Download PDF by Ross Honsberger: Ingenuity in mathematics

By Ross Honsberger

ISBN-10: 0883856239

ISBN-13: 9780883856239

The 19 essays the following illustrate many various points of mathematical pondering. the writer is especially famous for his best-selling books of difficulties; during this quantity he seeks to percentage his appreciation of the stylish and creative ways utilized in brooding about even uncomplicated arithmetic. normal highschool classes in algebra and geometry provide a enough foundation for knowing each one essay. subject matters contain quantity thought, geometry, combinatorics, common sense and chance, and the equipment used frequently contain an interplay among those disciplines. a number of the essays are effortless to learn, others more difficult; a number of the workouts are regimen, others lead the reader deeper into the topic.

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However, if one considers the superposition of two waves with the same amplitude but slightly different angular frequencies (! ) and (! 12 (a) Schematic diagram of the complete dispersion curve for a material which is transparent to the visible part of the EM spectrum. When dn/ d is positive, the material absorbs strongly. (b) More detailed dispersion curves for some materials often used for lenses and prisms are shown. Note that dn=d is negative over the visible spectrum (and therefore, dn=dk will be positive in this region).

There are many books giving detailed expositions of electromagnetic theory, optics and quantum mechanics (see the bibliography at the end of this chapter). Therefore, for brevity and practicality, the approach taken here will be to present only those concepts, which are relevant to the microscopical techniques discussed later in the book. 2 Electromagnetic waves A mathematical model of a wavelike motion is based upon our commonsense experience of typical waves, for example, ocean waves, ripples on ponds and vibrations on string instruments.

Fortunately, the study of materials using microscopical methods only requires that material structure is explored down to the atomic, nanoscale sizes (10À9 m) rather than the incredibly small String Theory domain (10À20 m). For most purposes, especially research into mesoscale material structure, a spatial resolution of 100 nm will be more than adequate. In order to understand how most optical microscopes work and to use them effectively, one can approximate the ‘true’ nature of light propagation (whatever that might be) by using the simple analogies of classical wave motion and geometrical ray tracing.

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Ingenuity in mathematics by Ross Honsberger

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