The Exterior Derivative

Introduction

The exterior derivative $d$ is the operator that raises the degree of a differential form by one and satisfies $d^2 = 0$; it is the coboundary of the de Rham complex, and it is the derivative that the smooth structure of a manifold supplies before any metric or connection is chosen. The forms, the coordinate formula for $d$, the statement $d^2 = 0$, the integral, Stokes' theorem and the cohomology are those of Differential Forms and Stokes' Theorem; this article reads $d$ as an operator and develops the properties that make it the operator of the complex.

The article characterises $d$ among the first-order operators on forms — the unique degree-one derivation of the graded algebra of forms whose value on a function is the differential, equivalently the unique operator with the stated symbol satisfying $d^2 = 0$ — and proves the naturality under the pullback that the characterisation expresses. It develops the de Rham complex generated by $d$: the cochain complex and its cohomology, its functoriality, the local exactness of the Poincaré lemma read as the statement that the operator is locally surjective onto the closed forms, and the homotopy invariance. It proves that the symbol sequence of the complex is exact off the zero section — the exterior algebra with the multiplication by a covector is acyclic — so the de Rham complex is an elliptic complex, and this is what makes it the setting of the Hodge and index theory. It closes with the Cartan calculus, the graded algebra generated by $d$, the interior products $\iota_X$ and the Lie derivatives $\mathcal{L}_X$.

The article assumes the differential forms, the wedge product, the exterior derivative, its coordinate formula, $d^2=0$, the de Rham cohomology and the theorem of de Rham of Differential Forms and Stokes' Theorem; the interior product, the Lie derivative and Cartan's formula of The Lie Derivative; and the order, the symbol and the elliptic complex of Differential Operators on a Manifold, all in this category. The Hodge star, the codifferential and the Laplace–de Rham operator are The Codifferential and The L2 Adjoint of a Differential Operator; the de Rham theorem and the algebraic topology of the cohomology are Part II's; the parametrix and the elliptic regularity are Pseudodifferential Operators and Partial Differential Equations. The article reproves nothing that the Theory article proves and states each reuse as a citation. No physics is invoked.

The Exterior Derivative as an Operator

The Operator and Its Symbol

Proposition. The exterior derivative is a first-order differential operator $d : \Omega^k(M) \to \Omega^{k+1}(M)$; its principal symbol, in the conventions of Differential Operators on a Manifold, is

$$ \sigma_1(d)(x, \xi) = \xi \wedge \cdot \ : \ \Lambda^kT^*_xM \longrightarrow \Lambda^{k+1}T^*_xM, $$

the exterior multiplication by the covector $\xi$; and its order is exactly one, because $\sigma_1(d)$ does not vanish identically.

Proof. In a chart the operator reads $d(\sum_I f_I\,dx^I) = \sum_{I,i}\partial_i f_I\,dx^i\wedge dx^I$, whose top-order part has the coefficient matrices $\xi_i$ contracted with the basis, which is the exterior multiplication by $\sum_i\xi_i dx^i$. The lower-order terms of an operator do not affect the symbol, so the class of $d$ modulo the zeroth-order operators is the multiplication by $\xi$, and the order is one.

Corollary. The symbol of $d$ depends on the point through the covector, not through any choice of coordinates: it is the universal algebraic operation, the multiplication by the coordinate functional, and it satisfies $\sigma_1(d)(x,\xi)\circ\sigma_1(d)(x,\eta)+\sigma_1(d)(x,\eta)\circ\sigma_1(d)(x,\xi)=0$ because the wedge product is graded-commutative; in particular $\sigma_1(d)(x,\xi)^2 = 0$, which is the symbol form of $d^2 = 0$.

Naturality under the Pullback

Proposition. For every smooth map $F : N \to M$ between smooth manifolds and every form $\omega$ on $M$,

$$ F^*(d\omega) = d(F^*\omega), $$

the pullback is a cochain map of the de Rham complexes, and it is the unique extension of the pullback of functions to the forms that is compatible with the wedge product and commutes with $d$ in this way.

Proof. Both sides are first-order operators on the components and are reduced to the coordinate expression; the chain rule gives $\sum_j\partial_j(f\circ F)\,dy^j = d(f\circ F)$ for a function, which is the case $k=0$, and the derivation property of the exterior derivative in the target, together with the compatibility of the pullback with the wedge product, gives the general case. Uniqueness: a map on forms compatible with the wedge product is determined by its values on the functions and the exact $1$-forms, and the requirement to commute with $d$ fixes the values on the exact forms.

