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Re: Questions & Discussions on Vectors & relationships to differe ntia l geometry



Jack,
I understand your comment now! I didn't realize you were just refering to
bumping down the dimension by one. Your right these issues that I mentioned
are confronted in S1.

I'd describe the tangent space of S1 as being the line, i.e. R^1.

What aspect of uniqueness were you refering to wrt S1?

S2's tangent space is R^2, S3's is R^3 etc and rather trivially I believe
one can say S0's tangent space is R^0 ??

I should be careful and in the above its the tangent space associated with a
specific point in SN. That is each point in S1 is endowed with a seperate
and unique tangent space which looks like R1. Admittedly isopmorphic to
every other point's tangent space.

I think I said this correctly.

Joel

-----Original Message-----
From: Jack Uretsky [mailto:jlu@HEP.ANL.GOV]
Sent: Thursday, March 15, 2001 5:26 PM
To: PHYS-L@lists.nau.edu
Subject: Re: Questions & Discussions on Vectors & relationships to
differe ntia l geometry


Hi Joel-
By S1 I mean the circle.
Position on S1 is not a vector (in S1 although it might be
described by a vector in R^2). Circular motion at constant speed -
the velocity vector lies in a tangent space and does not lie in S1,
right? That's exactly what you were asking about, I thought.
For homework, how do you describe the tangent space to S1.
You have to be a litte careful, since S1 is unique among the
spheres Sn of dimension n.
Regards,
Jack

On Thu, 15 Mar 2001, RAUBER, JOEL wrote:

Jack,

I'm not sure I understand your question.


Why do you want to make things so complicated? You
confront all
of these concepts when you consider circular motion.


But my question came precisely from some musing on
tranformations involving
rotation plus translation and trying unravel what is
geometric (coordinate
independent) about vectors and what isn't invariant about
them or the
vectors we use to represent physical concepts, velocity for example.

Joel Rauber


--
Franz Kafka's novels and novella's are so Kafkaesque that one has to
wonder at the enormity of coincidence required to have
produced a writer
named Kafka to write them.
Greg Nagan from "The Metamorphosis" in
<The 5-MINUTE ILIAD and Other Classics>