Inverse Acoustic and Electromagnetic Scattering Theory by David Colton & Rainer Kress

Author:David Colton & Rainer Kress
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

Integrating this identity over the domain and interchanging differentiation and integration, the Gauss integral theorem yields

(6.6)

Since on we have

straightforward calculations show that

(6.7)

Passing to the limit, with the help of the mean value theorem, it now follows from (6.7) that

and (6.5) is obtained from (6.6).

The case where E and H only belong to is treated by first integrating over parallel surfaces to the boundary of D and then passing to the limit .  □

In the case when E, H solve the Maxwell equations, it is convenient to transform the remaining boundary terms as described in the following theorem.

Theorem 6.2.

Let D be as in Theorem 6.1 and let be a solution to the Maxwell equations

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