Help with geometric interpretation of 1-form

[damnedcat]

I am currently reading the special relativity section in Goldstein's Classical, and there is an optional section on 1-Forms and tensors. However i am having a lot of trouble understanding the geometric interpretation of a 1-form.

Here is what I do understand: You take a regular vector (contravariant vector) and you act on it by the metric tensor, apparently this this gives you something called a 1-form (covariant vector) for example in spherical co ordinated acting on a vector (dr, d$$\theta$$, d$$\phi$$) with the metric tensor gives you (dr, r$$^{2}$$ d$$\theta$$, r$$^{2}$$sin$$^{2}$$$$\theta$$d$$\phi$$), ok I can accept this and all the raising and lowering of indecies by the metric tensor. what does baffle me is when Goldstein explains (ch 7.5 p289 3ed):

"If the vector is thought of as a directed line, the 1-form is a set of numbered surfaces through which the vector passes (there is a diagram). It is another functional similar to g (the metric tensor), except it turns a vector into a linear real valued scalar function. That is if n is a 1-form and v a vector then the quantity denoted by <n,v> is a number that tells us how many surfaces of n are pierced by v"

Can anyone help me find the link to this geometric interpretation and the piercing surfaces, or possibly offer an alternate geometric interpretation?

 

[Fredrik]

Introduction to smooth manifolds by John Lee has an excellent chapter on differential forms. Unfortunately some of the relevant pages aren't available for preview.

I passionately hate Goldstein's book, so I won't try to understand what he says.

 

[Altabeh]

"If the vector is thought of as a directed line, the 1-form is a set of numbered surfaces through which the vector passes (there is a diagram). It is another functional similar to g (the metric tensor), except it turns a vector into a linear real valued scalar function. That is if n is a 1-form and v a vector then the quantity denoted by <n,v> is a number that tells us how many surfaces of n are pierced by v"

Can anyone help me find the link to this geometric interpretation and the piercing surfaces, or possibly offer an alternate geometric interpretation?

I think there is just Goldstein himslelf who knows what the paragraph is all about! You have two choices here: 1- Ignore the paragraph and go on to the next one. 2- Read another textbook on the subject of 1-forms, which you can find many of them on the Internet! For example, Shcutz takes a short geometrical approach to 1-forms in his book "A first course in
general relativity" on pages 94-5 that, I think, has something in common with this paragraph but as you may know he is well-known for simplifying hard-to-be-understood things so it will be of a little help for you!

AB

 

[Hurkyl]

One-forms are to real-valued functions as tangent vectors are to curves.


If you like to think of a real-valued function in terms of its level curves, then Goldstein's picture is probably the corresponding way to think about a one-form. (well, this also depends on how you like to think of vectors)

But conversely, if you don't like thinking of a real-valued function that way, then you probably don't want to think about one-forms that way either.

 

[Rasalhague]

Gabriel Weinreich's Geometrical Vectors might help. Last time I looked it was possible to read enough of the first chapter on Google Books to get the gist.

http://books.google.co.uk/books?id=vwzG4ZHrMg8C
http://www.maa.org/reviews/vectors.html

He calls (tangent) vectors "arrow vectors", and 1-forms "stack vectors". While the magnitude of an "arrow vector" is traditionally pictured as its length, the magnitude of a "stack vector" can be pictured as the density of its level planes. I think the idea is that if a change of coordinate system results in a longer arrow representing the tangent vector, it will also result in bigger gaps between the level planes of the stack representing the 1-form. So if you picture the action of the stack on the arrow (the scalar product) as counting how many planes the arrow fits through, this number will be unaffacted by the change of coordinate system.

 

[jason12345]

I doubt Goldstein would have allowed his name on the 3rd edition Mechanics if he had known how tensors and 1-forms would be explained in the chapter on relativity. It's awful, the worst I have seen in any book. I'm still traumatised over it.

 

[damnedcat]

thanks for the suggestions guys, I'm checking out some of your recommendations, unfortunately I'm snowed in so no library till probably monday.

Gabriel Weinreich's Geometrical Vectors might help. Last time I looked it was possible to read enough of the first chapter on Google Books to get the gist.

http://books.google.co.uk/books?id=vwzG4ZHrMg8C
http://www.maa.org/reviews/vectors.html

He calls (tangent) vectors "arrow vectors", and 1-forms "stack vectors". While the magnitude of an "arrow vector" is traditionally pictured as its length, the magnitude of a "stack vector" can be pictured as the density of its level planes. I think the idea is that if a change of coordinate system results in a longer arrow representing the tangent vector, it will also result in bigger gaps between the level planes of the stack representing the 1-form. So if you picture the action of the stack on the arrow (the scalar product) as counting how many planes the arrow fits through, this number will be unaffacted by the change of coordinate system.

I cheecked out the links you sent and this seems along the lines of what Goldstein was talking about, unfortunately the preview doesn't contain that section so I'm trying to get a hard copy of the book as we speak. Things come to me very easy if i have a geometric intuition of them so this should help a lot. thanks if there are anymore suggestions or please let me know

 

[damnedcat]

I doubt Goldstein would have allowed his name on the 3rd edition Mechanics if he had known how tensors and 1-forms would be explained in the chapter on relativity. It's awful, the worst I have seen in any book. I'm still traumatised over it.

And here I was thinking it was only me.

 

[Rasalhague]

I cheecked out the links you sent and this seems along the lines of what Goldstein was talking about, unfortunately the preview doesn't contain that section so I'm trying to get a hard copy of the book as we speak.

Try p. 66 of Bernard Schutz's A First Course in General Relativity here:

http://books.google.co.uk/books?id=qhDFuWbLlgQC&printsec=frontcover#v=onepage&q=one-form&f=false

 

[damnedcat]

Try p. 66 of Bernard Schutz's A First Course in General Relativity here:

http://books.google.co.uk/books?id=qhDFuWbLlgQC&printsec=frontcover#v=onepage&q=one-form&f=false

thanks, i got my hand on a copy of the book and I'm checking it out. will let you know if i have any questions

 

[damnedcat]

Just to follow up, Schutz adeuately explained the geometric concept of a 1 form which I think Goldstein was trying to convey. Thanks to Rasalhague & Altabeh for recommending it.