Differential Geometry - Math History - NJ Wildberger

Explore the foundations and historical development of differential geometry in this comprehensive 52-minute lecture. Begin with an examination of planar curves, focusing on C. Huygens' work on involutes and evolutes, and the concepts of curvature and osculating circles. Investigate specific examples, including involutes of the catenary, cycloid, and parabola, as well as the evolute of the parabola. Delve into space curves, learning about tangent lines, osculating planes, principal normals, and binormals. Trace the evolution of surface studies through the contributions of Euler and Gauss, understanding how their work on curvatures led to new intrinsic notions of surface geometry. Discover how Riemann extended these concepts to higher dimensions, and explore the significant impact of curvature in modern physics, particularly in Einstein's work. Gain a solid foundation in differential geometry, preparing you for more advanced studies in this crucial field of mathematics.

Introduction
Evolute
Catenary
Space curves
Surface curves
Curves
Carl Friedrich Gauss
Gaussian curvature