Arc Length as a Parameter | Differential Geometry 3

After a long hiatus from Differential Geometry, I return to this highly-requested series as promised! The New Year is upon us and there is no better way to celebrate than with a discussion on reparametrizing with respect to arc length.

In this video, I begin by discussing how reparametrizing with respect to arc length is a valid choice, and why doing this procedure is important in the context of Differential Geometry (re: unit-speed curves). I then prove two theorems related to using arc length as the parameter and the resultant unit speed properties of the curve.

Questions/requests? Let me know in the comments!

#DifferentialGeometry #ArcLength #UnitSpeedCurve

[guiwenluo]

How did you present such an elegant writing script while talking? That's phenomenal!

[KevinS]

Yes! finally another 2 episodes! Thank you so much, haven't watched it yet, but I will asap and get back to taking notes :)
A bit late, but better than never..

[gonzalezm]

Bless! I’m starting differential geometry this quarter. Please save me :D

[Istillcantthinkofahandle]

Somebody call Rick, Birdperson is teaching math.

[mohammedal-haddad]

Thank God, I thought this channel is over.

[hevanderdacosta]

thank you its been a while since differential geometry video

[nestorwalters]

Question: how do you actually "parameterize by arc length?" I must have missed something because it seems we answered the question "when are we allowed to" but not how to actually do it. Do you just plug s(t) into gamma(s) so Gamma(s(t)) is the parameterized equation?

[ronouyang]

question. Should the reparametrization map be from arc length to time t?

[eulefranz]

I wish I took general relativity this semester instead of quantumfield theory
Edit: Taking a statistical physics course in addition and we are covering stochastic processes with the master equation, which is quite cool, will you explore some stochastic processes in the future?