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.
Definition
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.
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 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.[citation needed]
In terms of local coordinates
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-indexI = (i1, ..., ik) with 1 ≤ ip ≤ n for 1 ≤ p ≤ k (and denoting dxi1 ∧ ... ∧ dxik with dxI), the exterior derivative of a (simple) k-form
where each of the components of the multi-index I run over all the values in {1, ..., n}. Note that whenever i equals one of the components of the multi-index I then dxi ∧ dxI = 0 (see Exterior product).
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.
This result extends directly to the general k-form ω as
In particular, for a 1-form ω, the components of dω in
local coordinates are
Caution: There are two conventions regarding the meaning of . Most current authors[citation needed] have the convention that
while in older text like Kobayashi and Nomizu or Helgason
In terms of invariant formula
Alternatively, an explicit formula can be given [1] for the exterior derivative of a k-form ω, when paired with k + 1 arbitrary smooth
vector fieldsV0, V1, ..., Vk:
where Vi, Vj denotes the
Lie bracket and a hat denotes the omission of that element:
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:
Examples
Example 1. Consider σ = udx1 ∧ 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 = udx + vdy be a 1-form defined over ℝ2. 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.
de Rham cohomology
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.
Naturality
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. This follows from that f∗ω(·), by definition, is ω( f∗(·)), f∗ being the
pushforward of f. Thus d is a
natural transformation from Ωk to Ωk+1.
Exterior derivative in vector calculus
Most
vector calculus operators are special cases of, or have close relationships to, the notion of exterior differentiation.
Gradient
A
smooth functionf : 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
That is,
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.
Divergence
A vector field V = (v1, v2, ..., vn) on ℝn has a corresponding (n − 1)-form
where denotes the omission of that element.
(For instance, when n = 3, i.e. in three-dimensional space, the 2-form ωV is locally the
scalar triple product with V.) The integral of ωV over a hypersurface is the
flux of V over that hypersurface.
The exterior derivative of this (n − 1)-form is the n-form
Curl
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
Invariant formulations of operators in vector calculus
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.