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| The Geometry of physics An introduction |
Description :
which can be read before starting the text. This appears at the beginning of the text, before Chapter 1. The only prerequisites for reading this overview are sophomore courses in calculus and basic linear algebra. Many of the geometric concepts developed in the text are previewed here and these are illustrated by their applications to a single extended problem in engineering, namely the study of the Cauchy stresses created by a small twist of an elastic cylindrical rod about its axis.
The new shortened version of Appendix A, dealing with elasticity, requires the discussion of Cauchy stresses dealt with in the Overview. The author believes that the use of Cartan’s vector valued exterior forms in elasticity is more suitable (both in principle and in computations) than the classical tensor analysis usually employed in
engineering (which is also developed in the text.)
The new version of Appendix A also contains contributions by my engineering colleague Professor Hidenori Murakami, including his treatment of the Truesdell stress rate. I am also very grateful to Professor Murakami for many very helpful conversations.
AVANT PROPOS
This second edition differs mainly in the addition of three new appendices: C, D, and E. Appendices C and D are applications of the elements of representation theory of compact Lie groups.
Appendix C deals with applications to the flavored quark model that revolutionized particle physics. We illustrate how certain observed mesons (pions, kaons, and etas) are described in terms of quarks and how one can “derive” the mass formula of Gell-Mann/Okubo of 1962. This can be read after Section 20.3b.
Appendix D is concerned with isotropic hyperelastic bodies. Here the main result has been used by engineers since the 1850s. My purpose for presenting proofs is that the hypotheses of the Frobenius–Schur theorems of group representations are exactly met here, and so this affords a compelling excuse for developing representation theory, which had not been addressed in the earlier edition. An added bonus is that the group
theoretical material is applied to the three-dimensional rotation group SO(3), where these generalities can be pictured explicitly. This material can essentially be read after Appendix A, but some brief excursion into Appendix C might be helpful. Appendix E delves deeper into the geometry and topology of compact Lie groups.
Bott’s extension of the presentation of Morse theory that was given in Section 14.3c is sketched and the example of the topology of the Lie group U(3) is worked out in some detail.
Titre :The Geometry of physics An introduction
auteur(s) : Theodore Frankel
size : 5.7 Mb
file type : PDF
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