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Publisher:

Dover Publications

Publication Date:

1988

Number of Pages:

240

Format:

Paperback

Edition:

2

Price:

12.95

ISBN:

0486656098

Category:

Textbook

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by , on ]

Thomas F. Banchoff

02/15/2011

Dirk Struik’s *Lectures in Classical Differential Geometry*, first published in 1950, was the text I used in my first course in the subject in 1960. It is one of the main reasons I became a differential geometer. I especially appreciated the images, culled from many different sources, as well as the historical placement of the geometric discoveries. Although I was a bit confused at the time by the many different notations that showed up in the exercises taken from various sources, I realized later that multiple representations are a feature of the development of the subject, giving rise to the comical characterization: “Differential geometry is the study of properties invariant under change of notation.”

The book now listed in MAA Reviews is the second edition, published in 1961. It is even more valuable since it includes an appendix on “The Method of Pfaffians” which more aptly for present-day readers might be titled “Classical Differential Geometry from the Point of View of Differential Forms.” As such, the book now gives an introduction of the subject that leads into modern treatments such as the one preferred by my advisor, Shiing-Shen Chern, in his prize-winning essay on global differential geometry.

This relatively short book contains a concise introduction to a number of classical topics that are still important today. A wealth of exercises help the reader see the historical development of the subject and set the scene for modern investigations. The Dover paperback version makes this resource accessible to any student or school library and its placement on the list of essential books in mathematics is well deserved.

Thomas F. Banchoff is Professor of Mathematics at Brown University and a former president of the MAA.

PREFACE | ||||||||

BIBLIOGRAPHY | ||||||||

CHAPTER 1. CURVES | ||||||||

1-1 | Analytic representation | |||||||

1-2 | "Arc length, tangent " | |||||||

1-3 | Osculating plane | |||||||

1-4 | Curvature | |||||||

1-5 | Torsion | |||||||

1-6 | Formulas of Frenet | |||||||

1-7 | Contact | |||||||

1-8 | Natural equations | |||||||

1-9 | Helices | |||||||

1-10 | General solution of the natural equations | |||||||

1-11 | Evolutes and involutes | |||||||

1-12 | Imaginary curves | |||||||

1-13 | Ovals | |||||||

1-14 | Monge | |||||||

CHAPTER 2. ELEMENTARY THEORY OF SURFACES | ||||||||

2-1 | Analytical representation | |||||||

2-2 | First fundamental form | |||||||

2-3 | "Normal, tangent plane" | |||||||

2-4 | Developable surfaces | |||||||

2-5 | Second fundamental form | |||||||

2-6 | Euler's theorem | |||||||

2-7 | Dupin's indicatrix | |||||||

2-8 | Some surfaces | |||||||

2-9 | A geometrical interpretation of asymptotic and curvature lines | |||||||

2-10 | Conjugate directions | |||||||

2-11 | Triply orthogonal systems of surfaces | |||||||

CHAPTER 3. THE FUNDAMENTAL EQUATIONS | ||||||||

3-1 | Gauss | |||||||

3-2 | The equations of Gauss-Weingarten | |||||||

3-3 | The theorem of Gauss and the equations of Codazzi | |||||||

3-4 | Curvilinear coordinates in space | |||||||

3-5 | Some applications of the Gauss and the Codazzi equations | |||||||

3-6 | The fundamental theorem of surface theory | |||||||

CHAPTER 4. GEOMETRY ON A SURFACE. | ||||||||

4-1 | Geodesic (tangential) curvature | |||||||

4-2 | Geodesics | |||||||

4-3 | Geodesic coordinates | |||||||

4-4 | Geodesics as extremals of a variational problem | |||||||

4-5 | Surfaces of constant curvature | |||||||

4-6 | Rotation surfaces of constant curvature | |||||||

4-7 | Non-Euclidean geometry | |||||||

4-8 | The Gauss-Bonnet theorem | |||||||

CHAPTER 5. SOME SPECIAL SUBJECTS | ||||||||

5-1 | Envelopes | |||||||

5-2 | Conformal mapping | |||||||

5-3 | Isometric and geodesic mapping | |||||||

5-4 | Minimal surfaces | |||||||

5-5 | Ruled surfaces | |||||||

5-6 | lmaginaries in surface theory | |||||||

SOME PROBLEMS AND PROPOSITIONS | ||||||||

APPENDIX: The method of Pfaffians in the theory of curves and surfaces | ||||||||

ANSWERS TO PROBLEMS | ||||||||

INDEX | ||||||||

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