So, I recently moved in to a new research area. It's new to my advisor, too. Actually, generally speaking, it's pretty new period, first appearing some ten years ago or less. Anyway, this new field deals fairly heavily with Maxwell's Equations in curved spacetime, so to understand it we are needing to review differential geometry and general relativity, two fields which are not in the normal purview of my advisor's expertise. I was asked to prepare a chalk-talk that would introduce the key concepts of differential geometry to them, and another talk to segue in to Maxwells Equations in curved spacetime.
Not like I'm an expert on differential geometry, but I've studied it some privately and as an undergraduate.
While studying for this, it dawned on me suddenly, like the storm clouds that pile higher and higher until the first bolt of lightning strikes the ground, that vectors and 1-forms are different.
Every thing I have ever read in physics equates them. Or not really. Everything I have ever read in physics doesn't even demonstrate that it understands why those two should occupy different semantic domains.
What the heck am I even talking about?