
Re: How do we Evaluate This Form on S^1?
Posted:
Jan 20, 2013 7:28 PM


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 nonzero 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

