Definition of One-Forms & Their Action | Clarifying a Concept

[maverick280857]

Hi,

While reading Sean Carroll's book, I came across the following statement:

The canonical example of a one-form is the gradient of a function $f$, denoted $df$ as in (1.52). Its action on a vector $d/d\lambda$ is exactly the directional derivative of the function:

$$df\left(\frac{d}{d\lambda}\right) = \frac{df}{d\lambda}$$

It's tempting to ask, "why shouldn't the function $f$ itself be considered the one-form, and $df/d\lambda$ its action?". The point is that a one-form, like a vector, exists only at the point it is defined, and does not depend on information at other points on M. If you know a function in some neighborhood of a point, you can take its derivative, but not just from knowing its value at the point; the gradient, on the other hand, encodes precisely the information necessary to take the directional derivative along any curve through $p$, fulfilling its role as a dual vector.

Okay so this has me confused. Perhaps I am nitpicking, but isn't $f$ a scalar function, i.e. a 0-form? So shouldn't he really be saying "why shouldn't $df$ be considered the one-form..."?

If f is a scalar, then df (as he defines it) is

$$df = \frac{\partial f}{\partial x^\mu} dx^\mu$$

the 1-form (one way of thinking of d is the exterior derivative of a 0 form, which produces a 1 form). Now, the components of df in the coordinate basis are indeed dependent on the $\epsilon$ neighborhood of any point where you take the derivative. But in the limit of such an $\epsilon$ going to 0, the derivative (as a 1-form) does become local.

However what is the problem if the 1 form has a limiting expression that does depend on the neighborhood?

Am I missing something?

 

[pervect]

I'm not following what Sean means here either, I'm afraid. My impression was that you had a vector space and it generated naturally a dual space, but which one you called the vector space and which one you called the dual space was somewhat a matter of convention.

However, it's definitely conventional to call the tangent space the vector space.

Once you have defined the tangent space as the vector space, the dual of the vector space is a map from the vector space to a scalar. It turns out that the dual vector is also a one-form, as an n-form is an anti-symmetric map from n vectors to a scalar.

 

[maverick280857]

Thanks for the reply pervect. I am not sure I understand the distinction he's trying to put forth though about the alleged locality of the one-form as opposed to that of a derivative. If I consider the exterior derivative of a 0 form, I generate a 1 form, which is (very loosely speaking) similar to a 'differential'. So why does a 1 form have to be dependent only on the value at that point?