I've just come across one-forms for the first time

[Adriadne]

I've just come across one-forms for the first time. Everything I read makes them sound exactly like dual vectors, but nobody mentions them in the same breath. Why?
Is it that dual vectors are one-forms, but not all one-forms are dual vectors (e.g. covariant tensors etc) or is the difference more subtle? Or have I misunderstood?

 

[Hurkyl]

Well, a dual vector field, but yah, a one-form is a tensor.

 

[Adriadne]

I thank you for your response, but you're confusing me even more.
I have it here that a rank 1 tensor is a vector, a co-vector if it's type
(0,1). Schutz explicitly refers to "one-forms at a point P", which surely disqualifies them from having any field properties? He also refers to the vector space of one-forms, V*, as being dual to the vector space V. Again, spaces rather than fields.

Where have I taken a wrong turning?

 

[George Jones]

Physicists and physics books often use terms like "vector" and "vector field" interchangeably, letting context sort out what is really meant.

I prefer Hurkyl's terminology, i.e., a 1-form is a dual vector field. I think you'll find that Schutz uses "1-form" in both senses - as a field of dual vectors, and as a specific dual vector.

Regards,
George

 

[Adriadne]

Yes, but surely if they go to the trouble of writing "at the point P" they are not talking about a field? Certainly, if I consider all P, I can select a single vector, dual, one-form, whatever, and declare the collection to be some field. But in the absence of that relaxation, surely we're not talking fields?

I hate to quote books, as they may be wrong, but here's Schutz: "a one-form at P associates with a vector at P a real number...one-forms at P satisfy the axioms of a vector space which is called the dual vector space to {tangent vectors at the point P}"

I am in no sense suggesting that either you or Hurkyl are wrong, I am just not getting it

 

[George Jones]

Your quote from Schutz seems reasonable to me. For example, let $$\alpha$$ be a 1-form, i.e., a field of dual vectors. Denote $$\alpha$$ evaluated at $$p$$ by $$\alpha_{p}$$, so that $$\alpha_{p}$$ maps $$T_{p} \rightarrow R$$. If $$v$$ is in $$T_{p}$$, then $$\alpha_{p} \left( v \right)$$ is a real number.

Regards,
George

 

[Adriadne]

OK, that's progress for me, thanks. Nevertheless, I still don't quite see how I can turn a field (one thingy per point in space) into a space (all thingys at a point in space). Schutz (and Flanders I now see) insist that one-forms inhabit a vector space.

If I want to think about all spaces at all points then I'm talking about a bundle?

 

[George Jones]

Exactly. The tangent bundle is

$$\bigcup_{p} T_p,$$

the collection of all the tangent spaces. A vector field is a cross-section of the tangent bundle. Similarly, the cotangent bundle is the collection of all dual spaces, and a 1-form is a cross-section of the cotangent bundle.

Regards,
George

 

[Hurkyl]

If it helps...

In particular, sections of $\Omega^k(TM)$, which are just alternating covariant tensor fields of order k, are called k-forms on M. A 1-form is just a covariant vector field.

 

[Adriadne]

George, thanks, I appreciate your efforts, but you'll have to forgive me being a little slow here. Let's see if there's an early flaw in my thinking:
In any generalised space, a vector space V at the points P, Q... is all vectors at P, Q... (Yes, I'm aware this is a hokey defintion of a vector space. I use it for present purpose only)
Let me assume my vector space is an inner product space, where (v,w) = some a in R

Now if I want to form a vector field, I select one vector at P, Q... and declare it a field. OK so far?

Now let me find a set of functionals {f_i} such that f_v(w) = (v,w). I'll call these guys dual vectors, and say they inhabit the space V* which is dual to the vector space V. (I am also, dimly, aware that the reality of the dual/vector combination is not crucially dependent on there being an inner product)

Let me now concede that there is not (or not necessarily) a one to one correspondence between a vector and its dual. So am I to conclude that, to every single element of my vector field I can associate a dual space?
This (after a couple of beers) seems reasonable. So, I ask again, if 1-forms inhabit a dual vector space at P, Q...(Schutz, Flanders) am I to think of 1-forms as being a field of dual spaces?
Hmm.That doesn't feel right at all. Any simple ideas?

 

[George Jones]

Let me expand somewhat on these things.

Denote the manifold of all spacetime events by $$M$$. There is a separate vector space of tangent vectors at each $$p$$ in $$M$$, which is denoted $$T_{p} \left( M \right)$$. So if $$p$$ and $$q$$ are different points, $$T_{p} \left( M \right)$$ and $$T_{q} \left( M \right)$$ are different vector spaces.

