Ebook: Elementary Differential Geometry

Curves and surfaces are objects that everyone can see, and many of the questions that can be asked about them are natural and easily understood. Differential geometry is concerned with the precise mathematical formulation of some of these questions. It is a subject that contains some of the most beautiful and profound results in mathematics yet many of these are accessible to higher-level undergraduates.

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

Praise for the first edition:

"The text is nicely illustrated, the definitions are well-motivated and the proofs are particularly well-written and student-friendly…this book would make an excellent text for an undergraduate course, but could also well be used for a reading course, or simply read for pleasure."

Australian Mathematical Society Gazette

"Excellent figures supplement a good account, sprinkled with illustrative examples."

Times Higher Education Supplement

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Curves and surfaces are objects that everyone can see, and many of the questions that can be asked about them are natural and easily understood. Differential geometry is concerned with the precise mathematical formulation of some of these questions. It is a subject that contains some of the most beautiful and profound results in mathematics yet many of these are accessible to higher-level undergraduates.

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

Praise for the first edition:

"The text is nicely illustrated, the definitions are well-motivated and the proofs are particularly well-written and student-friendly…this book would make an excellent text for an undergraduate course, but could also well be used for a reading course, or simply read for pleasure."

Australian Mathematical Society Gazette

"Excellent figures supplement a good account, sprinkled with illustrative examples."

Times Higher Education Supplement

---

Curves and surfaces are objects that everyone can see, and many of the questions that can be asked about them are natural and easily understood. Differential geometry is concerned with the precise mathematical formulation of some of these questions. It is a subject that contains some of the most beautiful and profound results in mathematics yet many of these are accessible to higher-level undergraduates.

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

Praise for the first edition:

"The text is nicely illustrated, the definitions are well-motivated and the proofs are particularly well-written and student-friendly…this book would make an excellent text for an undergraduate course, but could also well be used for a reading course, or simply read for pleasure."

Australian Mathematical Society Gazette

"Excellent figures supplement a good account, sprinkled with illustrative examples."

Times Higher Education Supplement

Content:
Front Matter....Pages i-xi
Curves in the plane and in space....Pages 1-27
How much does a curve curve?....Pages 29-54
Global properties of curves....Pages 55-65
Surfaces in three dimensions....Pages 67-93
Examples of surfaces....Pages 95-119
The first fundamental form....Pages 121-157
Curvature of surfaces....Pages 159-177
Gaussian, mean and principal curvatures....Pages 179-213
Geodesics....Pages 215-245
Gauss’ Theorema Egregium....Pages 247-268
Hyperbolic geometry....Pages 269-304
Minimal surfaces....Pages 305-333
The Gauss–Bonnet theorem....Pages 335-377
Back Matter....Pages 379-473

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Curves and surfaces are objects that everyone can see, and many of the questions that can be asked about them are natural and easily understood. Differential geometry is concerned with the precise mathematical formulation of some of these questions. It is a subject that contains some of the most beautiful and profound results in mathematics yet many of these are accessible to higher-level undergraduates.

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

Praise for the first edition:

"The text is nicely illustrated, the definitions are well-motivated and the proofs are particularly well-written and student-friendly…this book would make an excellent text for an undergraduate course, but could also well be used for a reading course, or simply read for pleasure."

Australian Mathematical Society Gazette

"Excellent figures supplement a good account, sprinkled with illustrative examples."

Times Higher Education Supplement

Content:
Front Matter....Pages i-xi
Curves in the plane and in space....Pages 1-27
How much does a curve curve?....Pages 29-54
Global properties of curves....Pages 55-65
Surfaces in three dimensions....Pages 67-93
Examples of surfaces....Pages 95-119
The first fundamental form....Pages 121-157
Curvature of surfaces....Pages 159-177
Gaussian, mean and principal curvatures....Pages 179-213
Geodesics....Pages 215-245
Gauss’ Theorema Egregium....Pages 247-268
Hyperbolic geometry....Pages 269-304
Minimal surfaces....Pages 305-333
The Gauss–Bonnet theorem....Pages 335-377
Back Matter....Pages 379-473
....