1-Forms |

Fundamentally, a 1-form can be imagined to be a

Let our "scalar field" represent

The most natural way to represent a gradient is as a series of

Locally, the best linear approximation of the contours is a collection of straight lines (or hyperplanes). At a particular point on the manifold, the local linear approximation to the contours is a "covector" (or dual vector) and it's represented as a collection of hyperplanes in the tangent space at that point. On this page, all drawings are restricted to 2 dimensions, and the hyperplanes appear as straight lines.

Now, having given this description, I must hasten to add that, while every gradient is a 1-form, not all 1-forms are actually the gradients of functions. In particular, only "closed", or curl-free, 1-forms are actually the gradients of functions. But for the purposes of understanding 1-forms and their action on vectors, it's perfectly reasonable to think in terms of gradients. And the covector associated with a 1-form at each point is the same, whether or not the 1-form globally represents a gradient.

The gist of this page is in the pictures.

A covector provides a way to

Alternatively, we can view it as a "spatial frequency". It's a frequency

In Figure 2 we show a the covector being "applied" to a vector. The result is the number of hyperplanes the vector crosses. In the figure, the product of the covector and the vector is 5. In other words, the length of the vector

It's worth pointing out that, if we were to multiply the covector by 2, we would

Taking the analogy one step farther, a covector is a frequency, a vector is like a wavelength, and their "product" is analagous to a velocity.

This may sound confusing, but hopefully a picture will make it clearer.

To find the coordinates of a particular covector, all we need to do is apply it to each of the basis

Throughout this website, I tend to use "1-form" to refer to both the covector field on the manifold, and the covector associated with that field at each point.

The dot product of a vector and a covector, or the result of applying a 1-form to a vector, is just the number of lines of the 1-form which the vector crosses. It's the

If we look at the 1-form and the vector in a particular coordinate system, then we see that the total number of lines the vector crosses is just the number of lines it crosses in the direction of each basis vector in turn. It doesn't matter what path you follow -- you need to cross the same number of lines no matter how you go! The number of lines crossed in the direction of each basis vector is just the number of lines crossed by that basis vector,

Or, again in other words, it's just the dot product of the vector and the covector.

It doesn't matter how the axes are chosen.

In Cartesian space, under rigid rotations, the basis