The tangent bundle $$TM$$ is the union of all the tangent spaces, i.e,

$$TM = \cup_{p \in M} T_{p} \left( M \right).$$

$$TM$$ is not a vector space. For example there is no natural way to add a vector in $$T_{p} \left( M \right)$$ to a vector in $$T_{q} \left( M \right)$$.

A vector field $$v$$ on $$M$$ is, for each $$p \in M$$, the smooth assignment of a vector $$v_p \in T_{p} \left( M \right)$$.

Since each $$T_{p} \left( M \right)$$ is a vector space, the dual space $$T_{p} \left( M \right)*$$ of each $$T_{p} \left( M \right)$$ can be formed. An element of $$T_{p} \left( M \right)*$$ is called a cotangent vector at $$p$$. The cotangent bundle $$T*M$$ is the union of all the cotangent spaces, i.e,

$$T*M = \cup_{p \in M} T_{p} \left( M \right)*.$$

$$T*M$$ is not a vector space.

A one form$$\alpha$$ on $$M$$ is, for each $$p \in M$$, the smooth assignment of a covector $$\alpha_p \in T_{p} \left( M \right)*$$.

A finite-dimensional vector space $$V$$ is always isomorphic to its algebraic dual $$V*$$, but without additional structure, there is no natural isomorphism, i.e., there is no natural way to identify a particular element of $$V$$ with a particular element of $$V*$$. An "inner product" does give rise to a natural identification, in the way that you outlined.

Let $$g$$ be the (non-degenerate) "metric" for spacetime, and let $$v \in T_{p} \left( M \right)$$. (For now, drop the $$p$$ subscripts.) Define $$\tilde{v} \in T_{p} \left( M \right)*$$ by $$\tilde{v} \left( w \right) = g \left( v , w \right)$$ for every $$w \in T_{p} \left( M \right)$$. The mapping $$v \mapsto \tilde{v}$$ is a vector space isomorphism between $$T_{p} \left( M \right)$$ and $$T_{p} \left( M \right)*$$. This is the "lowering" (of component indices) map, and its inverse is the raising map.

Regards,
George

 

[Adriadne]

Thanks George (you too Hurkyl), that's helpful. I particularly like the notion of a field being a cross section of a bundle.

 

[Jimmy Snyder]

Is it that dual vectors are one-forms, but not all one-forms are dual vectors (e.g. covariant tensors etc) or is the difference more subtle?

a one-form is a tensor.

Hurkyl agrees with Schutz.

I am reading Schutz too. I found that I would read until I encountered the first thing I didn't understand (perhaps one-forms are the first thing you don't understand in Schutz) and keep on reading to the second thing I didn't understand, and keep on going until I was reading without understanding anything. At that point I went all the way back to the beginning and started again. The second reading got me further along, but the process had to start a third and then a fourth time. I have lost count. I suggest that you continue reading past this problem for the time being.

At the top of page 61 and substituting 1 for N, you find that a tensor of type (0,1) is a linear function from a vector space to the real numbers. And at the bottom of page 62 that a tensor of type (0,1) is called a one-form. Putting these two together we get:

A one-form is a linear function from a vector space to the real numbers. In this regard as you have pointed out, it is also known as a dual vector. Schutz also mentions other names for it such as covector, etc. I first encountered it with the name linear functional.

Indeed, given a vector space, there is a whole space of such one-forms. This is also known as a dual space.

What is more, if the vector space has a metric tensor g, then you can associate a particular one-form with a particular vector as follows: g(V, ). That is given V, there is a function g(V, ) that takes a vector, places it in the empty slot, and translates it linearly to a number, i.e. g(V, ) is a one-form. In this way, you get a function from vectors to one-forms. This function is a one-form field on the vector space.

This discussion has its own dual. That is vectors can be defined as linear functionals on one-forms. There is no circularity here. Its just a matter of which one you define first. The other one becomes the dual.

Schutz provides some images to help you visualize what a one-form looks like. It does not look like a vector, but rather looks like contour lines.

 

[Adriadne]

I am reading Schutz too. ...At the top of page 61 and substituting 1 for N, you find that a tensor of type (0,1) is a linear function from a vector space to the real numbers.

mm. My Schutz is Geometrical methods of mathematical physics I suspest you're reading his GR

Indeed, given a vector space, there is a whole space of such one-forms. This is also known as a dual space.

