> it's just interesting that all the images are of vector fields. Because they are! You can indeed "flatten" all these forms into vector and scalar fields, and

> it's just interesting that all the images are of vector fields. Because they are! You can indeed "flatten" all these forms into vector and scalar fields, and you lose some information but the data is the same. It's like forgetting the types of objects in a programming language and considering only their representation as byte arrays in memory. When you do multivariable calculus, you soon realize that there are two different kinds of vector fields: those that are "gradients" or "rates" and those that are "speeds", or "displacements" or "flows". They have different units, like 1/length or length/time. Notice that if you change the units of length from meters to centimeters, the arrows that represent gradients become shorter, and the arrows that represent speeds become longer (assuming that you keep the same data). Likewise, there are two different kinds of scalar fields: potentials, temperatures, etc, that are invariant to unit changes; and densities that change with the cube of the unit scaling. You don't need differential forms to understand any of this. But differential forms provide a nice formalization where all these objects are of different types, and it explains what differential operators can you apply to objects of one type to obtain another, and so on. Vectors are rates. Covectors are gradients. -- Luc Florack