Differential Form

1. Differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds.
2. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.

Differential forms are naturally dual to vector fields on a differentiable manifold, and the pairing between vector fields and forms is extended to arbitrary differential forms by the interior product. The algebra of differential forms along with the exterior derivative defined on it is preserved by the pullback under smooth functions between two manifolds. This feature allows geometrically invariant information to be moved from one space to another via the pullback, provided that the information is expressed in terms of differential forms. As an example, the change of variables formula for integration becomes a simple statement that an integral is preserved under pullback.

Sources:

Словник Webster's Dictionary https://www.merriam-webster.com/

Random House Unabridged Dictionary, Copyright © 1997, by Random House, Inc

Isaac Physics https://isaacphysics.org/