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This book is devoted to the rst acquaintance with the di erential geometry Therefore it begins with the theory of curves in three-dimensional Euclidean spac E. Then the vectorial analysis in E is stated both in Cartesian and curvilinea coordinates, afterward the theory of surfaces in the space E is considered. The newly fashionable approach starting with the concept of a di erentiabl manifold, to my opinion, is not suitable for the introduction to the subject. I this way too many e orts are spent for to assimilate this rather abstract notio and the rather special methods associated with it, while the the essential conten of the subject is postponed for. | RUSSIAN FEDERAL COMMITTEE FOR HIGHER EDUCATION BASHKIR STATE UNIVERSITY SHARIPOV R. A. COURSE OF DIFFERENTIAL GEOMETRY The Textbook Ufa 1996 2 MSC 97U20 UDC 514.7 Sharipov R. A. Course of Differential Geometry the textbook Publ. of Bashkir State University Ufa 1996. pp. 132. ISBN 5-7477-0129-0. This book is a textbook for the basic course of differential geometry. It is recommended as an introductory material for this subject. In preparing Russian edition of this book I used the computer typesetting on the base of the AmS-TEX package and I used Cyrillic fonts of the Lh-family distributed by the CyrTUG association of Cyrillic TEX users. English edition of this book is also typeset by means of the AmS-TEX package. Referees Mathematics group of Ufa State University for Aircraft and Technology yrATy Prof. V. V. Sokolov Mathematical Institute of Ural Branch of Russian Academy of Sciences MM ypO PAH . Contacts to author. Office Mathematics Department Bashkir State University 32 Frunze street 450074 Ufa Russia Phone 7- 3472 -23-67-18 Fax 7- 3472 -23-67-74 Home 5 Rabochaya street 450003 Ufa Russia Phone 7- 917 -75-55-786 E-mails RSharipov@ic.bashedu.ru r-sharipov@mail.ru ra_ sharipov@lycos.com ra_ sharipov@hotmail.com URL http www.geocities.com r-sharipov ISBN 5-7477-0129-0 English translation Sharipov R.A. 1996 Sharipov R.A. 2004 CONTENTS. CONTENTS. 3. PREFACE. 5. CHAPTER I. CURVES IN THREE-DIMENSIONAL SPACE. 6. x 1. Curves. Methods of defining a curve. Regular and singular points of a curve. 6. x 2. The length integral and the natural parametrization of a curve. 10. x 3. Frenet frame. The dynamics of Frenet frame. Curvature and torsion of a spacial curve. 12. x 4. The curvature center and the curvature radius of a spacial curve. The evolute and the evolvent of a curve. 14. x 5. Curves as trajectories of material points in mechanics. 16. CHAPTER II. ELEMENTS OF VECTORIAL AND TENSORIAL ANALYSIS. 18. x 1. Vectorial and tensorial fields in the space. 18. x 2. Tensor product
