# One-form is a linear map from a vector to a real number

1. Mar 7, 2006

### jwsiii

I'm trying to understand what one-forms are. The book I'm reading says a one-form is a linear map from a vector to a real number. It uses the gradient as an example but isn't the gradient a map from a function to a vector?

2. Mar 8, 2006

### Hurkyl

Staff Emeritus
No, the gradient is actually a map from a function to a one-form!

When people first learn about gradients, they don't know the difference between vectors and one-forms, and the Euclidean inner product gives rise to a canonical way to write one-forms as vectors. So, people first learn that the directional derivative is given by a dot product.

But in general, the gradient of a function is naturally a one-form: its purpose in life is to take a vector and spit out the value of the directional derivative in that direction -- this value doesn't depend on the inner-product (metric) you're using... and even works when your space doesn't have an inner-product at all!

As coordinates with respect to a basis, you may have seen that while vectors are supposed to be nx1 matrices ("column vectors"), the gradient is always written as an 1xn matrix ("row vectors"). This is true in general -- in coordinates, one-forms are supposed to be written as 1xn matrices, while vectors are as nx1 matrices.

3. Mar 8, 2006

4. Mar 8, 2006

### mathwonk

every real vector space V has a dual space V* consisting of linear functions from V to R.

every manifold, e.g. surface, is covered by a family of tangent spaces, e.g. the famnily of tangent planes to the sphere.

thus also on every manifold M there is both a family of tangent spaces, and also a family of duals of tangent spaces.

a tangent vector field on M is a choice of a tangent vector at each point of M, thus a family of tangent vectors.

a covector field on M is a choice of dual vector at each point of M, i.e. a family of dual vectors. this latter is also called a one form. i.e. a one form is a family of dual vectors, i.e. a family of linear functions on tangent spaces.

some people use the term (incorrectly in my view) to refer to an element of just one dual space, hence they use "one form" to name a linear functional from the space of tangent vectors at one point to numbers, but this is what I am calling a dual vector.

so given a vector space some people incorrectly use the term "one form" to refer to a linear function from that space to numbers. Others correctly use the term to refer to a family of such linear functions from a family of vector spaces to numbers.

hence given a function f on an open set U, the gradient of f, gradf, assigns to each tangent vector v(p) at each point p, a number, namely the directional derivative of f in the direction v(p).

hence grad(f) is [correctly called] a one form on U.

hence hurkyl [correctly] says that "grad" is a function from functions to one forms.

admission: even those who use the term "incorrectly" do so acceptably if they define their terms. they still differ from the mass of other users, but they cannot be blamed for choosing to differ from the rest of us, if they are clear about their terminology. it nonetheless happens that their use of the terminology will not serve their readers well except in their own books.

Last edited: Mar 8, 2006
5. Mar 8, 2006

### mathwonk

sadly, the terms "covariant" and "contravariant" are extremely unfortunate, as history has crowned with the term "covariant" those objects which behave in a [categorically] contravariant fashion, and vice versa. only god can forgive this idiocy, but classical differential,geometers, hence all physicists, cling to it.

as we say in the south, "bless their hearts".

6. Mar 8, 2006

### robphy

What probably needs to happen is for someone to write THE CANONICAL TEXT on the subject (and, of course, have it adopted)... then future generations will adopt the [correct] terminology, definitions, notations, points of view, and intuitions. (What I also wish would happen is that such [advanced] concepts get gradually introduced [in some way] much earlier in our education... so it will seem more "natural".)

7. Mar 8, 2006

### mathwonk

the book calculus on manifolds by mike spivak has been a standard now for over 40 years. how long does it take to get on the bandwagon?

8. Mar 8, 2006

### mathwonk

i confess to being puzzled at the discussion as to the meaning of the term "one forms" as all my books agree on this point:

hirzebruch; topological methoids in algebraic geometry, p.35;

phillip griffiths, introduction to algebraic curves : p.21;

analysis II, serge lang, p.406;

differential forms in algebraic topology, bott and tu, p.13;

complex manifolds and deformations of complex structures, kunihiko kodaira, p.76;

michael spivak, calculus on manifolds, p.88;

differential topology, guillemin and pollack, p.162;

notes on differential geometry, noel j. hicks, p.,50;

i could go on and on, but i will not.

my point here is that there is no difference of opinion whatsoever in the entire world of mathematics, no controversy at all; there is one and only one uniform agreement among "us" on what this definition means.

Last edited: Mar 8, 2006
9. Mar 9, 2006

### Haelfix

The difference is not manifest in modern treatments of differential forms and tensor analysis, everyone has been in agreement for 50-60 years about it. However before that point, for historical reasons the confusion was far more apparent (though probably well understood even then), and I guess for whatever reason (I would say classical general relativity being the main culprit) a few bigshots introduced poor wording into the lingo that persisted throughout the years and to this day. The prototypical example is contravariant vs covariant and so forth.

C'est la vie.

Operationally for applications so long as we know what we are doing, well, we know what we are doing.

Physicists are not the only ones incidentally who are guilty of throwing in ridiculous terminology. There are whole fields of mathematics (say Hodge theory and/or differential operators), where people heuristically tried to motivate the discussion by reusing linear algebra terminology when clearly it made no sense to do so, beyond formal manipulations.

In many ways, mathematics is confusing, precisely b/c we ran out of letters in the alphabet to represent objects and concepts. The same thing is true with words and phrases.

