Exterior derivative
Part of a series of articles about  
Calculus  





Specialized


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, metricindependent generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus.
If a kform is thought of as measuring the flux through an infinitesimal kparallelotope, then its exterior derivative can be thought of as measuring the net flux through the boundary of a (k + 1)parallelotope.
Contents
Definition
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 0form), then the exterior derivative of f is the differential of f . That is, df is the unique 1form such that for every smooth vector field X, df (X) = d_{X} f , where d_{X} 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 kform.
In terms of axioms
The exterior derivative is defined to be the unique ℝlinear mapping from kforms to (k + 1)forms satisfying the following properties:
 df is the differential of f for smooth functions f .
 d(df ) = 0 for any smooth function f .
 d(α ∧ β) = dα ∧ β + (−1)^{p} (α ∧ dβ) where α is a pform. That is to say, d is an antiderivation of degree 1 on the exterior algebra of differential forms.
The second defining property holds in more generality: in fact, d(dα) = 0 for any kform α; more succinctly, d^{2} = 0. The third defining property implies as a special case that if f is a function and α a kform, then d( fα) = d( f ∧ α) = df ∧ α + f ∧ dα because functions are 0forms, and scalar multiplication and the exterior product are equivalent when one of the arguments is a scalar.
In terms of local coordinates
Alternatively, one can work entirely in a local coordinate system (x^{1}, ..., x^{n}). The coordinate differentials dx^{1}, ..., dx^{n} form a basis of the space of oneforms, each associated with a coordinate. Given a multiindex I = (i_{1}, ..., i_{k}) with 1 ≤ i_{p} ≤ n for 1 ≤ p ≤ k (and denoting dx^{i1} ∧ ... ∧ dx^{ik} with an abuse of notation dx^{I}), the exterior derivative of a (simple) kform
over ℝ^{n} is defined as
(using Einstein notation). The definition of the exterior derivative is extended linearly to a general kform
where each of the components of the multiindex I run over all the values in {1, ..., n}. Note that whenever i equals one of the components of the multiindex I then dx^{i} ∧ dx^{I} = 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 kform φ as defined above,
Here, we have interpreted g as a 0form, and then applied the properties of the exterior derivative.
This result extends directly to the general kform ω as
In particular, for a 1form ω, the components of dω in local coordinates are
In terms of invariant formula
Alternatively, an explicit formula can be given for the exterior derivative of a kform ω, when paired with k + 1 arbitrary smooth vector fields V_{0},V_{1}, ..., V_{k}:
where [V_{i}, V_{j}] denotes the Lie bracket and a hat denotes the omission of that element:
In particular, for 1forms we have: dω(X, Y) = X(ω(Y)) − Y(ω(X)) − ω([X, Y]), where X and Y are vector fields, X(ω(Y)) is the scalar field defined by the vector field X ∈ Γ(TM) applied as a differential operator ("directional derivative along X") to the scalar field defined by applying ω ∈ Γ^{∗}(TM) as a covector field to the vector field Y ∈ Γ(TM) and likewise for Y(ω(X)).
Note: Some authors (e.g., Kobayashi–Nomizu and Helgason) use a formula that differs by a factor of 1/k + 1:
Examples
Example 1. Consider σ = u dx^{1} ∧ dx^{2} over a 1form basis dx^{1}, ..., dx^{n}. The exterior derivative is:
The last formula follows easily from the properties of the exterior product. Namely, dx^{i} ∧ dx^{i} = 0.
Example 2. Let σ = u dx + v dy be a 1form defined over ℝ^{2}. By applying the above formula to each term (consider x^{1} = x and x^{2} = y) we have the following sum,
Stokes' theorem on manifolds
If M is a compact smooth orientable ndimensional 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.
Further properties
Closed and exact forms
A kform 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 d^{2} = 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 d^{2} = 0, it can be used as the differential (coboundary) to define de Rham cohomology on a manifold. The kth de Rham cohomology (group) is the vector space of closed kforms modulo the exact kforms; 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 farreaching 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 Ω^{k} is the contravariant smooth functor that assigns to each manifold the space of kforms 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 Ω^{k} to Ω^{k+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 function f : M → ℝ on a real differentiable manifold M is a 0form. The exterior derivative of this 0form is the 1form 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 1form 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 = (v_{1}, v_{2}, ... v_{n}) on ℝ^{n} has a corresponding (n − 1)form
where denotes the omission of that element.
(For instance, when n = 3, i.e. in threedimensional space, the 2form ω_{V} is locally the scalar triple product with V.) The integral of ω_{V} over a hypersurface is the flux of V over that hypersurface.
The exterior derivative of this (n − 1)form is the nform
Curl
A vector field V on ℝ^{n} also has a corresponding 1form
 ,
Locally, η_{V} is the dot product with V. The integral of η_{V} along a path is the work done against −V along that path.
When n = 3, in threedimensional space, the exterior derivative of the 1form η_{V} is the 2form
Invariant formulations of grad, curl, div, and Laplacian
On any pseudoRiemannian manifold, the standard vector calculus operators can be generalized^{[1]} and written in coordinatefree 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.
See also
 Exterior covariant derivative
 de Rham complex
 Discrete exterior calculus
 Green's theorem
 Lie derivative
 Stokes' theorem
 Fractal derivative
Notes
 ^ This is a natural generalization of the operators from threedimensional Euclidean space to a pseudoRiemannian manifold of arbitrary dimension . Note that the expression for makes sense only in three dimensions since is a form of degree .
References
 Cartan, Élie (1899). "Sur certaines expressions différentielles et le problème de Pfaff". Annales Scientifiques de l'École Normale Supérieure. Série 3 (in French). Paris: GauthierVillars. 16: 239–332. ISSN 00129593. JFM 30.0313.04. Retrieved 2 Feb 2016.
 Conlon, Lawrence (2001). Differentiable manifolds. Basel, Switzerland: Birkhäuser. p. 239. ISBN 0817641343.
 Darling, R. W. R. (1994). Differential forms and connections. Cambridge, UK: Cambridge University Press. p. 35. ISBN 0521468000.
 Flanders, Harley (1989). Differential forms with applications to the physical sciences. New York: Dover Publications. p. 20. ISBN 0486661695.
 Loomis, Lynn H.; Sternberg, Shlomo (1989). Advanced Calculus. Boston: Jones and Bartlett. pp. 304–473 (ch. 7–11). ISBN 0486661695.
 Ramanan, S. (2005). Global calculus. Providence, Rhode Island: American Mathematical Society. p. 54. ISBN 0821837028.
 Spivak, Michael (1971). Calculus on Manifolds. Boulder, Colorado: Westview Press. ISBN 9780805390216.
 Warner, Frank W. (1983), Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, 94, Springer, ISBN 0387908943