Differential Geometry | Math History | NJ Wildberger

Differential geometry arises from applying calculus and analytic geometry to curves and surfaces. This video begins with a discussion of planar curves and the work of C. Huygens on involutes and evolutes, and the related notions of curvature and osculating circle. We discuss involutes of the catenary (yielding the tractrix), cycloid and parabola. The evolute of the parabola is a semi-cubical parabola. For space curves we describe the tangent line, osculating plane, principle normal and binormal.

Surfaces were studied by Euler, who investigated curvatures of planar sections and by Gauss, who realized that the product of Euler's two principal curvatures gave a new notion of curvature intrinsic to a surface. Curvature was ultimately extended by Riemann to higher dimensions, and plays today a major role in modern physics, due to the work of Einstein.

If you like this topic, and want to learn more, make sure you don't miss Wildberger's exciting new course on Differential Geometry! See the Playlist DiffGeom, at this channel.

************************

Here are the Insights into Mathematics Playlists:

Here are the Wild Egg Maths Playlists (some available only to Members!)

************************

[the_informative_edge]

Best visual explainations of involute, evolute, osculating circle etc. I have ever heard. No book can explain like this. Only such explaination are available at the stage of best teacher like Prof. Wildberger. Thank you so much respectable sir..

[kenroyadams]

This lecture was awe inspiring to say the least. Thanks for taking the time and effort to upload this. As a math major I truly appreciated this.

[stealthmonkeyG]

This is the most beautiful and surprising math I have learned yet. Thank you. Incredible job of explaining an incredibly difficult concept

[QuantumBunk]

This guy does a great job describing conceptually. Thanks!

[njwildberger]

But I will be starting a course on Differential Geometry shortly, which will be videoed.

[phononify]

i like your way of teaching - great videos, all of them. My father was prof. for experimental physics and very interested always in history ... i somehow inherited his interest ... your videos find their tangling bonds in my world picture therefore. Thanks a lot!

[fadesingh]

Brilliant presentation, extremely lucid as always. Thank you, Prof. Wildberger.

[njwildberger]

Even simpler! If these formulas are indeed correct, they are a very worthwhile addition to the literature. I strongly encourage you to write them both up---especially with some interesting examples.

[rodolforesende]

12:34 professor norman lost the opportunity to explain that "osculating" means "kissing"... It would be one of the rare moments where a poetic image is revealed to the mathematician to be.He did not insist on the technique that proves the unicity of the kissing circle. A monogamous relationship😂

[requerioayala]

Very informative and helpful for understanding some basic motivations behind differential geometry, thank you.

[TheAlison]

oh, thanks, history is awesome, and it has chapters too!!!
11:22 very interesting. Definition of the center of curvature. I always wondered how they found this out!!
20:38 he just wrote down a function there! Except it doesn't look all fancy in single letters.
35:00-37:40 my god, it feels like this can totally be used to map 2D planar images onto curves.
46:05 so this is what a Gaussian curvature is!!!

[njwildberger]

Sounds good, you could also use the Extended Spread law (Thm 82 of my book) to get the formula of the circumquadrance of a triangle (Exer.12.2 on the following page). How about some examples ---can you calculate the curvature of a parabola y=x^2 or some other curve, or perhaps recover the Huygens/Newton formula for curvature?

[zeeshanx]

Excellent lecture. I was just glued. it was like a story. The lecture unfolded differential geometry beautifully. looking for more. can I get such more lecture on differential geometry.

[jamesbra]

Yes very good geometric introduction. Thank you professor.

[jpdemont]

Another awesome video, Professor Wildberger. Thank you for sharing it.

[adamfrank]

This is incredible. Great lecture and great content.

[njwildberger]

We will be mentioning Riemann in our next lecture on Topology.

[AICodingAdventures]

Very good video! Thank you for the whole course.

[rionbreffeny]

I would say that this is one of the (and not just in maths) best lectures that I've ever heard. Oddly enough (other than several art lectures) i had an Australian chap for Thermodynamics who was one of the other excellent lecture-ers.

One of the worst (who happened to be one of my advisors) simply read to the class directly out of the textbook (which, naturally, we all had a copy of). 

Again, well done sir. -r.  (onward to Topology - or is it typography, topography, or teleology ;? -- ah jargon).

[gamigonyt]

great lecture, thanks for taking the time to uploading this video. really helped!