Corollary (homotopy invariance). If $F_0, F_1 : N \to M$ are smoothly homotopic, then $F_0^* = F_1^*$ on the de Rham cohomology: the difference is $dH + Hd$ for an operator $H$ of degree $-1$ lowering the degree, the homotopy operator of the pullback along the homotopy. Consequently the exterior derivative's cohomology depends only on the homotopy type of the manifold, and the two maps induce the same map on $H^\bullet_{dR}$.

Proof. The homotopy is a smooth map $H : N\times[0,1]\to M$; the classical chain homotopy is constructed by integrating the pullback of the contraction with $\partial_t$ over the parameter, and the identity $F_1^* - F_0^* = dK + Kd$ follows from Stokes' theorem on the strip, as in Differential Forms and Stokes' Theorem. The operator $K$ lowers the degree by one, so it is a cochain homotopy and the induced maps on cohomology agree.

Characterisations of the Exterior Derivative

The Axiomatic Characterisation

Theorem. On the graded algebra $\Omega^\bullet(M)$ of differential forms there is exactly one $\mathbb{R}$-linear operator $d$ of degree $+1$ with the two properties:

(i) $d$ is an antiderivation, $d(\alpha\wedge\beta) = d\alpha\wedge\beta + (-1)^p\,\alpha\wedge d\beta$ for $\alpha \in \Omega^p(M)$;

(ii) $df$ is the differential of $f$ for every function, so $df(X) = Xf$ for every vector field $X$.

Moreover this operator satisfies $d^2 = 0$.

Proof. Uniqueness: a form is locally a sum of terms $f_0\,df_1\wedge\cdots\wedge df_k$, since the coordinate forms $dx^i = d(x^i)$ are exact; the antiderivation property applied $k$ times expresses $d$ of such a term through $d$ of its factors, and property (ii) gives the values on both factors; hence $d$ is determined on a set of local generators and therefore everywhere, the local expressions agreeing on overlaps by the commutativity of the mixed partial derivatives. Existence is the coordinate formula of Differential Forms and Stokes' Theorem, which is checked to have the two properties. The property $d^2 = 0$ follows from the two: on a function, $d(df) = d(\sum_i\partial_if\,dx^i) = \sum_{i,j}\partial_j\partial_if\,dx^j\wedge dx^i = 0$ by the symmetry of the second derivatives against the antisymmetry of the wedge; on a general form, the antiderivation property reduces the verification to the generators.

Corollary. The exterior derivative is the unique operator on the forms that is natural under the pullback, in the sense of the previous section, and satisfies (i) and (ii); the two characterisations are the same characterisation read with $F$ and without.

The Operator Characterisation

Proposition. The class of $d$ modulo the zeroth-order operators is the multiplication by the tautological covector: $d$ is the unique first-order operator on forms with

$$ d(f\alpha) = df \wedge \alpha + f\,d\alpha, \qquad d(df) = 0 $$

for every function and form. The first identity is the Leibniz rule over the module structure and the second is the vanishing of the square on the exact forms.

Proof. The Leibniz rule over the multiplication by a function is exactly the statement that the commutator $[d,f]$ is multiplication by $df$, that is, the operator has order one with the symbol $\xi\wedge\cdot$; and the vanishing on the exact forms is a second-order condition on the coefficients, which together with the symbol pins the operator. Equivalently, the requirements are the statement that $d$ is the unique extension of the differential of functions to a derivation of degree one whose square vanishes.

The de Rham Complex

The Complex and Its Cohomology

Definition. The de Rham complex of a smooth manifold $M$ of dimension $n$ is the cochain complex of real vector spaces

$$ 0 \longrightarrow \Omega^0(M) \xrightarrow{\ d\ } \Omega^1(M) \xrightarrow{\ d\ } \cdots \xrightarrow{\ d\ } \Omega^n(M) \longrightarrow 0, $$

with cohomology $H^k_{dR}(M) = \ker(d : \Omega^k \to \Omega^{k+1}) \big/ \operatorname{im}(d : \Omega^{k-1} \to \Omega^k)$; the elements of the kernel are the closed forms, those of the image the exact forms, and the quotient measures the failure of the closed forms to be exact.

The complex and its cohomology are those of Differential Forms and Stokes' Theorem, where the theorem of de Rham $H^k_{dR}(M) \cong H^k(M;\mathbb{R})$ is proved; what belongs to the present article is the operator reading of the complex, its functoriality and the exactness of its symbol. The cohomology is a contravariant functor: a smooth map $F$ induces $F^* : H^k_{dR}(M) \to H^k_{dR}(N)$ because the pullback commutes with $d$, and homotopic maps induce the same map by the corollary above.

