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

Michael Spivak

Hardcover

ISBN 10: 0914098705 ISBN 13: 9780914098706

Publisher: Publish or Perish, 1999

Synopsis

A Comprehensive Introduction to Differential Geometry, Volume One, Third Edition by Michael Spivak is an essential resource for anyone interested in the field of differential geometry. Published by Publish or Perish, Inc. in Houston, Texas, this volume is part of a five-volume series that has been meticulously crafted to bridge the gap between classical and modern differential geometry. This third edition includes significant updates and corrections, along with the inclusion of Gauss' paper in Volume 2. The book provides a thorough introduction to the theory of differentiable manifolds, emphasizing both historical context and modern language. It covers a wide range of topics, including immersion and functions, the tangent bundle, integral curves and vector fields, differential equations, integration of differential forms, inner product on vector spaces, critical points and conditions, exact sequence of complexes and cohomology vector spaces, and Euler characteristic and related theorems. Unique features of this book include detailed explanations and propositions related to integral curves, vector fields, and coordinate systems, comprehensive coverage of solving differential equations and integration of differential forms, in-depth discussion on inner product on vector spaces, including Euclidean and Riemannian metrics, exploration of critical points, variational approach, and geodesics, extensive treatment of exact sequences, cohomology vector spaces, and Euler characteristic, and application of Poincaré duality theorem, Thom class, Euler class, vector fields, and indices. With its rigorous approach and clear presentation, A Comprehensive Introduction to Differential Geometry, Volume One, Third Edition is an invaluable addition to the library of mathematicians, physicists, and anyone with a keen interest in differential geometry.

About this title

About the Author

Michael David Spivak, born on May 25, 1940, is an American mathematician with a profound impact on the field of differential geometry. As an expositor of mathematics, he has skillfully bridged the gap between complex theory and accessible explanations. Spivak’s legacy extends beyond academia—he is also the visionary behind Publish-or-Perish Press. 📚 Author of the Five-Volume Masterpiece: Spivak’s magnum opus, “A Comprehensive Introduction to Differential Geometry,” stands as a testament to his mathematical prowess. This five-volume work is a treasure trove for those seeking a deep understanding of geometric concepts. 🔍 Why Choose Spivak? 🌟 Clarity and Precision: Spivak’s writing style demystifies intricate topics, making them approachable for learners at all levels. 🌟 Problem-Solving Focus: Dive into challenging problems guided by a master mathematician. 🌟 Mathematical Rigor: Immerse yourself in proofs, theorems, and applications—the hallmark of Spivak’s approach. 📖 About the Mathematician: Hailing from Queens, New York, Spivak earned his Ph.D. from Princeton University. His contributions have left an indelible mark on mathematics, and his passion for clarity continues to inspire generations of learners. 🛒 Explore Spivak’s Work: Whether you’re a student, educator, or lifelong learner, delve into the intricacies of calculus, geometry, and mathematical reasoning with Michael Spivak. 📚🔢

Excerpt. © Reprinted by permission. All rights reserved.

The third edition of A Comprehensive Introduction to Differential Geometry, Volume One by Michael Spivak includes significant updates and corrections, along with the inclusion of Gauss' paper translation in Volume 2. The book begins with a preface that discusses the evolution from mimeographed notes to a comprehensive book, highlighting the challenges in understanding differential geometry due to the gap between classical and modern treatments. The first volume focuses on differentiable manifolds, while the second volume covers fundamental papers by Gauss and Riemann. The content includes chapters on manifolds, differential structures, the tangent bundle, tensors, vector fields and differential equations, integral manifolds, differential forms, integration, Riemannian metrics, Lie groups, and an excursion in algebraic topology. The text emphasizes the importance of understanding the geometric aspect of differential geometry through historical development and presents major approaches in the field. The document section discusses advanced concepts in differential geometry, focusing on vector bundles, particularly the tangent bundle (TM) and cotangent bundle (TM) of a manifold M. Key points include the definition and properties of vector bundles, including the tangent bundle TM and cotangent bundle TM, construction of dual bundles and their properties, local triviality and equivalence of vector bundles, and sections of TM (vector fields) and their properties. The document also covers the properties and applications of differential forms, focusing on alternating tensors, wedge products, and the exterior derivative. It provides a detailed exploration of differential forms, their algebraic properties, and their significance in differential geometry and topology. The document concludes with a discussion on the integration in polar coordinates and the calculation of de Rham cohomology vector spaces for specific manifolds.

Edition details

• Publisher: Publish or Perish
• Publication date: 1999
• ISBN 10: 0914098705
• ISBN 13: 9780914098706
• Binding: Hardcover
• Language: English
• Edition number: 3
• Number of pages: 509