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This note presents geometric and physical interpretations of the sufficient condition for a critical point to be a strict relative extremum: f[subscript xx]f[subscript yy] - f[superscript 2][subscript xy] greater than 0. The role of the double derivative f[subscript xy] in this inequality will be highlighted in these interpretations. (Contains 14 figures.)

Here, we address the problem of the scattering of a plane, TM-polarized electromagnetic wave by a two-dimensional rectangularly corrugated surface. The height of the corrugation is measured by a characteristic length D and its period by A. We take the ordering of these scales to be D [similar to] [Lambda] [similar to] [Lambda] where [Lambda] is the wavelength of the incident wave. We apply a radiation condition (strictly valid only in the far-field) in the aperture above a rectangular notch. This approximation allows the determination of an explicit representation of the reflection coefficients, the scattered field, and the field within a corrugation. Numerical results are presented which are in excellent agreement with a published finite difference approximation for diffraction gratings.

A comprehensive treatment of Rayleigh-Schr&oum;'inger perturbation theory for the symmetric matrix eigenvalue problem is furnished with emphasis on the degenerate problem. The treatment is simply based upon theMoore-Penrose pseudoinverse thus distinguishing it from alternative approaches in the literature. In addition to providing a concise matrixtheoretic formulation of this procedure, it also provides for the explicit determination of that stage of the algorithm where each higher-order eigenvector correction becomes fully determined. The theory is built up gradually with each successive stage appended with an illustrative example. [ABSTRACT FROM AUTHOR]