By Irving Adler
This richly special assessment surveys the evolution of geometrical principles and the improvement of the innovations of contemporary geometry from precedent days to the current. themes contain projective, Euclidean, and non-Euclidean geometry in addition to the position of geometry in Newtonian physics, calculus, and relativity. Over a hundred workouts with solutions. 1966 edition.
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Extra resources for A New Look at Geometry (Dover Books on Mathematics)
55) for the skin depth we obtain the approximation o '" (e 2pF/Ne 2w)1/3. Using the condition [ > 0 and this estimation we can find the frequency requirements necessary to observe the anomalous skin effect, w > e 2 pF/Ne 2[3. For good metals we have N '" 1022 cm- 3 , PF'" 10- 19 gm cm 20 1. The Electronic Liquid of Metals S-I, which give us w > 10- 2 /- 3 S-1 cm3 . When the mean free path is of the order of 10-4 cm we find w > 1010 S-I. To analyze the behavior of a metal placed in a high-frequency electromagnetic field one can introduce a new quantity-the so-called surface impedance Z.
The magnetic field as usual is supposed to be directed along the "z" -axis. 17) that for the electron moving along the cyclotron orbit of the maximum diameter, Vy = VF sin Qt and x = R sin Qt. e l lT E· vdt = lelEyvF lT sin(qR sin rU)sinQtdt. 76) To proceed we expand the factor sin(q R sin Qt) in the integrand in Bessel functions =2L 00 sin(qR sin Qt) k=O hk+l(qR)sin[(2k + l)Qt]. 76) gives the result lei loT E· vdt = 2rrleIEyR11(qR). 78) The geometric resonances occur when the Bessel function 11 (q R) gets its maxima.
For a fixed wavelength of the sound the absorption will oscillate as the magnetic field is varied. This phenomenon is known as a geometric resonance. The theory of this effect was proposed by Pippard . Consider an isotropic metal where the projection of a cyclotron path to the plane perpendicular to the magnetic field is a circle. Assume that the transverse ultrasonic wave propagates along the "x" -axis of the chosen coordinate system with a polarization vector along the "y"-axis. The magnetic field as usual is supposed to be directed along the "z" -axis.
A New Look at Geometry (Dover Books on Mathematics) by Irving Adler