ABSTRACT
Differential forms are integrands, the things one integrates. So dx is a dif-
ferential form (a 1-form), and so is dx dy (a 2-form). However, orientation
matters; think about change of variables, where for instance
du dv = ∂(u, v)
∂(x, y) dx dy, (6.1)
where ∂(u,v)∂(x,y) denotes the determinant of the Jacobian matrix of (u, v) with
respect to (x, y). One often puts an absolute value sign around this determi-
nant, but that’s misleading. For example, flux depends on the orientation
of the surface, so it’s a feature, not a bug, for integrals to depend on the
ordering (“handedness”) of the coordinates.
