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source: njwildberger 2013年8月1日

The first lecture of a beginner's course on Differential Geometry! Given by Assoc Prof N J Wildberger of the School of Mathematics and Statistics at UNSW. Differential geometry is the application of calculus and analytic geometry to the study of curves and surfaces, and has numerous applications to manufacturing, video game design, robotics, physics, mechanics and close connections with classical geometry, algebraic topology, the calculus of several variables and mostly notably Einstein's General Theory of Relativity.

This lecture summarizes the basic topics of the course, the unique point of view of the lecturer, and then heads straight into a survey of classical curves, starting with the line, then the conic sections (ellipse, parabola, hyperbola), then moving to classical ways of generating new curves from old ones. These techniques include the Conchoid construction of Nicomedes, the Cissoid construction of Diocles, the Pedal curve construction and the evolute and involute introduced by Huygens. This lecture should be viewed in conjunction with MathHistory16: Differential Geometry.

My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

**Differential Geometry**

A beginner's course on Differential Geometry. We present a systematic and sometimes novel development of classical differential differential, going back to Euler, Monge, Dupin, Gauss and many others. Our approach is more algebraic than usual; we are interested in making the theory more elementary and at the same time more general; extending quite a few results to finite fields for example, and minimizing the role of all but basic calculus. The geometrical side of things will be emphasized, and includes some pertinent theory from projective geometry. We also want to avoid the logical difficulties, usually completely ignored, in basing the subject on the so-called `real numbers': our theory has the rational numbers firmly at the heart of the subject, with quadratic extensions employed when necessary. In this sense the course is much more elementary than the usual treatments.

Topics include looking at classical curves and surfaces, in particular both parametrized and algebraic curves and surfaces. We approach the local study via tangent lines/planes and tangent conics/quadrics following the algebraic approach to calculus championed by Lagrange (and Euler before him). Curvature is the key to the subject, and we present a careful step by step treatment of this subject starting from curves to surfaces, with a variety of interesting classical applications and results.

DiffGeom1: Classical curves 44:11

DiffGeom2: Introduction to GeoGebra 14:20

DiffGeom3: Parametrized curves and algebraic curves 41:14

DiffGeom4: The differential calculus for curves, via Lagrange! 48:27

DiffGeom5: Tangent conics and tangent quadrics 49:16

DiffGeom6: Visualizing the folium surface with GeoGebra 23:51

DiffGeom7: Differential geometry with finite fields 49:43

DiffGeom8: The differential calculus for curves (II) 48:15

DiffGeom9: Projective view of conics and quadrics 38:31

DiffGeom10: Duality, polarity and projective linear algebra 37:27

DiffGeom11: Duality, polarity and projective linear algebra (II) 48:26

DiffGeom12: Metrical structure and curvature of a parabola 44:07

DiffGeom13: Curvature for the general parabola 47:55

DiffGeom14: Quadratic curvature for algebraic curves 33:08

DiffGeom15: Quadratic curvature for algebraic curves (cont) 36:11

DiffGeom16: Curvature, turning numbers and winding numbers 48:24

DiffGeom17: Curvature, turning numbers and winding numbers (cont) 48:01

DiffGeom18: The Frenet Serret equations 50:31

DiffGeom19: The Frenet Serret equations (example) 23:02

DiffGeom20: Geometric and algebraic aspects of space curves 54:56

DiffGeom21a: An introduction to surfaces 42:44

DiffGeom21b: A tutorial: some differential geometry problems 46:32

DiffGeom22: More general surfaces 52:35

DiffGeom23: Paraboloids and associated quadratic forms 46:38

DiffGeom24: Topological spaces and manifolds 50:14

DiffGeom25: Manifolds, classification of surfaces and Euler characteristic 46:17

DiffGeom26: Classification of 2-manifolds and Euler characteristic 45:50

DiffGeom28: Curvature for the general paraboloid 46:14

DiffGeom29: Curvature for general algebraic surfaces 49:05

DiffGeom30: Examples of curvatures of surfaces 42:49

DiffGeom31: Meusnier, Monge and Dupin I 49:04

DiffGeom32: Meusnier, Monge and Dupin II 48:16

DiffGeom33: Meusnier, Monge and Dupin III 54:38

DiffGeom34: Gauss, normals and fundamental forms 51:27

DiffGeom35: Gauss's view of curvature and the Theorema Egregium 43:46

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