Differential form ($\alpha \in {\Omega }^k(M)$)

A differential form is a smooth field of alternating multilinear maps that can be integrated over an appropriately oriented manifold of matching dimension. A one-form measures a directed displacement; a two-form measures an oriented area element. The point is to express what is being accumulated along a curve or over a surface without tying the answer to one set of coordinates. This makes line integrals, surface integrals, and the integral theorems of vector calculus parts of one construction.

For a concrete one-form, take $\alpha =xdy$ in the plane. Travel from $(0,0)$ to $(1,1)$ along the bottom and then the right edge of the unit square. The horizontal leg contributes zero because $dy=0$; the vertical leg has $x=1$ and contributes ${\int }_0^{1}dy=1$. Taking the left and then the top edge instead gives zero: the vertical leg has $x=0$, and the horizontal leg again has $dy=0$. The two paths have the same endpoints but different integrals.

Now replace $\alpha$ with the exact differential $\beta =d(xy)=ydx+xdy$. Both paths give $1$, because integrating an exact differential along a curve gives the endpoint difference of its potential. Here that difference is $xy{|}_{(1,1)}-xy{|}_{(0,0)}=1$. This is the fundamental theorem of calculus applied to a function along a parametric curve, and is the basic reason conservative vector fields have path-independent work.

The exterior derivative records the local obstruction in the first example: $d(xdy)=dx\wedge dy$. The Generalized Stokes theorem turns the difference between the two path integrals into the integral of this two-form over the square, whose oriented area is $1$. Interior edges cancel when adjoining pieces are added; only the outer boundary survives. That cancellation connects a local calculation with a global measurement.

More precisely, a degree-$k$ differential form assigns to each point $p$ a multilinear scalar-valued function of $k$ tangent vectors at $p$. Interchanging two arguments changes its sign. A one-form is therefore a covector field, while a two-form vanishes on a pair of parallel vectors. Degree zero simply means a smooth function. The space ${\Omega }^k(M)$ is a vector space; the wedge product of differential forms combines degrees, and the exterior derivative raises degree by one.

The usefulness goes beyond bookkeeping. Pullbacks of a differential form supply the right transformation law automatically under a coordinate change. The Generalized Stokes theorem unifies the fundamental theorem of calculus, Green theorem, Stokes theorem, and divergence theorem. Meanwhile, the distinction between closed differential forms and exact differential forms detects holes through de Rham cohomology. Reyer Sjamaar develops these connections in sections 5.4 and 9.3 of his university notes. ^[ref].

A differential one-form on a specified domain is inexact if it is not the differential of a single-valued scalar potential there. For $Pdx+Qdy$, unequal cross derivatives $P_y\ne Q_x$ prove inexactness locally. Equality is sufficient on a simply connected plane domain under the usual smoothness hypotheses, but holes can obstruct a global potential. The form $42ydx+xdy$ has cross derivatives $42$ and $1$ and counterclockwise unit-square integral $-41$, so it is inexact. A conservative vector field has an exact differential as its work form.

On a complex manifold of complex dimension $n$, a holomorphic volume form is a nowhere-vanishing holomorphic differential form of type $(n,0)$. On ${\mathbb{C}}^n$, the form $d{z}_1\wedge \cdots \wedge d{z}_n$ transforms under a complex linear coordinate change by the determinant. It is consequently preserved by the special unitary group $SU(n)$.

A compactly supported top-degree differential form on an oriented smooth manifold is integrated using orientation-preserving coordinates and a partition of unity. The coordinate change of variables formula makes the result independent of the choices. More generally, a form is integrated over a lower-dimensional oriented submanifold by its pullback. The Generalized Stokes theorem states ${\int }_{M}d\beta ={\int }_{\partial M}\beta$ with the induced boundary orientation. Integrating a closed differential form over a closed cycle depends only on its homology class, since its integral over a boundary vanishes.

