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. The resulting calculus, known as exterior calculus, allows for a natural, metric-independent generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus. If a differential k-form is thought of as measuring the flux through an infinitesimal k-parallelotope at each point of the manifold, then its exterior derivative can be thought of as measuring the net flux through the boundary of a (k + 1)-parallelotope at each point. The exterior derivative of a differential form of degree k (also differential k-form, or just k-form for brevity here) 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. The exterior product of differential forms (denoted with the same symbol ∧) is defined as their pointwise exterior product. 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 that has the following properties: df is the differential of f for a 0-form f . d(df ) = 0 for a 0-form f . d(α ∧ β) = dα ∧ β + (−1)^p (α ∧ dβ) where α is a p-form. That is to say, d is an antiderivation of degree 1 on the exterior algebra of differential forms (see the graded product rule). The second defining property holds in more generality: d(dα) = 0 for any k-form α; more succinctly, d^2 = 0. The third defining property implies as a special case that if f is a function and α is a k-form, then d( fα) = d( f ∧ α) = df ∧ α + f ∧ dα because a function is a 0-form, and scalar multiplication and the exterior product are equivalent when one of the arguments is a scalar.