One of the definitions of a $1$-form is that it is a vectorfield picking a covector for each point. Or in other words picking one member of the cotangent space at some point, in the same way a vector field is a pick from the tangent space.
Presumably the 1 forms is something we use to study differentail equations, which in turn are related to picking a vector of the tangent space.
I am puzzled how we are suppose to use the $1$-forms to study the equations, we usually have $ 1$ vector from the tangent space given an equation. But to "get info" on this vector it seams we want more then just $1 $ covector or equivalently $ 1$ functional on the tangent space.
Does someone understand what I don't understand?
A one-form defines a hyperplane, a co-dimension 1 subspace, in the tangent space. If your space is the euclidean plane, then these subspaces resp. their projection from the tangent space to the point space are lines. Thus a one-form in a plane defines a direction field. No orientation or length of vector is fixed. However, if you select them, with having quite the freedom in this, you get a vector field on which you then can solve a differential equation.