By Nadir Jeevanjee
The moment variation of this hugely praised textbook presents an creation to tensors, team idea, and their purposes in classical and quantum physics. either intuitive and rigorous, it goals to demystify tensors via giving the marginally extra summary yet conceptually a lot clearer definition present in the mathematics literature, after which connects this formula to the part formalism of physics calculations. New pedagogical gains, resembling new illustrations, tables, and boxed sections, in addition to extra “invitation” sections that offer obtainable introductions to new fabric, provide elevated visible engagement, readability, and motivation for students.
Part I starts with linear algebraic foundations, follows with the trendy component-free definition of tensors, and concludes with purposes to physics by utilizing tensor items. half II introduces workforce thought, together with summary teams and Lie teams and their linked Lie algebras, then intertwines this fabric with that of half I via introducing illustration conception. Examples and workouts are supplied in every one bankruptcy for reliable perform in utilizing the offered fabric and techniques.
Prerequisites for this article comprise the traditional lower-division arithmetic and physics classes, although huge references are supplied for the inspired pupil who has now not but had those. complicated undergraduate and starting graduate scholars in physics and utilized arithmetic will locate this textbook to be a transparent, concise, and fascinating advent to tensors and groups.
Reviews of the 1st Edition
“[P]hysicist Nadir Jeevanjee has produced a masterly booklet that would aid different physicists comprehend these topics [tensors and teams] as mathematicians comprehend them… From the 1st pages, Jeevanjee exhibits extraordinary ability to find clean, compelling phrases to carry ahead the perception that animates the trendy mathematical view…[W]ith compelling strength and readability, he offers many rigorously worked-out examples and well-chosen particular difficulties… Jeevanjee’s transparent and forceful writing offers generic instances with a freshness that might attract and reassure even a frightened pupil. [This] is a masterpiece of exposition and rationalization that may win credits for even a professional author.”
"Jeevanjee’s [text]is a precious piece of labor on numerous counts, together with its convey pedagogical provider rendered to fledgling physicists and the truth that it does certainly provide natural mathematicians the way to come to phrases with what physicists are asserting with an analogous phrases we use, yet with an ostensibly varied that means. The e-book is so easy to learn, very easy, packed with examples...and workouts, and should do the task the writer desires it to do with style.”
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Extra resources for An Introduction to Tensors and Group Theory for Physicists
12. 27) are linearly independent. 17. 8. ei / D ıi so the f j are just the dual vectors j e described above. A/ D Aij ; these vectors are clearly dual to the Eij and thus form the corresponding dual basis. Œ a; a/ so we clearly cannot yet properly treat its dual, but we would like to point out here that in infinite-dimensions, dual spaces get much more interesting. In finite-dimensions, we saw above that a basis fei g for V induces a dual basis fe i g for V , so in a sense V “looks” very much like V .
Let v; v 0 be two nonzero vectors in V . v 0 jw/ 8 w 2 W , then condition 3 above is violated. 18. w1 ; : : : ; wn / 2 Rn . Define . vjw/ Á n X v i wi : i D1 This is sometimes written as v w. You can check that . 8 is an orthonormal basis. 19. w1 ; : : : ; wn / 2 Cn . Define . 30) i D1 Again, you can check that . 8 is an orthonormal basis. Such inner products on complex vector spaces are sometimes referred to as Hermitian scalar products and are present on every quantum-mechanical vector space.
42) are referred to as the covariant components of v. We see now that the contravariant components of a vector are just its usual components, while its covariant components are actually the components of the associated dual vector v. Q For a dual vector f , the situation is reversed—the covariant components f are its actual components, and the contravariant components are the components of fQ. Since L allows us to turn vectors into dual vectors and vice-versa, we usually don’t bother trying to figure out whether something is “really” a vector or a dual vector; it can be either, depending on which components we use.