The distinction between the two main kinds of vectors should be taught in high school:
• A train moves along a track with a velocity in, say, metres/second, but described with *bigger* coords if you use mm/sec.
• A temp gradient in °/metre has *smaller* coords if you use °/mm.
And you can, beautifully, combine the two: a train, with a velocity in whatevers/second, moves across a temperature gradient given in °/whatevers, and if you add up the products of the components, you get the same temperature rate of change in °/sec, regardless of your choice.
@gregeganSF That does seem like a nice way of approaching vectors vs. one-forms without a bunch of jargon/machinery. Certainly I found the way I was originally taught these concepts to be very opaque.
