By Michel Cessenat

Presents the reader with easy instruments to unravel difficulties of electromagnetism in thier ordinary practical frameworks due to smooth mathematical equipment: imperative floor equipment, and in addition semigroups, variational tools, and so forth. good tailored to a numerical procedure. As examples of purposes of those instruments and ideas, we resolve numerous basic difficulties of electromagentism, desk bound or time-dependent: scattering of an incident wave via a drawback, bounded or no longer, by means of gratings; wave propogation in a waveguide, with junctions and cascades. it's the goal to offer mathematical notions that may let a greater realizing of modelization in electromagnetism and emphasize the essentail beneficial properties regarding the geometery and nature of fabrics.

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**Additional info for Mathematical Methods in Electromagnetism: Linear Theory and Applications**

**Example text**

3 . DIFFERENTIAL OPERATORS ON A "REGULAR" SURFACE T We define here some differential operators on a surface T in Rn, with fairly weak regularity hypotheses, that is T Lipschitzian. But we often have to suppose more regularity, for example that r is Cl,ot, for 0 < a < 1 at least on pieces Ij whose union is T (see App. 1 for these notations). The ideas on differential operators are in fact more natural from the point of view of differential geometry, so we refer to the appendix at the end of the book.

We denote by: (68) IT 1 7 2 ^) the dual space of H 172 ^), (H^ 2 ^))' the dual space of H*' 2 ^), that is also: (68)' ( H ^ ) ) ' = {f = f0 + fx, f0 € H" 1 ' 2 ^), p1/2fj € L2(TX)}. Let T2 = I\f j be the (regular) complementary sheet in T. For every function v on T, we denote by Vj the restriction of v to Tj for i = 1, 2. Lemma 6. We have the following equivalences: I i) w € H1/2(r)~Wi € H1/2(I\), i = 1, 2, and w{ - R > 2 € H ^ f l U (69) ii) f € H - 1 / 2 ( D ^ fi€(H*/2(ri))\ i= 1, 2, a n d V i +f 2 €H" 1/2 (r 2 ), (orf^R^cH"172^)), ^AR^^R^R;1.

F o r tangent vector fields (or 1-forms) we have matching properties similar to the scalar case i n the framework of H spaces, as i n Lemma 6; this may be verified on each component of the field. F r o m now on, we have to specify the behavior of "fields" under orientation change, so using the language of differential geometry (see the Appendix) we consider the elements of space H ~ ( d i v , r ) as odd 1-currents (or also as generalized odd 1-forms), and the elements of H " " ( c u r l , r ) as (generalized) even 1-forms.