Local Exactness and the Poincaré Lemma

Theorem (Poincaré lemma). Let $U \subseteq \mathbb{R}^n$ be star-shaped with respect to the origin. Then every closed form on $U$ of positive degree is exact: for each $k \geq 1$ there is an operator $K : \Omega^k(U) \to \Omega^{k-1}(U)$ with

$$ dK + Kd = \mathrm{id} \quad \text{on } \Omega^k(U), \ k \geq 1, $$

the homotopy operator obtained by integrating the contraction with the radial field along the segment from the origin; consequently $H^k_{dR}(U) = 0$ for $k \geq 1$ and $H^0_{dR}(U) = \mathbb{R}$.

Proof. The operator is $K\alpha = \int_0^1 \iota_{R}\bigl(\alpha(tx)\bigr)\,\frac{dt}{t}$-type, the classical construction of Differential Forms and Stokes' Theorem; the identity $dK + Kd = \mathrm{id}$ is the fundamental theorem of calculus applied along the radial direction, and it is the operator statement that closed forms of positive degree on a star-shaped set are exact. The identity is the exact analogue, on the complex, of the statement that the complex is contractible.

The lemma is local: on a general manifold every point has a contractible coordinate neighbourhood, so closed forms are locally exact, and the global de Rham cohomology measures the obstruction to patching the local primitives. The disjointness of the local and global exactness is the content of the cohomology.

The Symbol Sequence and Ellipticity

Theorem. For every $x \in M$ and every nonzero $\xi \in T^*_xM$ the symbol sequence

$$ 0 \longrightarrow \Lambda^0T^*_xM \xrightarrow{\ \xi\wedge\ } \Lambda^1T^*_xM \xrightarrow{\ \xi\wedge\ } \cdots \xrightarrow{\ \xi\wedge\ } \Lambda^nT^*_xM \longrightarrow 0 $$

is exact. Consequently the de Rham complex is an elliptic complex in the sense of Differential Operators on a Manifold.

Proof. Choose $\eta \in T_xM$ with $\langle\xi,\eta\rangle = 1$ and let $\iota_\eta$ be the contraction with $\eta$ on the exterior algebra. The identity

$$ \iota_\eta(\xi\wedge\alpha) + \xi\wedge\iota_\eta\alpha = \langle\xi,\eta\rangle\,\alpha = \alpha $$

holds for every $\alpha$, because $\iota_\eta\xi = \langle\xi,\eta\rangle$ in the graded Leibniz rule for the contraction; it says that $\iota_\eta$ is a contracting homotopy for the sequence with differential $\xi\wedge\cdot$. A complex with a contracting homotopy is exact: if $\xi\wedge\alpha = 0$, then $\alpha = \iota_\eta(0) + \xi\wedge\iota_\eta\alpha$ lies in the image of $\xi\wedge\cdot$; and the map $\Lambda^0\to\Lambda^1$ is injective because $\xi\neq0$. This proves exactness at every spot, and the exactness at the symbol level is exactly the definition of the ellipticity of the complex.

Corollary. The Laplace-type operator of the de Rham complex, the Laplace–de Rham operator $\Delta = d\delta + \delta d$, is elliptic; the Hodge theory represents the de Rham cohomology by the harmonic forms, and the index of the complex — the alternating sum of the Betti numbers — is computed from the symbol by the index theorem of Part IV. The regularity theory that makes these statements analytic belongs to Partial Differential Equations and The L2 Adjoint of a Differential Operator; what the present article supplies is the exactness of the symbol and hence the ellipticity.

The Cartan Calculus

The exterior derivative is not an isolated operator: with the interior products and the Lie derivatives of The Lie Derivative it generates a graded algebra, the Cartan calculus, whose structure is the source of most of the identities used on forms.

Theorem. Let $\iota_X$ be the interior product with a vector field, of degree $-1$, and let $\mathcal{L}_X$ be the Lie derivative, of degree zero. The graded commutators among these operators satisfy

$$ [d,d] = 0, \qquad [\iota_X,\iota_Y] = 0, \qquad [d,\iota_X] = \mathcal{L}_X, \qquad [d,\mathcal{L}_X] = 0, \qquad [\mathcal{L}_X,\iota_Y] = \iota_{[X,Y]}, $$

where the graded commutator is $[A,B] = AB - (-1)^{\deg A\deg B}BA$. In particular $d$ is a differential, the interior products anticommute, and the whole algebra generated by $d$, the $\iota_X$ and the $\mathcal{L}_X$ is closed under the graded commutator.

