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Differential Geometry and Lie Groups for Physicists de Marián Fecko

Differential Geometry and Lie Groups for Physicists

Autor Marián Fecko
en Limba Engleză Hardback – 11 oct 2006

Recomandăm Differential Geometry and Lie Groups for Physicists ca resursă fundamentală pentru nivelul de licență avansată și masterat, fiind un manual care acoperă puntea dintre calculul matematic standard și formalismul utilizat în fizica teoretică modernă. Marián Fecko propune o abordare care demistifică subiecte complexe precum varietățile, formele diferențiale, conexiunile și geometria simplectică, structurând materia pentru a servi direct înțelegerii mecanicii hamiltoniene, relativității și teoriilor de gauge.

Subliniem dimensiunea pedagogică a volumului publicat de Cambridge University Press. Spre deosebire de tratatele pur matematice, stilul este unul informal și discursiv, punând accent pe intuiție fără a sacrifica rigoarea. Elementul distinctiv major este includerea a peste 1000 de exerciții integrate în text, însoțite de soluții complete sau indicații detaliate, ceea ce transformă cele peste 700 de pagini într-un instrument de lucru activ, nu doar un suport de curs pasiv.

Considerăm această lucrare o alternativă la Geometrical Methods of Mathematical Physics de Bernard F. Schutz pentru cursurile de fizică matematică, cu avantajul unui volum mult mai mare de aplicații practice și o acoperire extinsă a grupurilor Lie și a spinorilor. În timp ce Modern Differential Geometry for Physicists de C. J. Isham păstrează un format de note de curs sintetice, Marián Fecko oferă o expunere mult mai detaliată, ideală pentru studiul individual (self-study) sau pentru studenții care au nevoie de o tranziție mai lină de la analiza matematică elementară la geometria diferențială coordonată-free.

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Specificații

ISBN-13: 9780521845076
ISBN-10: 0521845076
Pagini: 714
Ilustrații: 95 b/w illus. 1100 exercises
Dimensiuni: 176 x 253 x 39 mm
Greutate: 1.54 kg
Ediția:New.
Editura: Cambridge University Press
Colecția Cambridge University Press
Locul publicării:Cambridge, United Kingdom

De ce să citești această carte

Această carte este esențială pentru studenții la fizică sau matematică ce doresc să stăpânească limbajul geometriei moderne. Cititorul câștigă o înțelegere profundă prin practică directă, grație celor 1000 de exerciții rezolvate. Este resursa ideală pentru a trece de la calculul multivariat la studiul avansat al relativității generale și al particulelor elementare.

Descriere scurtă

Differential geometry plays an increasingly important role in modern theoretical physics and applied mathematics. This textbook gives an introduction to geometrical topics useful in theoretical physics and applied mathematics, covering: manifolds, tensor fields, differential forms, connections, symplectic geometry, actions of Lie groups, bundles, spinors, and so on. Written in an informal style, the author places a strong emphasis on developing the understanding of the general theory through more than 1000 simple exercises, with complete solutions or detailed hints. The book will prepare readers for studying modern treatments of Lagrangian and Hamiltonian mechanics, electromagnetism, gauge fields, relativity and gravitation. Differential Geometry and Lie Groups for Physicists is well suited for courses in physics, mathematics and engineering for advanced undergraduate or graduate students, and can also be used for active self-study. The required mathematical background knowledge does not go beyond the level of standard introductory undergraduate mathematics courses.

Cuprins

Introduction; 1. The concept of a manifold; 2. Vector and tensor fields; 3. Mappings of tensors induced by mappings of manifolds; 4. Lie derivative; 5. Exterior algebra; 6. Differential calculus of forms; 7. Integral calculus of forms; 8. Particular cases and applications of Stoke's Theorem; 9. Poincaré Lemma and cohomologies; 10. Lie Groups - basic facts; 11. Differential geometry of Lie Groups; 12. Representations of Lie Groups and Lie Algebras; 13. Actions of Lie Groups and Lie Algebras on manifolds; 14. Hamiltonian mechanics and symplectic manifolds; 15. Parallel transport and linear connection on M; 16. Field theory and the language of forms; 17. Differential geometry on TM and T*M; 18. Hamiltonian and Lagrangian equations; 19. Linear connection and the frame bundle; 20. Connection on a principal G-bundle; 21. Gauge theories and connections; 22. Spinor fields and Dirac operator; Appendices; Bibliography; Index.

Recenzii

Review of the hardback: 'All basic material that is necessary for a young scientist in the field of geometrical formulation of physical theories is included … ordered and represented in a very appropriate manner … with a great respect to the reader. … I truly believe that reading this book will bring a real pleasure to all physically inclined young mathematicians and mathematically inclined young physicists … a very good high level textbook … [I] recommend it to all young scientists being interested in finding correspondence between harmony in the physical world and harmony in geometrical structures. … well written, very well ordered and the exposition is very clear.' Journal of Geometry and Symmetry in Physics
Review of the hardback: 'the contents of this book covers a lot (if not most) of what a theoretical physicist might wish to know about differential geometry and Lie groups. particularly useful may be that the modern formalism is always related to the classical one with tensor indices still mostly used in the physics literature.' American Mathematical Society
Review of the hardback: '… the presentation is almost colloquial and this makes reading rather pleasant. The author has made a concerted effort to give intuitive interpretations of complicated ideas such as: the Lie derivative, tensors, the Hodge star operator, Lie group representations, Hamiltonian and Lagrangian mechanics, parallel transport, connections, curvature, gauge theories, spinors and Dirac operators. This will be much appreciated by students (and even researchers, I think). … an excellent reference for geometers.' Zentralblatt MATH
Review of the hardback: 'The contents of this book cover a lot (if not most) of what a theoretical physicist might wish to know about differential geometry and Lie groups.' Mathematical Reviews Hans-Peter Künzle
Review of the hardback: 'Marian Fecko deftly guides you through the material step-by-step, with all the rigour, but without the pain. When going through the chapters, definition by definition, proof by proof and hint by hint, you get an impression of a caring, experienced (and often quirkily funny, but never boring) tutor who really, really wants you to succeed.' Physics in Canada
Review of the hardback: 'From the point of view of presentation the book has a lot going for it. It is written in a pedagogically discursive, conversational style with numerous workable examples and exercises distributed through the text. … From the UK perspective a student undertaking a level 4 (level 5 in Scotland) or MSc course in theoretical physics would find this book well-pitched to his or her needs.' Contemporary Physics