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Topic: How do we Evaluate This Form on S^1?
Replies: 11   Last Post: Jan 22, 2013 4:17 PM

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W. Dale Hall

Posts: 71
Registered: 2/11/05
Re: How do we Evaluate This Form on S^1?
Posted: Jan 20, 2013 7:28 PM
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hbertaz@gmail.com wrote:
> Hi, All:
>
> Just curious:
>
> How does one evaluate the form d(theta) on the circle?
>
> I know a form takes a tangent vector X and spits out a real number ,
> in a linear way. The tangent space to S^1 is R^1, so that a tangent
> vector is of the form c ; any non-zero real number can be a basis.
>
> But, how do we get a number from d(theta)[c]?
>
> I tried using a change of coordinate, changing from cartesian to
> polar, but I don't see much difference.
>
> Any suggestions, please?
>
> Thanks.
>


At any point x on the circle S^1, d(theta) is the linear function on the
tangent space T_x(S^1) that sends the vector \partial_theta in
T_x(S^1) to 1.

In order to identify the reals R with the tangent space T_x(S^1), you
need to map r in R to the vector r*\partial_theta. A tangent vector
isn't *really* a real number [after all, tangent vectors to different
points on S^1 aren't quite identifiable with one another since they
belong to different tangent spaces], but is uniquely expressible as a
basis element times a real number (where the basis element is chosen
freely, but once chosen, it's fixed). So, if you've chosen the vector
\partial_theta as the basis element for T_x(S^1), the map d(theta) sends
r*\partial_theta to r in R.

Dale



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