Why sure. The problem I had, and have only resolved in a Micky Mouse way (with help here) is that Schutz introduces 1-forms as a dual space at the point P, whereas George and Hurkyl say they are a field. Yet later Schutz introduces the notion of a 1-form field, which sounds kind of tautological in George's and Hurkyl's terminology.
OK, here's my moron's entry into this (think of it as getting used to the concept). I consider the 1-dimensional manifold, the curve C. At one point P on C I define a vector space V of (possibly infinitely many) tangent vectors. I now imagine a field of dual vectors V*(F), each v* in V*(F) being assigned to all possible points on C, such that each v in V "interacts" - being MM I'll say intersects - with a definite "number" of v*. Let me refer to the dual field as 1-forms.
To make it easy for myself, I think of elements of V*(F) as being "perpendicular" to C. Now, the "number" of intersects of any v in V with any subset of v* in V*(F) gives me a notion of magnitude of my vectors in V.
And for manifolds of dim < 1, this also gives me a notion of angle. Which is what I need, of course.
So I can now say that those duals in the field V*(F), my 1-forms, that "intersect" a vector v at P are elements of the dual field evaluated at the point P.
In other words, these intersections are a property of vectors in the space
V at P, not of those in V*(F), so I can think of the set of 1-forms so intersected as a space of 1-forms over the point P of C.

 

[Jimmy Snyder]

My Schutz is Geometrical methods of mathematical physics I suspest you're reading his GR

You are right. Sorry for the interruption.

 

[Adriadne]

Jimy, don't be silly, you helped me loads. Thanks

 

[Jimmy Snyder]

I went to the library to get a copy of Schutz's "Geometrical Methods of Mathematical Physics". I have the first edition, so page and section numbers may not agree with yours. The first paragraph in section 2.16 on page 49 says "we define a one-form as a linear, real-valued function of vectors". He never deviates from this definition, so there should be no confusion over the difference between a one-form, a vector space of one-forms, and a one-form field.
If you start with a manifold, then for each point P in the manifold, you can define a vector space called the tangent space and denoted $T_P$. For instance, for a sphere imbedded in Euclidean 3-space, the tangent vector space at a point P on the sphere would be the plane that touches the sphere tangentially at the point P. As the tangent space is a vector space, we can define one-forms on it. In fact we can define an entire vector space of one-forms for this vector space. That dual space is the space of one-forms at P.
However, there is a typo in his description. I wonder if the typo is the source of your problems. I will quote him exactly, and then I will provide the corrected version. Of course, this may already be fixed in your edition of the book. On page 50, below eqn. (2.16a)
Thus one-forms at the point P satisfy the axioms of a vector space, which is called the dual vector to $T_P$, and is denoted by $T^*{}_P$.
It should read:
Thus one-forms at the point P satisfy the axioms of a vector space, which is called the dual (vector) space to $T_P$, and is denoted by $T^*{}_P$.
Please let me know if this helps. If you have other questions about this book, please ask them sooner rather than later as I will only have the book for a limited time.

 

[Adriadne]

Jimmy, thanks for that. Last things first (as always!). My Dover edn. of Schutz's book has indeed corrected that typo. Had it not, I would have binned it by now.

But to return to the core of my problem. Schutz, you, me have no problem in thinking about a vector space T*p at P which is dual to the vector space Tp at P. That doesn't differ in any way from what I was taught before I'd ever heard of 1-forms. And I was quite content to assume that 1-forms were a generalization from dual vectors (tpye (0,1) tensors), to tensors of higher rank, aka covariant tensors.

But George and Hurkyl insist that 1-forms are a field not a space, and these are guys whose opinions command respect. Now, Shutz, section 2.19, p52, subsequently introduces the notion of a field of 1-forms, and then (p53) says that 1-form fields are cross-sections of the cotangent bunndle T*M. Which is exactly what these two guys told me.

Maybe I'm quibbling, but it seems to me that George and Hurkyl want to define a 1-form as a field, whereas Schutz is saying there is a space of 1-forms at P, a bundle of 1-forms on M and 1-form fields formed by taking cross sections of the bundle.

Geuss it's not big deal at the end of the day, we all arrive at the same place.

 

[Jimmy Snyder]

But George and Hurkyl insist that 1-forms are a field not a space

But Schutz insists that a one-form is a function that maps a vector to a number. I don't see how any other definition is going to help you understand that book. In section 2.19 Schutz consistently uses the term one-form field and, never the term one-form, to describe a field of one-forms. At any given time, Schutz speaks of three different things and never confuses the terminology.

