In order to explain to non-experts what a vector field is, one usually describes an assignment of an arrow to each point of space. And this works quite well also when moving to manifolds, where a generalized arrow will be a tangent vector.
My question is: What are similar objects that can help with imagining differential forms?
Positive qualities for such an object would be, for example:
- it helps justify change-of-coordinate formulas and formulas for the pullback via a function;
- it is "easily drawable";
- it helps understand more complicated differential-form-based concepts, e.g. connections, cohomology groups, etc.