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Hi, All:
It's been a long time since I did this, and I have some basic doubts; please bear with me:
In Lee's Riemannian Mflds, p.48, he states that , given a parametrization :
γ:(a,b)→M, of a curve , "the velocity vector γ'(t) has a coordinate-independent meaning
for each t in M" (this should be for each t in (a,b).
Now, Lee goes on to give an example of two parametrizations of S1 , one
of which is γ(t)=(cost,sint), and the other is the polar-coordinate expression:
γ 2(t)=(r(t),θ(t))=(1,t) .
Now, in the first parametrization, the velocity is given by:
γ'(t)=(-sint, cost) , while
in the second one, we get:
γ'2(t)=(0,1)
So, in what sense is the velocity coordinate-independent then?
Thanks.
EDIT: Moreover, the reason given for the "ambiguity" in defining acceleration seems to apply to the definition
of velocity too:
In the difference quotient LimΔt→0 [f(x+Δt)-f(x)]/Δt
the vectors x and x+Δt live in tangent spaces that are not naturally isomorphic
to each other, right?
Lastly --hope this is not too long of a question-- I understand at an informal level
that a connection is a device used to define/select a choice of isomorphism between
vector spaces that are not naturally-isomorphic to each other, but I do not see anywhere
in this chapter where/how those isomorphisms are defined. Any Ideas/Suggestions?
Thanks.
It's been a long time since I did this, and I have some basic doubts; please bear with me:
In Lee's Riemannian Mflds, p.48, he states that , given a parametrization :
γ:(a,b)→M, of a curve , "the velocity vector γ'(t) has a coordinate-independent meaning
for each t in M" (this should be for each t in (a,b).
Now, Lee goes on to give an example of two parametrizations of S1 , one
of which is γ(t)=(cost,sint), and the other is the polar-coordinate expression:
γ 2(t)=(r(t),θ(t))=(1,t) .
Now, in the first parametrization, the velocity is given by:
γ'(t)=(-sint, cost) , while
in the second one, we get:
γ'2(t)=(0,1)
So, in what sense is the velocity coordinate-independent then?
Thanks.
EDIT: Moreover, the reason given for the "ambiguity" in defining acceleration seems to apply to the definition
of velocity too:
In the difference quotient LimΔt→0 [f(x+Δt)-f(x)]/Δt
the vectors x and x+Δt live in tangent spaces that are not naturally isomorphic
to each other, right?
Lastly --hope this is not too long of a question-- I understand at an informal level
that a connection is a device used to define/select a choice of isomorphism between
vector spaces that are not naturally-isomorphic to each other, but I do not see anywhere
in this chapter where/how those isomorphisms are defined. Any Ideas/Suggestions?
Thanks.
Last edited: