- #1
bres gres
- 18
- 1
- TL;DR Summary
- i get stuck when i watch the tensor video in this link:
https://www.youtube.com/watch?v=Af9JUiQtV1k&list=PLJHszsWbB6hpk5h8lSfBkVrpjsqvUGTCx&index=21
hope someone can help me as it is in my head for several days
thank you
1.how can i estimate the effect of the "straight component" in different small circles on the sphere and plot them on the graph
2.there is contradiction between the graph and the definition?
question1 :
if you draw a small circle around the north pole (it should be the same at every points because of the symmetry of the sphere),then it is approximately a flat space ,then we can translate the vector on sphere just like what we have done in flat space(which translate the vector "passively" because we just move our feet but the vector is hold in the same direction)(case1) .
There is no straight line on sphere,however the straightest line is the path lie on the great circle,therefore we can bend the vector actively to change the direction such that it seems like the vector is unchanged when we move forward in a "straight line" therefore we should subtract the normal component(and there is no change in tangential component since the vector seems to be unchanged in the observers frame). (case2)
Between these two extreme case, the person moving on the Earth will inevitably bend his fit into some direction in his frame(because the path on the great circle is straight one in his frame and other paths will be non-straight one even in his frame),at the same time there is "straight component" as well just like as case (2) mentioned,if we combine the effect in case1 and case 2 in a miraculous way,we will gradually bend the vector into some direction.here is what i understand the only thing i get stuck is i cannot imagine how two things combined in a way that i can plot the vectors on the graph,it seems hard to me. how can i estimate the effect of the "straight component" in different small circles.
question2:
according to the definition of the covariant derivative (which is the ordinary vector derivative in R^3 space subtract the normal component),it seems the graph in 3:40 fits the definition as well since there is no rate of change of the vector in R^3 space,then if we use x-y-z coordinate to calculate the rate of change of the vector,then it should be zero. The normal component should be zero as well(since the normal component of the rate of change of this vector is zero),therefore the final value will be zero. because the author uses extrinsic perspective in the graph,therefore we can use normal xyz coordinate to treat the vector as normal,if every point on the sphere assigned with a vector that is pointed into x direction,then the "ordinary part" would be zero(there is no normal part as well),therefore this fits the definition as well. However this cannot be true since in 3:40 ,the author said that the vector will be pointed into the sky geometrically(after the vector is moved after a quarter of great circle) ,but algebraically we can get zero which fits the definition. there is contradiction between his explanation and the definition?thank for help!
this is my first time to ask physics question
if you draw a small circle around the north pole (it should be the same at every points because of the symmetry of the sphere),then it is approximately a flat space ,then we can translate the vector on sphere just like what we have done in flat space(which translate the vector "passively" because we just move our feet but the vector is hold in the same direction)(case1) .
There is no straight line on sphere,however the straightest line is the path lie on the great circle,therefore we can bend the vector actively to change the direction such that it seems like the vector is unchanged when we move forward in a "straight line" therefore we should subtract the normal component(and there is no change in tangential component since the vector seems to be unchanged in the observers frame). (case2)
Between these two extreme case, the person moving on the Earth will inevitably bend his fit into some direction in his frame(because the path on the great circle is straight one in his frame and other paths will be non-straight one even in his frame),at the same time there is "straight component" as well just like as case (2) mentioned,if we combine the effect in case1 and case 2 in a miraculous way,we will gradually bend the vector into some direction.here is what i understand the only thing i get stuck is i cannot imagine how two things combined in a way that i can plot the vectors on the graph,it seems hard to me. how can i estimate the effect of the "straight component" in different small circles.
question2:
according to the definition of the covariant derivative (which is the ordinary vector derivative in R^3 space subtract the normal component),it seems the graph in 3:40 fits the definition as well since there is no rate of change of the vector in R^3 space,then if we use x-y-z coordinate to calculate the rate of change of the vector,then it should be zero. The normal component should be zero as well(since the normal component of the rate of change of this vector is zero),therefore the final value will be zero. because the author uses extrinsic perspective in the graph,therefore we can use normal xyz coordinate to treat the vector as normal,if every point on the sphere assigned with a vector that is pointed into x direction,then the "ordinary part" would be zero(there is no normal part as well),therefore this fits the definition as well. However this cannot be true since in 3:40 ,the author said that the vector will be pointed into the sky geometrically(after the vector is moved after a quarter of great circle) ,but algebraically we can get zero which fits the definition. there is contradiction between his explanation and the definition?thank for help!
this is my first time to ask physics question