Incidentally, I'd argue that the determinant is a paradigm of a covolume element: you feed it a volume element and get out a number. (More technically, it liv

Incidentally, I'd argue that the determinant is a paradigm of a covolume element: you feed it a volume element and get out a number. (More technically, it lives in the top exterior power of the cotangent bundle, not of the tangent bundle.)