75. Serret-Frenet Equations | Chapter 5 | Theory of Curves | Differential Geometry

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Serret-Frenet Equations

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Introduction
Theory of Space Curves
• Introduction, index notation and summation convention
• Space curves, arc length, tangent, normal and binormal
• Osculating, normal and rectifying planes
• Curvature and torsion
• The Frenet-Serret theorem
• Natural equation of a curve
• Involutes and evolutes, helices
• Fundamental existence theorem of space curves
Theory of Surfaces
• Coordinate transformation
• Tangent plane and surface normal
• The first fundamental form and the metric tensor
• Christoffel symbols of first and second kinds
• The second fundamental form
• Principal, Gaussian, mean, geodesic and normal curvatures
• Gauss and Weingarten equations
• Gauss and Codazzi equations
Recommended Books
1. R. S. Millman and G.D. Parker, Elements of Differential Geometry (Prentice-Hall,
New Jersey, 1977).
2. A. Goetz, Introduction to Differential Geometry (Addison-Wesley, 1970).
3. E. Kreyzig, Differential Geometry (Dover, 1991).
4. M. M. Lipschutz, Schaum's Outline of Differential Geometry (McGraw Hill, 1969).
5. D. Somasundaram, Differential Geometry (Narosa Publishing House, New Delhi,
2005).