Electromagnetic Boundary Problems

7    Integral Equations in Scattering Problems

7.1    GREEN’S THEOREM AND GREEN’S FUNCTIONS

7.1.1    SCALAR PROBLEMS

The Lorentz reciprocity theorem is representative of a type of reciprocity relationship that holds for scalar, vector, or dyadic functions satisfying certain kinds of governing differential equations. For example, let f and g be some scalar functions (or generalized functions) of r. Consider the expression

$\nabla \cdot \left(f\nabla g-g\nabla f\right)=f{\nabla }^{2}g-g{\nabla }^{2}f$

(7.1)

which is true by elementary vector identities. Integrating this over a volume V bounded by a closed surface S (Figure 7.1) and applying the divergence theorem in the usual way, we obtain Green’s theorem:

${\displaystyle {\oint }_S\left(f\nabla g-g\nabla f\right)\cdot {\mathbf{u}}_{n}dS={\displaystyle {\int }_V\left(f{\nabla }^{2}g-g{\nabla }^{2}f\right)dV}}$

(7.2)

Suppose, moreover, that the functions ...