Video of the lecture.

I explained that a differential 0-form w(x,y,z) in 3 variables, in conjunction with some constant k, can be thought of as the surface S defined by w(x,y,z) = k. I went on to explain that, when the 0-form w is viewed in this way, the derivative dw is just a unit normal vector to the surface S.

The explanation began with a review of the dot product of vectors. It also included the alternative terminology grad(w), or gradient of w, for derivative dw of the 0-form w.