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 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.
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:
The second defining property holds in more generality: d(dα) = 0 for any k-form α; more succinctly, d2 = 0. The third defining property implies as a special case that if f is a function and α a is 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.
Alternatively, one can work entirely in a local coordinate system (x1, ..., xn). The coordinate differentials dx1, ..., dxn form a basis of the space of one-forms, each associated with a coordinate. Given a multi-index I = (i1, ..., ik) with 1 ≤ ip ≤ n for 1 ≤ p ≤ k (and denoting dxi1 ∧ ... ∧ dxik with an abuse of notation dxI), the exterior derivative of a (simple) k-form
The definition of the exterior derivative in local coordinates follows from the preceding definition in terms of axioms. 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.
In particular, for a 1-form ω, the components of dω in local coordinates are
In particular, when ω is a 1-form we have that dω(X, Y) = dX(ω(Y)) − dY(ω(X)) − ω([X, Y]).
Note: With the conventions of e.g., Kobayashi–Nomizu and Helgason the formula differs by a factor of 1/k + 1:
Example 1. Consider σ = u dx1 ∧ dx2 over a 1-form basis dx1, ..., dxn for a scalar field u. The exterior derivative is:
The last formula, where summation starts at i = 3, follows easily from the properties of the exterior product. Namely, dxi ∧ dxi = 0.
Example 2. Let σ = u dx + v dy be a 1-form defined over ℝ2. By applying the above formula to each term (consider x1 = x and x2 = y) we have the following 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. For smooth manifolds, integration of forms gives a natural homomorphism from the de Rham cohomology to the singular cohomology over ℝ. 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
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 0-form is the 1-form df.
When an inner product ⟨·,·⟩ is defined, the gradient ∇f of a function f is defined as the unique vector in V such that its inner product with any element of V is the directional derivative of f along the vector, that is such that
where ♯ denotes the musical isomorphism ♯ : V∗ → V mentioned earlier that is induced by the inner product.
The 1-form df is a section of the cotangent bundle, that gives a local linear approximation to f in the cotangent space at each point.
A vector field V = (v1, v2, ..., vn) on ℝn has a corresponding (n − 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
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.