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Differential Geometry


Introduction to Smooth Manifolds (Graduate Texts in Mathematics, Vol. 218)

Introduction to Smooth Manifolds (Graduate Texts in Mathematics, Vol. 218) Lowest new price: $63.41
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Author: John Lee
Brand: Brand: Springer

This book is an introductory graduate-level textbook on the theory of smooth manifolds. Its goal is to familiarize students with the tools they will need in order to use manifolds in mathematical or scientific research--- smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology, vector fields, flows, foliations, Lie derivatives, Lie groups, Lie algebras, and more. The approach is as concrete as possible, with pictures and intuitive discussions of how one should think geometrically about the abstract concepts, while making full use of the powerful tools that modern mathematics has to offer.

This second edition has been extensively revised and clarified, and the topics have been substantially rearranged. The book now introduces the two most important analytic tools, the rank theorem and the fundamental theorem on flows, much earlier so that they can be used throughout the book. A few new topics have been added, notably Sard’s theorem and transversality, a proof that infinitesimal Lie group actions generate global group actions, a more thorough study of first-order partial differential equations, a brief treatment of degree theory for smooth maps between compact manifolds, and an introduction to contact structures.

Prerequisites include a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis.

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Geometry Part 1 (Quickstudy: Academic)

Geometry Part 1 (Quickstudy: Academic) Lowest new price: $1.33
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Author: Inc. BarCharts
Brand: QuickStudy

Coverage of the fundamental structure of geometry.

Part 1 of two guides on geometry.

 

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Geometric Formulas (Quickstudy: Academic)

Geometric Formulas (Quickstudy: Academic) Lowest new price: $2.83
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Author: Inc. BarCharts
Brand: QuickStudy

6-page laminated guide includes:
·general terms
·lines
·line segments
·rays
·angles
·transversal line angles
·polygons
·circles
·theorems & relationships
·postulates
·geometric formulas

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Geometry Part 2 (Quickstudy: Academic)

Geometry Part 2 (Quickstudy: Academic) Lowest new price: $2.12
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Author: Inc. BarCharts
Brand: QuickStudy
Model: FBA-|283672

Part 2 of our coverage of the fundamental structure of geometry.

 

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Differential Geometry (Dover Books on Mathematics)

Differential Geometry (Dover Books on Mathematics) Lowest new price: $9.80
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Author: Erwin Kreyszig

An introductory textbook on the differential geometry of curves and surfaces in three-dimensional Euclidean space, presented in its simplest, most essential form, but with many explanatory details, figures and examples, and in a manner that conveys the theoretical and practical importance of the different concepts, methods and results involved. With problems at the end of each section, and solutions listed at the end of the book. Includes 99 illustrations.

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Riemannian Geometry

Riemannian Geometry Lowest new price: $30.00
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Author: Manfredo P. do Carmo

Riemannian Geometry is an expanded edition of a highly acclaimed and successful textbook (originally published in Portuguese) for first-year graduate students in mathematics and physics. The author's treatment goes very directly to the basic language of Riemannian geometry and immediately presents some of its most fundamental theorems. It is elementary, assuming only a modest background from readers, making it suitable for a wide variety of students and course structures. Its selection of topics has been deemed "superb" by teachers who have used the text.

A significant feature of the book is its powerful and revealing structure, beginning simply with the definition of a differentiable manifold and ending with one of the most important results in Riemannian geometry, a proof of the Sphere Theorem. The text abounds with basic definitions and theorems, examples, applications, and numerous exercises to test the student's understanding and extend knowledge and insight into the subject. Instructors and students alike will find the work to be a significant contribution to this highly applicable and stimulating subject.

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A Comprehensive Introduction to Differential Geometry, Vol. 1, 3rd Edition

A Comprehensive Introduction to Differential Geometry, Vol. 1, 3rd Edition Lowest new price: $47.50
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Author: Michael Spivak
Brand: Brand: Publish or Perish

Book by Michael Spivak, Spivak, Michael

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Curvature in Mathematics and Physics (Dover Books on Mathematics)

Curvature in Mathematics and Physics (Dover Books on Mathematics) Lowest new price: $11.87
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Author: Shlomo Sternberg

This original text for courses in differential geometry is geared toward advanced undergraduate and graduate majors in math and physics. Based on an advanced class taught by a world-renowned mathematician for more than fifty years, the treatment introduces semi-Riemannian geometry and its principal physical application, Einstein's theory of general relativity, using the Cartan exterior calculus as a principal tool.
Starting with an introduction to the various curvatures associated to a hypersurface embedded in Euclidean space, the text advances to a brief review of the differential and integral calculus on manifolds. A discussion of the fundamental notions of linear connections and their curvatures follows, along with considerations of Levi-Civita's theorem, bi-invariant metrics on a Lie group, Cartan calculations, Gauss's lemma, and variational formulas. Additional topics include the Hopf-Rinow, Myer's, and Frobenius theorems; special and general relativity; connections on principal and associated bundles; the star operator; superconnections; semi-Riemannian submersions; and Petrov types. Prerequisites include linear algebra and advanced calculus, preferably in the language of differential forms.

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Elementary Differential Geometry (Springer Undergraduate Mathematics Series)

Elementary Differential Geometry (Springer Undergraduate Mathematics Series) Lowest new price: $29.66
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Author: A.N. Pressley

Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject. Prerequisites are kept to an absolute minimum – nothing beyond first courses in linear algebra and multivariable calculus – and the most direct and straightforward approach is used throughout.

New features of this revised and expanded second edition include:

  • a chapter on non-Euclidean geometry, a subject that is of great importance in the history of mathematics and crucial in many modern developments. The main results can be reached easily and quickly by making use of the results and techniques developed earlier in the book.
  • Coverage of topics such as: parallel transport and its applications; map colouring; holonomy and Gaussian curvature.
  • Around 200 additional exercises, and a full solutions manual for instructors, available via www.springer.com

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Introduction to Tensor Analysis and the Calculus of Moving Surfaces

Introduction to Tensor Analysis and the Calculus of Moving Surfaces Lowest new price: $50.81
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Author: Pavel Grinfeld

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 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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