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..
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?
That was pretty much the question that brought me here. It turns out that you actually have to carry out the path integral s(t). For example: Let's assume the parameter t has the meaning of time, and assume t_0 is 0. Then you have to actually compute the derivatives of x(t) with respect to t for all components of x(t). Then you need to sum up their squares and take the square root. And then you need to integrate this thing from 0 to t, just like he wrote. This is usually, even for the simplest examples, a pain in the neck. But if this integral from 0 to t can be computed then you have the parameter s(t) as function of t. The integral s(t) is basically a function that maps the old parameter, t --> s(t), the new parameter. But now there may be new trouble: Suppose you had initially all the coordinates of x expressed as functions of t, e.g., x(t)=(f(t), g(t)) = (cos(t), sin(t^2)). Now you have to express them as functions of s(t), which could mean that you have to also invert the function s(t), which is already very complicated, because it was based on the time derivatives of these two functions f(t) and g(t). Good luck with that :-) BTW: In this case it would be xdot(t) = ( -sin(t), 2 t cos(t^2)) . So the integral to get s(t) is over Sqrt[ sin(t)^2 + 4 t^2 cos(t^2)^2 ] . And that function is initially sort of linear rising, but at about t=1 it reaches a maximum and then goes to quickly down to 0 (at t ~ 1.3), which means that the path won't get longer. So apart from the problem to actually solve this integral, it won't work as a reparametrization after that.
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?
Hopefully, statistical physics has been on my to-do list for a while; I just haven't had the chance to get to it with all my other series going on. It's something I've planned to do later for sure!
Bless! I’m starting differential geometry this quarter. Please save me :D
where do you go to school and what are you studying
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..
How did you present such an elegant writing script while talking? That's phenomenal!
Thank you man.great job as always.
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?
That was pretty much the question that brought me here. It turns out that you actually have to carry out the path integral s(t).
For example: Let's assume the parameter t has the meaning of time, and assume t_0 is 0. Then you have to actually compute the derivatives of x(t) with respect to t for all components of x(t). Then you need to sum up their squares and take the square root. And then you need to integrate this thing from 0 to t, just like he wrote. This is usually, even for the simplest examples, a pain in the neck. But if this integral from 0 to t can be computed then you have the parameter s(t) as function of t. The integral s(t) is basically a function that maps the old parameter, t --> s(t), the new parameter.
But now there may be new trouble: Suppose you had initially all the coordinates of x expressed as functions of t, e.g., x(t)=(f(t), g(t)) = (cos(t), sin(t^2)). Now you have to express them as functions of s(t), which could mean that you have to also invert the function s(t), which is already very complicated, because it was based on the time derivatives of these two functions f(t) and g(t). Good luck with that :-)
BTW: In this case it would be xdot(t) = ( -sin(t), 2 t cos(t^2)) . So the integral to get s(t) is over Sqrt[ sin(t)^2 + 4 t^2 cos(t^2)^2 ] . And that function is initially sort of linear rising, but at about t=1 it reaches a maximum and then goes to quickly down to 0 (at t ~ 1.3), which means that the path won't get longer. So apart from the problem to actually solve this integral, it won't work as a reparametrization after that.
Thank God, I thought this channel is over.
Somebody call Rick, Birdperson is teaching math.
thank you its been a while since differential geometry video
question. Should the reparametrization map be from arc length to time t?
Just letting you know I subscribed :)
Ok, great
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?
Hopefully, statistical physics has been on my to-do list for a while; I just haven't had the chance to get to it with all my other series going on. It's something I've planned to do later for sure!
Hi from reddit r/math
Le reddit armie has arrived!