A Generalization of Rolle's Theorem with Application to by Dieudonne J.

By Dieudonne J.

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61) then reduces to A(s) = {~rn Equlphose Surface (n)-Conn,. 13 Geometrical picture of two-dimensional ray tube. 2D fields. Plane waves with appropriate polarizations also will be two-dimensional TE or TM fields. An arbitrarily polarized electric field always can be written as a linear combination of the TE and TM states. These cases therefore can be dealt with separately, and the advantage gained is that the problem becomes scalar. 8. For convenience, we stress again the complete set of conventions here, and then immediately study some instructive examples.

We begin at the bottom of the figure and work our way up through the numbered rays singled out for consideration, (of course, there are infinitely many of them). Ray 1 is not reflected from the strip because it does not strike it at all. Ray 2 is the "first" to impact the conducting strip and is reflected in accordance with the law of reflection. , ray 6) will not be reflected, as it travels straight past the strip. Thus, only in the region of space between the reflected rays labeled RSB will there exist a reflected GO field, the latter rays forming the reflection shadow boundaries for this particular obstacle and specific incident wavefront, incident at the given angle with respect to the obstacle.

6 Ray-fixed coordinate system used to define the polarization of the incident field in the GOGTD schema. In addition, given si and s1', we can write A x ii x en = (ii - ii)ei % = (ii - i"pL . 8) This system is a local ray-fixed coordinate system because it is inextricably linked to the ray directions si and sr , and the unit normal A at each Qr. These results from differential geometry will be usefully employed in the remainder of this text. In Chapter 4 an additional geometrical concept, that of the edge-fixed coordinate system, will be introduced.

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