Exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899; it allows for a natural, metric-independent generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus.
If a k-form is thought of as measuring the flux through an infinitesimal k-parallelotope, then its exterior derivative can be thought of as measuring the net flux through the boundary of a (k + 1)-parallelotope.
The exterior derivative of a differential form of degree k is a differential form of degree k + 1.
If f is a smooth function (a 0-form), then the exterior derivative of f is the differential of f . That is, df is the unique 1-form such that for every smooth vector field X, df (X) = dX f , where dX f is the directional derivative of f in the direction of X.
There are a variety of equivalent definitions of the exterior derivative of a general k-form.
The exterior derivative is defined to be the unique ℝ-linear mapping from k-forms to (k + 1)-forms satisfying the following properties:
...
Wikipedia