The comass is the operator norm of a differential form restricted to simple unit tangent multivectors. Reversing orientation gives the equivalent two-sided bound. It differs from the Euclidean norm of all the form's components. The calibration bound uses comass because tangent planes are represented by simple multivectors.

For a smooth form $\theta =gd{z}_1\wedge \cdots \wedge d{z}_n$ of type $(n,0)$, the complex orientation gives $i^{n^2}\theta \wedge \overline{\theta }=2^n|g{|}^{2}d{x}_1\wedge d{y}_1\wedge \cdots \wedge d{x}_n\wedge d{y}_n$. Consequently its integral on a compact complex manifold is positive unless $\theta$ vanishes identically. Applying Stokes theorem to $d(\psi \wedge \overline{d\psi })$ proves that a holomorphic differential form of degree $n-1$ on a compact complex $n$-manifold is closed, without requiring a Kähler metric.

A differential three-form is a smooth, totally antisymmetric covariant tensor of rank three. For a Killing-Yano two-form, its covariant derivative is such a tensor, because it is antisymmetric in the first two and last two slots.

For a differential three-form $A$ with $F=dA$ on an oriented eleven-dimensional manifold, varying $\int (\frac{1}{2}F\wedge \star F+F\wedge F\wedge A)$ with fixed metric gives $\delta S=\int \delta A\wedge (d\star F+3F\wedge F)$ up to a boundary term. Indeed $d(\delta A\wedge F\wedge A)=d\delta A\wedge F\wedge A-\delta A\wedge F\wedge F$, so the two differentiated $F$ factors and the differentiated $A$ factor contribute equally. The shift $A\mapsto A+d\Lambda$ changes the density by $d(F\wedge F\wedge \Lambda )$, preserving the field equation.

A smooth section of ${\bigwedge }^k(T^{\ast }X\otimes \mathbb{C})\otimes E$ is a differential form with values in the complex vector bundle $E$. A connection on a vector bundle extends to these forms by the graded Leibniz rule.

For a holomorphic vector bundle $E$, this sheaf assigns to $U$ the smooth $E$-valued differential forms of type (p, q) on $U$. Its Dolbeault operator acts coefficientwise in a holomorphic local frame and squares to zero.

On a smooth manifold, ${\mathcal{A}}^k(U)$ is the space of smooth complex-valued degree-$k$ differential forms on $U$. Restriction of forms supplies the sheaf maps. Smooth partitions of unity make this a fine sheaf.

On a complex manifold, this sheaf consists of smooth differential forms of type (p, q). The type decomposition of the complexified tangent bundle gives ${\mathcal{A}}^k={⨁}_{p+q=k}{\mathcal{A}}^{p,q}$.

A volume form is a nowhere-vanishing differential form of top degree on an oriented manifold. A metric supplies the canonical local density $\sqrt{|\det g|}$ and hence a metric volume form after choosing orientation.

On an oriented Riemannian manifold, the metric volume form is the volume form taking value one on positively oriented orthonormal frames. In an oriented manifold chart it has the displayed expression. Under an orientation-preserving coordinate change with Jacobian $A$, $\det g$ acquires the factor $(\det A{)}^2$, precisely matching the transformation of the top exterior product. Thus the expressions agree on chart overlaps. Orientation is needed to make a globally positive form, though the volume density exists without it.

The oriented metric volume form on a two-dimensional surface. On the unit sphere with standard orientation, $\omega =\sin \theta d\theta \wedge d\phi$ and $\int \omega =4\pi$. Its pullback of a differential form under a map measures signed local area.

On an oriented $d$-dimensional pseudo-Riemannian manifold, the metric volume tensor is the totally antisymmetric tensor with ${ϵ}_{1\dots d}=\sqrt{|\det g|}$ in positively oriented coordinates. It supplies the components of the volume form. It differs from the coordinate Levi-Civita symbol, whose entries are just $0,1,-1$. Raising indices with a Lorentzian metric can change signs; the metric, dimension and orientation must be specified.

