Exterior derivative

The exterior derivative was first described in its current form by Élie Cartan in 1899.

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.

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: If

Given a multi-index I = (i1, ..., ik) with 1 ≤ ip ≤ n for 1 ≤ p ≤ k (and denoting dxi1 ∧ ... ∧ dxik with dxI), the exterior derivative of a (simple) k-form over ℝn is defined as (using the Einstein summation convention).

Note that whenever j equals one of the components of the multi-index I then dxj ∧ dxI = 0 (see Exterior product).

Indeed, with the k-form φ as defined above, Here, we have interpreted g as a 0-form, and then applied the properties of the exterior derivative.

By applying the above formula to each term (consider x1 = x and x2 = y) we have the sum If M is a compact smooth orientable n-dimensional manifold with boundary, and ω is an (n − 1)-form on M, then the generalized form of Stokes' theorem states that Intuitively, if one thinks of M as being divided into infinitesimal regions, and one adds the flux through the boundaries of all the regions, the interior boundaries all cancel out, leaving the total flux through the boundary of M. A k-form ω is called closed if dω = 0; closed forms are the kernel of d. ω is called exact if ω = dα for some (k − 1)-form α; exact forms are the image of d. Because d2 = 0, every exact form is closed.

The Poincaré lemma states that in a contractible region, the converse is true.

Because the exterior derivative d has the property that d2 = 0, it can be used as the differential (coboundary) to define de Rham cohomology on a manifold.

The k-th de Rham cohomology (group) is the vector space of closed k-forms modulo the exact k-forms; as noted in the previous section, the Poincaré lemma states that these vector spaces are trivial for a contractible region, for k > 0.

The theorem of de Rham shows that this map is actually an isomorphism, a far-reaching generalization of the Poincaré lemma.

As suggested by the generalized Stokes' theorem, the exterior derivative is the "dual" of the boundary map on singular simplices.

The exterior derivative is natural in the technical sense: if  f : M → N is a smooth map and Ωk is the contravariant smooth functor that assigns to each manifold the space of k-forms on the manifold, then the following diagram commutes so d( f∗ω) =  f∗dω, where  f∗ denotes the pullback of  f .

Most vector calculus operators are special cases of, or have close relationships to, the notion of exterior differentiation.

A smooth function  f : M → ℝ on a real differentiable manifold M is a 0-form.

The exterior derivative of this (n − 1)-form is the n-form A vector field V on ℝn also has a corresponding 1-form Locally, ηV is the dot product with V. The integral of ηV along a path is the work done against −V along that path.

When n = 3, in three-dimensional space, the exterior derivative of the 1-form ηV is the 2-form The standard vector calculus operators can be generalized for any pseudo-Riemannian manifold, and written in coordinate-free notation as follows: where ⋆ is the Hodge star operator, ♭ and ♯ are the musical isomorphisms,  f  is a scalar field and F is a vector field.

Note that the expression for curl requires ♯ to act on ⋆d(F♭), which is a form of degree n − 2.

A natural generalization of ♯ to k-forms of arbitrary degree allows this expression to make sense for any n.