Lectures on Classical Differential Geometry: Second Edition

✍ Scribed by Dirk J. Struik

• Publisher: Dover Publications
• Year: 1988
• Tongue: English
• Leaves: 243
• Edition: 2
• Category: Library

✦ Synopsis

Excellent brief introduction presents fundamental theory of curves and surfaces and applies them to a number of examples. Topics include curves, theory of surfaces, fundamental equations, geometry on a surface, envelopes, conformal mapping, minimal surfaces, more. Well-illustrated, with abundant problems and solutions. Bibliography.

✦ Table of Contents

Cover......Page 1
Contents......Page 4
Preface......Page 6
Preface to the Second Edition......Page 7
Bibliography......Page 8
1.1 Analytic representation......Page 12
1.2 Arc length, tangent......Page 16
1.3 Oscula ting plane......Page 21
1.4 Curvature......Page 24
1.5 Torsion......Page 26
1.6 Formulas of Frenet......Page 29
1.7 Contact......Page 34
1.8 Natural equations......Page 37
1.9 Helices......Page 44
1.10 General solution of the natural equations......Page 47
1.11 Evolutes and involutes......Page 50
1.12 Imaginary curves......Page 55
1.13 Ovals......Page 58
1.14 Monge......Page 64
2.1 Analytical representation......Page 66
2.2 First fundamental form......Page 69
2.3 Normal, tangent plane......Page 73
2.4 Developable surfaces......Page 77
2.5 Second fundamental form Meusnier's theorem......Page 84
2.6 Euler's theorem......Page 88
2.7 Dupin's indicatrix......Page 94
2.8 Some surfaces......Page 97
2.9 A geometrical interpretation of asymptotic and curvature lines......Page 104
2.10 Conjugate directions......Page 107
2.11 Triply orthogonal systems of surfaces......Page 110
3.1 Gauss......Page 116
3.2 The equations of Gauss-Weingarten......Page 117
3.3 The theorem of Gauss and the equations of Codazzi......Page 121
3.4 Curvilinear coordinates in space......Page 126
3.5 Some applications of the Gauss and the Codazzi equations......Page 131
3.6 The fundamental theorem of surface theory......Page 135
4.1 Geodesic (tangential) curvature......Page 138
4.2 Geodesics......Page 142
4.3 Geodesic coordinates......Page 147
4.4 Geodesics as extremals of a variational problem......Page 151
4.5 Surfaces of constant curvature......Page 155
4.6 Rotation surfaces of constant curvature......Page 158
4.7 Non-Euclidean geometry......Page 161
4.8 The Gauss-Bonnet theorem......Page 164
5.1 Envelopes......Page 173
5.2 Conformal mapping......Page 179
5.3 Isometric and geodesic mapping......Page 186
5.4 Minirual surfaces......Page 193
5.5 Ruled surfaces......Page 200
5.6 Imaginaries in surface theory......Page 207
Some Problems and Propositions......Page 212
Appendix: The method of Pfaffians in the theory of curves and surfaces......Page 216
Answers To Problems......Page 228
Index......Page 237