I’ve had a moderate amount of exposure to the study of differential forms in the context of pure differential geometry, as well as in the background of studies in hypercomplex analysis, abstract algebra, etc. Despite all that, I apparently never learned the actual definition of a differential form.
Recall that the covector space is the dual vector space for the vector space of (tangent) vectors to at . Elements of are called covectors. An assignment of a covector at each point of a differentiable manifold is called a differential form of degree 1.
Seriously, guys: Who knew?