Direct integration of field equations

Abstract

Summary form only given. The integral equations of the field equations in electrostatics, magnetostatics, and electrodynamics are derived with the aid of a scalar-vector Green's theorem which can be obtained by several different approaches. One of them is to start with the vector Green's theorem first formulated by Stratton. The steps lead to the scalar-vector Green's theorem is outlined. The integration of the field equations can then be executed readily using that theorem. The results can be interpreted in terms of potential functions if so desired but they are not necessary. In fact, these interpretations are not unique. The second part of this work consists of a review of Stratton and Chu's (1939) famous formula. It is shown that the diffraction integrals obtained by Franz (1948) with the aid of the dyadic Green's functions can be derived readily from Stratton and Chu's formula by means of some simple vector identities. Schelkunoff's (1951) claim that his formulas are stronger than Stratton and Chu's formula is without foundation. In the first place he never demonstrated where his strength is. In fact, it can be shown that his expressions are equivalent to Franz's expressions which are derivable from Stratton and Chu's formula. Finally, it should be pointed out that the line integral deliberately added by Stratton and Chu to compensate for the discontinuity of the fields across the edge of an aperture is really not necessary because this line integral as a result of Meixner's edge condition is a null integral.