Introduction to Tensor Analysis and the Calculus of Moving Surfaces
(Sprache: Englisch)
Tensor calculus is a must for researchers dealing with natural phenomena as well as for highly qualified engineers working with man-made technological equipment. This textbook was conceived as a self-contained introduction to tensor calculus and grown out...
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Produktinformationen zu „Introduction to Tensor Analysis and the Calculus of Moving Surfaces “
Tensor calculus is a must for researchers dealing with natural phenomena as well as for highly qualified engineers working with man-made technological equipment. This textbook was conceived as a self-contained introduction to tensor calculus and grown out of the lecture notes for a course continually taught by the author at Drexel University. The text contains over 150 exercises and is divided into four parts.
The first part focuses on tensors in Euclidean spaces, and includes chapters on subjects such as covariant differentiation. The second part focuses on tensors in embedded surfaces, and includes chapters on the curvature tensor and Gauss's theorem. The third part covers applications of tensor calculus and contains chapters on equations of classical mechanics, equations of continuum mechanics, and Einstein's Theory of Relativity. The fourth and final chapter explains the rules and applications of the calculus of moving surfaces.
Though the book's approach is informal and avoids a formalization of the subject, it maintains a respectable level of rigor and reflects the author's deep passion for the subject. Moreover, it focuses on concrete objects and appeals to the reader's intuition with regard to such fundamental concepts as the Euclidean space, surface, and length. It is intended for advanced undergraduate students (and first-year graduate students) in technical fields, and assumes a solid understanding of linear algebra and multivariable calculus.
The first part focuses on tensors in Euclidean spaces, and includes chapters on subjects such as covariant differentiation. The second part focuses on tensors in embedded surfaces, and includes chapters on the curvature tensor and Gauss's theorem. The third part covers applications of tensor calculus and contains chapters on equations of classical mechanics, equations of continuum mechanics, and Einstein's Theory of Relativity. The fourth and final chapter explains the rules and applications of the calculus of moving surfaces.
Though the book's approach is informal and avoids a formalization of the subject, it maintains a respectable level of rigor and reflects the author's deep passion for the subject. Moreover, it focuses on concrete objects and appeals to the reader's intuition with regard to such fundamental concepts as the Euclidean space, surface, and length. It is intended for advanced undergraduate students (and first-year graduate students) in technical fields, and assumes a solid understanding of linear algebra and multivariable calculus.
Klappentext zu „Introduction to Tensor Analysis and the Calculus of Moving Surfaces “
This textbook is distinguished from other texts on the subject by the depth of the presentation and the discussion of the calculus of moving surfaces, which is an extension of tensor calculus to deforming manifolds.Designed for advanced undergraduate and graduate students, this text invites its audience to take a fresh look at previously learned material through the prism of tensor calculus. Once the framework is mastered, the student is introduced to new material which includes differential geometry on manifolds, shape optimization, boundary perturbation and dynamic fluid film equations.
The language of tensors, originally championed by Einstein, is as fundamental as the languages of calculus and linear algebra and is one that every technical scientist ought to speak. The tensor technique, invented at the turn of the 20th century, is now considered classical. Yet, as the author shows, it remains remarkably vital and relevant. The author's skilled lecturing capabilities are evident by the inclusion of insightful examples and a plethora of exercises. A great deal of material is devoted to the geometric fundamentals, the mechanics of change of variables, the proper use of the tensor notation and the discussion of the interplay between algebra and geometry. The early chapters have many words and few equations. The definition of a tensor comes only in Chapter 6 - when the reader is ready for it. While this text maintains a consistent level of rigor, it takes great care to avoid formalizing the subject.
The last part of the textbook is devoted to the Calculus of Moving Surfaces. It is the first textbook exposition of this important technique and is one of the gems of this text. A number of exciting applications of the calculus are presented including shape optimization, boundary perturbation of boundary value problems and dynamic fluid film equations developed by the author in recent years. Furthermore, the
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moving surfaces framework is used to offer new derivations of classical results such as the geodesic equation and the celebrated Gauss-Bonnet theorem.
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Inhaltsverzeichnis zu „Introduction to Tensor Analysis and the Calculus of Moving Surfaces “
Preface.- Why Tensor Calculus?.- 1. Rules of the Game.- 2. Coordinate Systems and the Role of Tensor Calculus.- 3. Change of Coordinates.- 4. Tensor Description of Euclidean Spaces.- 5. The Tensor Property.- 6. Covariant Differentiation.- 7. Determinants and the Levi-Civita Symbol.- 8. Tensor Description of Surfaces.- 9. Covariant Derivative of Tensors with Surface Indices.- 10. The Curvature Tensor.- 11. Covariant Derivative of Tensors with Spatial Indices.- 12. Integration and Gauss's Theorem.- 13. Intrinsic Features of Embedded Surfaces.- 14. Further Topics in Differential Geometry.- 15. Classical Problems in the Calculus of Variations.- 16. Equations of Classical Mechanics.- 17. Equations of Continuum Mechanics.- 18. Einstein's Theory of Relativity.- 19. The Rules of Calculus of Moving Surfaces.- 20. Applications of the Calculus of Moving Surfaces.
Autoren-Porträt von Pavel Grinfeld
Pavel Grinfeld is currently a professor of mathematics at Drexel University, teaching courses in linear algebra, tensor analysis, numerical computation, and financial mathematics. Drexel is interested in recording Grinfeld's lectures on tensor calculus and his course is becoming increasingly popular. The author has published in a number of journals including 'Journal of Geometry and Symmetry in Physics' and 'Numerical Functional Analysis and Optimization'. Grinfeld received his PhD from MIT under Gilbert Strang.
Bibliographische Angaben
- Autor: Pavel Grinfeld
- 2013, 2013, XIII, 302 Seiten, 4 farbige Abbildungen, Masse: 16 x 24,1 cm, Gebunden, Englisch
- Verlag: Springer, Berlin
- ISBN-10: 1461478669
- ISBN-13: 9781461478669
- Erscheinungsdatum: 14.10.2013
Sprache:
Englisch
Pressezitat
From the book reviews:"The textbook is meant for advanced undergraduate and graduate audiences. It is a common language among scientists and can help students from technical fields see their respective fields in a new and exiting way." (Maido Rahula, zbMATH, Vol. 1300, 2015)"This book attempts to give careful attention to the advice of both Cartan and Weyl and to present a clear geometric picture along with an effective and elegant analytical technique ... . it should be emphasized that this book deepens its readers' understanding of vector calculus, differential geometry, and related subjects in applied mathematics. Both undergraduate and graduate students have a chance to take a fresh look at previously learned material through the prism of tensor calculus." (Andrew Bucki, Mathematical Reviews, November, 2014)
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