1. A one-form

2. A vector space of one-forms

3. A field of one-forms

Perhaps there is some confusion over notation since he does elide the dependence of a one-form field on the underlying points. In other words he uses the notation $\tilde{\omega}$ to denote a one-form and while it might make some things more clear to use the notation $\tilde{\omega}(p)$ to denote a one-form field, he uses the notation $\tilde{\omega}$ for that too. Perhaps because my background is in math and not physics, I prefer Schutz's practice of suppressing the argument because it differentiates between the field and the value of the field at a point.

 

[Adriadne]

So, I have from Hurkyl (I paraphrase, I hope accurately): A 1-form is a dual vector field and is a tensor. Implication: a tensor is a vector field.

From George: I agree, but don't worry, we can evaluate the 1-form = dual vector field, at the point P, and get our linear functional as usual.

Schutz: A 1-form at P associates to a vector at P a real number. 1-forms at the point P satisfy the axioms of a vector space...called the dual vector space...A field of 1-forms...is a rule giving a 1-form at every point...Cross sections of the cotangent bundle are 1-form fields.

My education pre-1-form: To any vector space V one can associate a dual vector space V* such that elements of V* associate to elements of V a number. (This is so close to S., it was the basis of my OP).

Putting this all together I have: A 1-form is a tensor, is a covector field. A 1-form field is a tensor field, is a field of covector fields.

Hmm

I freely admit I came to this site as a shortcut vs. book-learning. Oh well, hello books...

(EDIT: Sorry guys, that came out as ungracious, it was meant to be self-deprecating)

 

[Jimmy Snyder]

Adriadne, I apologize for taking so long to figure out what you are saying. Now I suggest that you may have taken some of Hurkyl's and George's words out of context. Here is a quote from George:

Physicists and physics books often use terms like "vector" and "vector field" interchangeably, letting context sort out what is really meant.

George is not saying that the concept of vector is somehow blurred with that of vector field, these are two separate concepts. It is only the terms that are used interchangably. By the way, I might quibble about that 'interchangeably' part since I am familiar with the practice in one direction only.

He goes on:

I prefer Hurkyl's terminology, i.e., a 1-form is a dual vector field. I think you'll find that Schutz uses "1-form" in both senses - as a field of dual vectors, and as a specific dual vector.

Well, he comes out and says what you are focused on "a one-form is a dual vector field". However, that is not a direct quote from Hurkyl and is considerably softened by what comes next. Again, there are TWO senses, and these are not to be confused even if the same term is used for both of them.

I had said that Schutz never mixes the terminology, but that was based on reading only a few paragraphs of the book. In those paragraphs, he was meticulous in his vocabulary, but I can't say whether he relaxes his usage later on as George suggests.

 

[Adriadne]

Then I give up! If there's no agreed definition, only a general agreement that we don't need to specify (even to a beginner) whether we're talking about fields, spaces or vectors, then it's a lost cause.
I'm really cross! (but only at myself).

Thanks all for trying.

 

[George Jones]

Then I give up! If there's no agreed definition, only a general agreement that we don't need to specify (even to a beginner) whether we're talking about fields, spaces or vectors, then it's a lost cause.

No, it isnt.

Terminology in most math books:

1) cotangent vector = dual vector;

2) 1-form = field of cotangent vectors.


Terminology in some math books:

1') 1-form = dual vector;

2') 1-form field.

I much prefer 1) and 2). Apparently, Schutz uses 1') and 2').

Regards,
George

 

[Adriadne]

Phew! Thanks George, I was beginning to think I should study sociology instead.

 

[Jimmy Snyder]

Terminology in most math books:
1) cotangent vector = dual vector;
2) 1-form = field of cotangent vectors.
Terminology in some math books:
1') 1-form = dual vector;
2') 1-form field.

Now I realize why it took me so long to get it. The only books using the term 'one-form' that I have read were both written by Schutz. I kept thinking that some people use 'one-form' and 'one-form field' to mean field of dual vectors, but that everyone used the term one-form to mean dual vector. Now I see that most authors do not use the term in that sense.

Adriadne, my take on this is that you need to store this information away for when you read other books on the subject. Schutz seems to use a terminology different from the others. This is by no means uncommon in math books and is so commonplace in physics books that MTW's 'Gravitation' includes a translation chart for various notational convensions. Unfortunately it is the responsibility of the reader to be familiar with the notational convensions of the author. The author's only responsibility is to be self consistent or at least unambiguous about the meaning when not. But some authors duck this responsibility too so caveat lector.

 

[mathwonk]

some people use the words "one form" for a dual vector, and some people like spivak reserve that usage for a family of dual vectors. that's all.