- Tapa blanda: 400 páginas
- Editor: Dover Publications Inc.; Edición: New edition (19 de marzo de 1990)
- Colección: Dover Books on Mathematics
- Idioma: Inglés
- ISBN-10: 0486658406
- ISBN-13: 978-0486658407
- Valoración media de los clientes: 1 opinión de cliente
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nº23.403 en Libros en idiomas extranjeros (Ver el Top 100 en Libros en idiomas extranjeros)
- n.° 19 en Libros en idiomas extranjeros > Ciencias, tecnología y medicina > Matemáticas > Álgebra
- n.° 43 en Libros en idiomas extranjeros > Ciencias, tecnología y medicina > Matemáticas > Cálculo
- n.° 103 en Libros en idiomas extranjeros > Ciencias, tecnología y medicina > Tecnología e ingeniería > Cuestiones generales
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Tensors, Differential Forms and Variational Principles (Dover Books on Mathematics) (Inglés) Tapa blanda – 19 mar 1990
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Incisive, self-contained account of tensor analysis and the calculus of exterior differential forms, interaction between the concept of invariance and the calculus of variations. Emphasis is on analytical techniques, with large number of problems, from routine manipulative exercises to technically difficult assignments.
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The first 238 pages of "Tensors, differential forms, and variational principles", by David Lovelock and Hanno Rund, are metric-free. This book is very heavily into tensor subscripts and superscripts. If you don't like "coordinates", you won't like this book. Here's a round-up of the chapters.
Chapter 1 (17 pages) has some interesting examples which demonstrate how tensors arise naturally, namely the (symmetric) stress tensor in elasticity, the (antisymmetric) inertia tensor for rigid bodies, and cross-product vectors (which arise in electromagnetism). Also discussed are vector components and the properties of determinants.
Chapter 2 (36 pages) presents "affine tensor algebra in Euclidean geometry", which means basic tensor algebra in flat Euclidean spaces, including non-linear coordinate transformations. There's a very interesting explanation of how a metric tensor and Christoffel symbols naturally arise in flat space when parallel vector fields are subjected to non-linear transformations.
Chapter 3 (47 pages) introduces manifolds (using an atlas of charts), including tensor algebra on manifolds and the derivatives of tensor fields, where once again the Christoffel symbol is introduced to make the derivatives tensorial, thereby motivating Christoffel symbols. Then there's more on absolute differentials (i.e. covariant derivatives) of tensor fields, the effects of multiple covariant differentiation (which motivates the definition of Riemannian curvature), parallelism on manifolds, and properties of the Riemannian curvature tensor.
Chapter 4 (29 pages) has some miscellaneous tensor calculus topics, namely scalar densities (with transformation-invariant integrals), normal coordinates, and the Lie derivative.
Chapter 5 (51 pages) is about differential forms, including exterior products, the exterior derivative, Poincaré's lemma, systems of total differential equations, the Stokes theorem, and curvature forms.
Chapter 6 (58 pages) is concerned with "invariant problems in the calculus of variations".
Chapter 7 (59 pages) introduces Riemannian geometry. This includes Finsler spaces and Riemannian and pseudo-Riemannian spaces. Topics include geodesics, Riemannian curvature tensor properties in the presence of a metric, and a divergence theorem for Riemannian manifolds.
Chapter 8 (33 pages) is titled "invariant variational principles and physical field theories". This includes Lagrangians, vector field theory, metric field theory, and Einstein's equations.
The authors have made great efforts to explain and motivate everything.
This is an addition to my previous comment. The author could have used matrix algebra to simply many derivations. The book could be much shorter and easier to read, in my opinion.
One of my pet peeves is that this book certainly lacks structure. It could go on for several tens of pages non-stop, like old fashion Fotran code. There are no boldface titles such as definition, axiom, lemma, theorem, corollary ... Readers have to figure them out in the context.
It's well written in general context. Everything is motivated and follows logically. I think the book is perfect so far.
It presents the "mathematics of GR" in a very procedural manner. In a sense that it is always clear what we are trying to define, why we are doing so, and how to do it. Start by defining then requiring "orthogonal transformations" for all coordinate transformations. Define scalers, vectors, co-vectors, etc. by how they respond under these transformations. Build calculus and whatever else from these definitions. Everything else follows accordingly.
Again, it's an excellent text written with clarity. Definitely a must if looking at GR and tensor analysis for the first time (or 2nd time. Or maybe even 3rd).
The materials seems quite comprehensive, mind you.