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

1. Oct 22, 2005

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?

2. Oct 22, 2005

### Hurkyl

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

3. Oct 22, 2005

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?

4. Oct 22, 2005

### George Jones

Staff Emeritus
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

5. Oct 22, 2005

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, Im just not getting it

6. Oct 22, 2005

### George Jones

Staff Emeritus
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

7. Oct 22, 2005

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?

Last edited: Oct 22, 2005
8. Oct 22, 2005

### George Jones

Staff Emeritus
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

9. Oct 22, 2005

### Hurkyl

Staff Emeritus
If it helps...

10. Oct 22, 2005

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?

Last edited: Oct 22, 2005
11. Oct 23, 2005

### George Jones

Staff Emeritus
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

Last edited: Oct 24, 2005
12. Oct 24, 2005

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

13. Oct 24, 2005

### Jimmy Snyder

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.

Last edited: Oct 24, 2005
14. Oct 24, 2005

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

15. Oct 24, 2005

### Jimmy Snyder

You are right. Sorry for the interruption.

16. Oct 24, 2005

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

17. Oct 25, 2005

### 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 $T^*{}_P$.
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 $T^*{}_P$.

18. Oct 26, 2005

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.

Last edited: Oct 26, 2005
19. Oct 26, 2005

### Jimmy Snyder

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.

20. Oct 26, 2005

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)

Last edited: Oct 26, 2005