 9.2 to hold. Given the fact that in the neighborhood of a point in phase space other points are inaccessible via solution curves of the form X, dxi — 0, the Pfaffian form is integrable. 

Maxwell s equations can also be put in integral form. Specifically, integrating both sides of each equation, over a volume enclosed by a surface S for the first two equations and over a surface enclosed by a curve C for the last two equations, and using Gauss s theorem or Stokes s theorem (see Appendix G) as appropriate, we find that the equations in polarizable matter take the form... 

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