Proof. The first two identities are $d^2=0$ and $\iota_X^2=0$ with the anticommutativity of the contractions. The third is Cartan's formula, $\mathcal{L}_X = d\iota_X+\iota_Xd = [d,\iota_X]$, of The Lie Derivative. The fourth is the commutation $\mathcal{L}_Xd = d\mathcal{L}_X$ deduced from it and $d^2=0$. The fifth is the identity $[\mathcal{L}_X,\iota_Y]=\iota_{[X,Y]}$ of the same article. That the generated algebra with the graded commutator is closed is the content of the Jacobi identity for the graded commutator; the identities above show that the commutator of any two generators is again in the span of the generators.

Corollary (the transgression identity). The contractions give the identity

$$ [d, \iota_X\iota_Y + \iota_Y\iota_X] = 0, \qquad d(\iota_X\iota_Y\omega) = \mathcal{L}_X\iota_Y\omega - \iota_Y\mathcal{L}_X\omega + \iota_X\iota_Yd\omega, $$

and the contraction with two fields on a $2$-form recovers the evaluation, $\iota_Y\iota_X\omega = \omega(X,Y)$; these are the elementary identities by which the closedness and the exactness of the invariant forms of the Hermitian articles are checked.

Summary

The exterior derivative is the unique degree-one antiderivation of the graded algebra of forms with $df(X) = Xf$ on functions; it is a first-order operator with the symbol $\xi\wedge\cdot$, it is natural under the pullback, and it satisfies $d^2=0$, so it makes the forms into the de Rham cochain complex. The cohomology of the complex is the de Rham cohomology of Differential Forms and Stokes' Theorem; the pullback is a cochain map, homotopic maps induce the same map in cohomology, and on a star-shaped open set the closed forms of positive degree are exact by the homotopy operator $K$ with $dK+Kd=\mathrm{id}$ — the local exactness that fails globally exactly by the cohomology.

The symbol sequence of the complex, the exterior algebra with the multiplication by a covector, is exact off the zero section: the contraction with a vector $\eta$ dual to $\xi$ is a contracting homotopy, $\iota_\eta(\xi\wedge\alpha) + \xi\wedge\iota_\eta\alpha = \alpha$ for $\langle\xi,\eta\rangle=1$. Consequently the de Rham complex is elliptic, its Laplace-type operator $\Delta = d\delta+\delta d$ is elliptic, and the Hodge and index theory of the Part applies. With the interior product and the Lie derivative, $d$ generates the Cartan calculus $[d,d]=0$, $[\iota_X,\iota_Y]=0$, $[d,\iota_X]=\mathcal{L}_X$, $[d,\mathcal{L}_X]=0$, $[\mathcal{L}_X,\iota_Y]=\iota_{[X,Y]}$.

Summary of Notation

Symbol	Meaning
$d : \Omega^k(M) \to \Omega^{k+1}(M)$	Exterior derivative; the coboundary of the de Rham complex
$\sigma_1(d)(x,\xi) = \xi\wedge\cdot$	Principal symbol; multiplication by the covector
$F^d = dF^$	Naturality under the pullback; $F^*$ is a cochain map
$d(\alpha\wedge\beta) = d\alpha\wedge\beta + (-1)^p\alpha\wedge d\beta$	Antiderivation property
$d^2 = 0$	The square vanishes; the complex property
$0 \to \Omega^0 \xrightarrow{d} \cdots \xrightarrow{d} \Omega^n \to 0$	de Rham complex
$H^k_{dR}(M)$	de Rham cohomology; closed modulo exact forms
$dK + Kd = \mathrm{id}$	Homotopy operator of the Poincaré lemma on a star-shaped set
$\xi\wedge\cdot$, $\iota_\eta$, $\langle\xi,\eta\rangle=1$	Symbol sequence and its contracting homotopy; exactness
$\Delta = d\delta + \delta d$	Laplace–de Rham operator; elliptic
$[d,d]=0$, $[\iota_X,\iota_Y]=0$, $[d,\iota_X]=\mathcal{L}_X$	The Cartan calculus
$[\mathcal{L}_X,\iota_Y]=\iota_{[X,Y]}$	Commutation of the Lie derivative and the contraction
$\iota_Y\iota_X\omega = \omega(X,Y)$	Contraction on a $2$-form; the evaluation