As a physics student who is studying differential geometry with no knowledge of something like manifolds, this "topology, manifolds & differential geometry" playlist helped me quite a lot :D Thank you! Looking forward for more!
Are you going to do more videos on Einstein-Cartan theory? I would like to know how it is different from GR. I'm also interested in the letters between Cartan and Einstein.
I often lose the general picture during my physics classes when they "review" the maths background (w/o previous context) your videos are wonderful to understand how all these build up more advanced topics, thank you maybe you could consider a more in depth review of some topics for future videos, really looking forward for your future content
maybe. caution.college teaching is easy.our man here does not have to deal with disruptive Johnny in the back row. That is another teaching strategy and that is not easy.
Two things: Fisrt is that you changed directions of the moving vector on the sphere but not the cylinder so I don't see the comparison. If you took the vector from the north pole, then down to the equator and kept going like in the cylinder you would go through the south pole and continue back up to the north pole and the vector would be pointing in the same direction as at the start. Second, the constraint to the surface of the sphere doesn't make sense to me as an example of intrinsic curvature because the constraint itself is an embedding from 3d to 2d. The closest distance between Paris and New York City is through the earth. If we constrain that to a geodesic on the surface it is still a 3d path. I find it impossible to ignore. The only way for "intrinsic" curvature to make sense to me that I have found is Feynman's idea (also Einstein) of thinking of the surface with varying temperature. If the parallel transported vector is affected by temperature then one path can change the vector as opposed to another path.
Difficult subject to explain 'in a light way'. On the cylinder ANY path gives you back your starting vector. That's not case on the sphere (like is shown in the video)! But taking a walk along the equator does give you back your starting vector. It's all about the word ANY.
True, I probably should have emphasised this point! Tracing out a geodesic in full will in general not have any affect on the vector (since a geodesic is defined as a curve whose velocity vector is unaffected by parallel transport along that curve). However, following some other arbitrary path (that may consist of piecewise smooth combinations of several geodesics) will in result in a shift due to the curvature, since the vector being transported is not always aligned with the tangent velocity vector to that portion of the curve - in this example the shifting happens when the vector is parallel transported along the equator, since it is not aligned in the direction of the velocity vector of the equator. Hope that helps!
@@WHYBmaths I'm trying to understand GR for some time now. In general I have a good idea about the concepts (I think). But when I start doing the math I get lost after some time. Thanks for the reply and stay healthy! P.S. I really like this video: czcams.com/video/LD6Y9_Q-xo8/video.html
Thanks a lot for this interesting video! Since equator is a closed geodesic on a sphere, parallel transporting a vector along it will not result in any dispersion between the vector, right? However, if we were to parrallel transport along a circle of a costant latitute or longitude , what would we see? Are the circles of constant latitude and longitude geodesics on a sphere?
As a physics student who is studying differential geometry with no knowledge of something like manifolds, this "topology, manifolds & differential geometry" playlist helped me quite a lot :D
Thank you!
Looking forward for more!
Are you going to do more videos on Einstein-Cartan theory? I would like to know how it is different from GR. I'm also interested in the letters between Cartan and Einstein.
I often lose the general picture during my physics classes when they "review" the maths background (w/o previous context) your videos are wonderful to understand how all these build up more advanced topics, thank you maybe you could consider a more in depth review of some topics for future videos, really looking forward for your future content
more! you're an excellent teacher
More coming soon! Just moved so it's a bit hectic right now but once I get everything set up I'll be starting relativity
maybe. caution.college teaching is easy.our man here does not have to deal with disruptive Johnny in the back row. That is another teaching strategy and that is not easy.
Two things: Fisrt is that you changed directions of the moving vector on the sphere but not the cylinder so I don't see the comparison. If you took the vector from the north pole, then down to the equator and kept going like in the cylinder you would go through the south pole and continue back up to the north pole and the vector would be pointing in the same direction as at the start.
Second, the constraint to the surface of the sphere doesn't make sense to me as an example of intrinsic curvature because the constraint itself is an embedding from 3d to 2d. The closest distance between Paris and New York City is through the earth. If we constrain that to a geodesic on the surface it is still a 3d path. I find it impossible to ignore. The only way for "intrinsic" curvature to make sense to me that I have found is Feynman's idea (also Einstein) of thinking of the surface with varying temperature. If the parallel transported vector is affected by temperature then one path can change the vector as opposed to another path.
6:17 holonomia
Difficult subject to explain 'in a light way'. On the cylinder ANY path gives you back your starting vector. That's not case on the sphere (like is shown in the video)! But taking a walk along the equator does give you back your starting vector. It's all about the word ANY.
True, I probably should have emphasised this point!
Tracing out a geodesic in full will in general not have any affect on the vector (since a geodesic is defined as a curve whose velocity vector is unaffected by parallel transport along that curve). However, following some other arbitrary path (that may consist of piecewise smooth combinations of several geodesics) will in result in a shift due to the curvature, since the vector being transported is not always aligned with the tangent velocity vector to that portion of the curve - in this example the shifting happens when the vector is parallel transported along the equator, since it is not aligned in the direction of the velocity vector of the equator. Hope that helps!
@@WHYBmaths I'm trying to understand GR for some time now. In general I have a good idea about the concepts (I think). But when I start doing the math I get lost after some time. Thanks for the reply and stay healthy! P.S. I really like this video:
czcams.com/video/LD6Y9_Q-xo8/video.html
Awesom Bro, great video!
Thanks a lot for this interesting video!
Since equator is a closed geodesic on a sphere, parallel transporting a vector along it will not result in any dispersion between the vector, right?
However, if we were to parrallel transport along a circle of a costant latitute or longitude , what would we see? Are the circles of constant latitude and longitude geodesics on a sphere?
Please proceed.
Nice playlist to understand manifolds better, than you!