Lectures on Classical Differential Geometry Second Edition
by Struik, Dirk J.Edition: 2nd
Format: Paperback
Pub. Date: 1988-04-01
Publisher(s): Dover Publications
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Summary
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
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
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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