Noun
differential form (plural differential forms)
(differential geometry, tensor calculus, sometimes as "differential p-form") A completely antisymmetric tensor (of order p) that is defined on a Riemannian manifold; an expression, derived by applying a formalism to said tensor, that represents an integrand over the manifold.
The notion of differential form combines the concepts of multilinear form (itself an extension of linear form) and smooth function.
The choice of a Riemannian manifold - roughly speaking, a differentiable manifold whose every point has a tangent space with a defined metric - means it is possible to define a differential form over it.
Differential forms provide a unified approach to defining integrands over curves, surfaces and higher-dimensional manifolds, as well as providing an approach to multivariable calculus that is independent of coordinates.