Working through Differential Forms by David Bachman

[revprez]

A quick question about Bachman's A Geometric Approach to Differential Forms. I've read:

 

1. [imath]T_{p}\mathbb{R}^n[/imath] to mean "the tangent space of some path/surface in [imath]\mathbb{R}^n[/imath]" and,

 

2. [imath]T_{p}\mathbb{R}^n \times \mathbb{R}^m[/imath] to mean "some (n-1)*m dimensional space of real numbers."

 

I haven't run into any problems using this definition but I think it would make the text easier to read if I actually understood what was going on (especially in the second example.

 

Rev Prez

[revprez]

Okay, we're just talking about a Cartesian product, so the coordinate space is just given by some tangent space in Rⁿ and R^m.

 

Rev Prez

[fuhrerkeebs]

I don't really know what you want to understand better, but I'll give it a shot:

 

Think of a tangent space as the set of all vectors that are tangents to some point on an n-manifold. Therefore, the tangent space is an (n-1)-dimensional surface. Since the tangent space is an (n-1)-dimensional surface, the product of Rⁿ and the tangent space of some point on a surface in R^m is an n(m-1)-dimensional space. I don't really know if that's what you want to know, but...

[Tom Mattson]

A quick question about Bachman's

1. [imath]T_{p}\mathbb{R}^n[/imath] to mean "the tangent space of some path/surface in [imath]\mathbb{R}^n[/imath]" and' date='

[/quote']

 

For the time being in this book, the only difference between Rⁿ and T_pRⁿ is the origin. The latter is a carbon copy of the former, but the origin is relocated to point p. All points that exist in one space also exist in the other, but they have different coordinates.

 

2. [imath]T_{p}\mathbb{R}^n \times \mathbb{R}^m[/imath] to mean "some (n-1)*m dimensional space of real numbers."

 

I think I know the part of the book to which you are referring. It's where he describes the graph of the 1-form, and he says that it lives in T_pR²XR. Note that the dimension of the space is not (2-1)*1. Rather, the dimension is 3.

 

From your remarks above, I think that you may be holding 2 misconceptions.

 

1. That the dimension of T_pRⁿ is n-1. As you can see from the discussion on pp 48-49, this is not the case. On p. 48 he explicitly writes down the basis of the tangent space to the plane P, and there are 2 basis vectors. And on p 49 he draws a picture representing T_pR², and as you can see there are 2 coordinate axes.

 

2. That the dimension of a space RⁿXR^m is n*m. That is not the dimension; rather it is n+m.

 

By the way, I've just revived my own thread on this book at PF, which has been dormant for about 1.5 months. When I first started it I got Bachman himself to come and answer some of our questions. I'll drop him an email and see if he'll come back.

[revprez]

By the way, I've just revived my own thread on this book at PF, which has been dormant for about 1.5 months. When I first started it I got Bachman himself to come and answer some of our questions. I'll drop him an email and see if he'll come back.

 

Thanks, that clears a hell of a lot up for me. I'll take a look over at PF.

 

Rev Prez

[Tom Mattson]

Good, glad to hear it.

 

Don't mean to be cheeky, but I see you poking around the General Math Forum at PF. My thread is in Tensor Analysis and Differential Geometry, in case that's what you are looking for.

[revprez]

Good' date=' glad to hear it.

 

Don't mean to be cheeky, but I see you poking around the General Math Forum at PF. My thread is in Tensor Analysis and Differential Geometry, in case that's what you are looking for. [/quote']

 

Thanks man, I was wondering where the hell it was. I passed out looking for it.

 

Rev Prez