Differential Forms and Applications - Paperback

Differential Forms and Applications

This textbook by Manfredo P. Do Carmo provides a comprehensive application of differential forms for studying local and global aspects of differential geometry of surfaces. The book is designed to make differential forms accessible and attractive to practical users of mathematics through a straightforward, elementary approach.

Key Mathematical Topics Covered

The text introduces differential forms in a simple, user-friendly manner before building toward more advanced concepts. A brief and elementary introduction to differentiable manifolds provides the necessary foundation for understanding Stokes' theorem, which is presented in its natural mathematical setting as the main theorem of the book.

The applications section develops E. Cartan's method of moving frames to study the local differential geometry of immersed surfaces in R3, as well as the intrinsic geometry of surfaces. This systematic approach culminates in the final chapter, which presents Chern's proof of the Gauss-Bonnet theorem for compact surfaces.

Ideal for Advanced Mathematics Students

Part of Springer's Universitext series, this paperback edition is suited for graduate students and researchers in differential geometry, topology, and mathematical analysis. The book balances theoretical rigor with practical application, making complex concepts in differential topology and surface theory accessible to readers with appropriate mathematical background.

Content Structure

The book progresses logically from foundational concepts to advanced applications. Readers gain understanding of differential forms as computational tools, learn to work with differentiable manifolds, master Stokes' theorem and its applications, and explore the method of moving frames for surface geometry. The text concludes with the elegant proof of the Gauss-Bonnet theorem, demonstrating the power of differential forms in global differential geometry.

About This Edition

Published by Springer in September 1994 as part of the Universitext series, this paperback edition has become a standard reference for students and researchers studying differential geometry and its applications. The clear exposition and systematic development make it valuable both as a textbook and reference work.