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This book gives an introduction to the basic concepts which are used in differential topology, differential geometry, and differential equations. A certain number of concepts are essential for all three of these areas, and are so basic and elementary, that it is worthwhile to collect them together so that more advanced expositions can be given without having to start from the very beginning. The concepts are concerned with the general basic theory of differential manifolds. As a result, this book can be viewed as a prerequisite to Fundamentals of Differential Geometry. Since this book is intended as a text to follow advanced calculus, manifolds are assumed finite dimensional. In the new edition of this book, the author has made numerous corrections to the text and he has added a chapter on applications of Stokes' Theorem.
List of contents
Differential Calculus.- Manifolds.- Vector Bundles.- Vector Fields and Differential Equations.- Operations on Vector Fields and Differential Forms.- The Theorem of Frobenius.- Metrics.- Integration of Differential Forms.- Stokes' Theorem.- Applications of Stokes' Theorem.
Summary
This book gives an introduction to the basic concepts which are used in differential topology, differential geometry, and differential equations. The author's book, Fundamentals of Differential Geometry, can be viewed as a continuation of the present book. Since this book is intended as a text to follow advanced calculus, manifolds are assumed finite dimensional. The author has made numerous corrections to this new edition, and he has also added a chapter on applications of Stokes' Theorem.
Additional text
From the reviews:
"This volume is an introduction to differential manifolds which is intended for post-graduate or advanced undergraduate students. … Basic concepts are presented, which are used in differential topology, differential geometry, and differential equations. Charts are used systematically … . The book is well readable, and it is of interest not only for mathematicians, but also for theory-oriented researchers in applied sciences, who need an introduction to this important topic." (I. Troch, Internationale Mathematische Nachrichten, Issue 196, 2004)
"The author recommends his text to ‘the first year graduate level or advanced undergraduate level’ … . his explanation is very precise, with rich formalism and with maximum generality … . In summary, this is an ideal text for people who like a more general and abstract approach to the topic." (EMS, June, 2003)
"The book offers a quick introduction to basic concepts which are used in differential topology, differential geometry and differential equations. … The bibliography contains important new titles in studying differential geometry. A large index is also included. This is an interesting Universitext (for students – the first year graduate level or advanced undergraduate level), with important concepts concerning the general basic theory of differential manifolds." (Corina Mohorianu, Zentralblatt MATH, Vol. 1008, 2003)
Report
From the reviews:
"This volume is an introduction to differential manifolds which is intended for post-graduate or advanced undergraduate students. ... Basic concepts are presented, which are used in differential topology, differential geometry, and differential equations. Charts are used systematically ... . The book is well readable, and it is of interest not only for mathematicians, but also for theory-oriented researchers in applied sciences, who need an introduction to this important topic." (I. Troch, Internationale Mathematische Nachrichten, Issue 196, 2004)
"The author recommends his text to 'the first year graduate level or advanced undergraduate level' ... . his explanation is very precise, with rich formalism and with maximum generality ... . In summary, this is an ideal text for people who like a more general and abstract approach to the topic." (EMS, June, 2003)
"The book offers a quick introduction to basic concepts which are used in differential topology, differential geometry and differential equations. ... The bibliography contains important new titles in studying differential geometry. A large index is also included. This is an interesting Universitext (for students - the first year graduate level or advanced undergraduate level), with important concepts concerning the general basic theory of differential manifolds." (Corina Mohorianu, Zentralblatt MATH, Vol. 1008, 2003)