10. Mar 9, 2006

### George Jones

Staff Emeritus
Well, an algebraist once, after I mentioned categories, said to me something like "Category theory should be functored out of existence." :tongue2:

I prefer to use vector for an element of a tangent space and covector for an element of a cotangent space, dropping the "variant" altogether.

Mathematicans don't always agree on terminology. For example, the text (Munkres) for the point-set topology course that I took defines a neighbourhood of x as an open set that contains x, while many other books define a neighbourhood of x as any set that contains an open set that contains x. I as I recall, these differing definitions make proofs and other definitions look a little different.

I promote strongly the "mathematician's" definition of one-form. However, there are a couple possible reason for the different meaning of one-forms - one in mathematics and one in physics.

One yellow-and-white book which I have says that a natural extension of the concept of a bilinear form on a vector space is a multilinear form. The book hints that this terminology is consistent with calling an element of the dual space a one-form. In any case, I think that you might like this book - Tensor Geometry by Dodson and Poston. From its preface:'The title of this book is mileading. Any possible title would mislead somebody. "Tensor Analysis" suggests an ungeometric, manipulative, debauch of indices, with tensors ill-defined as "quantitities that transform acording to" unspeakable fromulae. "Differential Geometry" would leave many a physicist unaware that the book is about matters withwich he is very much concerned.'

Physicists sometimes conflate f, where f: X -> Y, with f(x), an element of Y. An extension of this is that physcists often conflate a cross section on a cotangent bundle with an element of a cotangent space. Usually, context sorts things out.

Try not to shudder too violently.

Regards,
George

11. Mar 9, 2006

### mathwonk

i have just written and lost a detailed study of this topic to the vagaries of the browser software used here.

here is some of it:
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basically george is right. but the terminology he uses is out of date. true algebraists, out of respect for gauss, used to call homogeneous polynomials "forms" (see zariski samuel, page 35, commutative algebra, 1955 or so) but this persists today almost exclusively to refer to just "quadratic forms". (I have used the term symplectic form myself in papers.)

hardly anyone calls linear functions "linear forms" anymore.

unfortunately the desire to abbreviate the common term "differential form", has led to the common occurrence of "form", or "p form" as short for "differential p - form"; since the old algebraists term is little used any more, this confuses only people still using it in the old sense.

david bachman has attempted to satisfy boith camps in his book by referring to the pointwise concepot as "p form" and to the glovbal one as "differential p form".

this is however not much more logical mathematically since the word "differential" does not clearly contain the global nature of the geometers concept as oppsoed to the pointwise notion of the algebraist.

for example there are global forms that are not differentiable, merely continuous, or not even that.

the more universal word for a global collection of objects is the physicists "field" so it would be clearer to speak of vector fields or covector fields, although the covector fields are the contyravariant ones.

I would advise all young students as well to learn the basic ideas of categories and functors or risk being innocent of the underlying structure and unity of mathematics, despite the opinions of disgruntled senior workers, tired of the excessive abstraction of some practicioners of the art.

everyone should ask himself e.g. how a construction which turns one sopace into another, acts on the amps of those spaces.

the chain rule e.g. is a primary example of the fact that the derivative is a functor. having the functorial point of view would signal that learning the definition of a derivative is useless without knowing the chain rule, a point often lost on young students.

12. Mar 9, 2006

### mathwonk

the communication problem persists, aided by the fact that many physicists seem to study mathematics from very old books.

but in the discussion above, context is important: if talking algebra or number theory, "form" usually means multilionear fucntion. if talking differential geometry, it usually means a field of tangent vectors or dual tangent vectors.

except in bachman's book, but he does define his terms. thus his readers are ok until they venture into anmother book, and there too they are ok if they read the definitions.

which makes me wonder why the original questioner had the question in the first place. presumably the book he is reading defiend the term "one form". if so it means whatever they said it did. i.e. like in alice and wonderland, "when i use a word it means whatever i choose for it to mean."

13. Mar 9, 2006

### mathwonk

so the answer to the original question is: not everyone agrees on the use of words, but your book is to me somewhat out of the main stream, although it would not have been 50-150 years ago.

14. Mar 29, 2006

### kryptyk

Reciprocal Frame

Say we have a frame $$\{ \mathbf{e}_1, \mathbf{e}_2, \ldots, \mathbf{e}_n\}\;$$. Even if we have not defined an inner product,

$$\mathbf{e}_i \cdot \mathbf{e}_k = ?$$

we can still construct a reciprocal frame $$\{ \mathbf{e}^1, \mathbf{e}^2, \ldots, \mathbf{e}^n\}\;$$, no? We would just require that it satisfy:

$$\mathbf{e}^i \cdot \mathbf{e}_k = \delta_{k}^{i}$$

15. Mar 30, 2006

### dextercioby

That reciprocal frame is actually a basis in the cotangent space and is made up of linear functionals over the tangent space (made up of vectors).

You should write

$$e^{i}\left(e_{k}\right) =\delta^{i}_{k}$$

Daniel.

16. Apr 15, 2006

### javanse

Interesting to see, that if you have an orthonormal frame there really do exist a dual basis which is simply the projection on the ith coordinate.

There is a confusion about notation:

The reciprocal frame is not simply the basis of the cotangent space.

Its like if you say a covector is a one-form.
A one-form is a differential-form, but a covector is in generell a linear map from a vector space to the reals. So a one-form is a special case of a covector, if you work in tangent space.