A New Look at Differential 1-Forms

When Arnold introduced the notion of differential 1-forms in Chapter 1, I had accepted it simply as linear functions of vectors. But there is more. Differential 1-forms has changed the way I understand lines and planes; it is also a powerful tool in creating new structures in space.

I will provide examples below:

Notice that by describing a line in the (t, x)-plane using the form at point (t,x), we are seeing the line as a collection of velocity vectors leaving point (t, x) heading in the direction p (up to sign).

Similarly, a plane defined by at point (t, x, p) are the collection of velocity vectors (in 3-dim. space) leaving (t, x, p) such that its motion in the x-direction and its motion in the t-direction satisfy this relation. And if we are to ignore coordinates, then that plane can be seen as a class of motions satisfying a certain relation.

And notice how this differential form defines not only one plane but a field of planes. It is essentially describing the relation between the velocity vectors and their point of departure at every single location in space.