A vector field preserves a chosen volume form $\omega$ when its local flow pulls $\omega$ back to itself. Infinitesimally this is ${\mathcal{L}}_X\omega =0$, using the Lie derivative. The condition depends on the chosen volume form.

Suppose a nonempty compact $(n+1)$-manifold with a volume form has $n$ pointwise independent volume-preserving vector fields that commute and are tangent to its boundary. If restriction $H_{\mathrm{dR}}^1(M)\to H_{\mathrm{dR}}^1(\partial M)$ is injective, the boundary must have at least two connected components.

Indeed $\eta ={\iota }_{X_1}\cdots {\iota }_{X_n}\omega$ is nowhere zero. Cartan's magic formula and commutation show $d\eta =0$, while boundary tangency gives $j^{\ast }\eta =0$. Injectivity in de Rham cohomology makes $\eta =df$. On a connected boundary $f$ is constant; a maximum or minimum differing from that boundary value would be an interior critical point, contradicting $df\ne 0$. Empty boundary is also impossible.

A 2-form is an alternating covariant tensor field of rank two. Curvature forms and electromagnetic field strengths are standard examples.

The wedge product combines a degree-$p$ differential form and a degree-$q$ differential form into one of degree $p+q$. It is associative and graded-commutative:In particular, $dx\wedge dy=-dy\wedge dx$ and $dx\wedge dx=0$. For planar vectors $v,w$,This determinant is the signed area of their parallelogram. Exchanging the edges reverses the sign; parallel edges give zero. Thus antisymmetry records the geometry of oriented area rather than adding an arbitrary algebraic convention. The wedge product of differential forms requires no Riemannian metric to define it. It obeys the graded Leibniz rule under the exterior derivative.

Exterior multiplication by a fixed differential form is the linear operator taking the wedge product of differential forms on the left. For a one-form $\omega$, it raises degree by one. Anticommutativity gives $e(\omega )e(\eta )+e(\eta )e(\omega )=0$ for one-forms. On the exterior algebra of an inner product space, pairing this operation with contraction produces Clifford multiplication $c(v)=e(v^{♭})-{\iota }_v$.

A smooth map $\varphi :M\to N$ pulls a differential form on $N$ back to one on $M$ by applying its differential $d\varphi$ to each tangent vector argument. In particular, a parametrization of a curve pulls a one-form back to an ordinary integrand on an interval.

For $\alpha =xdy$ and the parabola $\varphi (t)=(t,t^2)$, $0\le t\le 1$, substitute $x=t$ and $dy=2tdt$ to obtainThe same endpoints gave $0$ and $1$ along the two square-edge paths in the introduction, so this third answer is consistent with path dependence. By contrast, ${\varphi }^{\ast }d(xy)=d(t^3)=3t^{2}dt$ gives integral $1$, the same endpoint difference as before.

The pullback of a differential form commutes with the wedge product of differential forms and the exterior derivative: ${\varphi }^{\ast }(\alpha \wedge \beta )={\varphi }^{\ast }\alpha \wedge {\varphi }^{\ast }\beta$ and ${\varphi }^{\ast }(d\alpha )=d({\varphi }^{\ast }\alpha )$. These identities make changes of coordinates and parametrizations compatible with differentiation and integration.

It is the alternating-tensor case of pullback.

The interior product contracts a differential form with a vector field in its first argument:

A differential 1-form is a smooth section of the cotangent bundle; it assigns a covector to every point of a smooth manifold. In planar coordinates, $\alpha =P(x,y)dx+Q(x,y)dy$ evaluates a tangent vector $v$ as $P(x,y)v_x+Q(x,y)v_y$.

Along a parametric curve $\gamma (t)=(x(t),y(t))$, its line integral isReversing the direction reverses the integral. A force expressed as a one-form therefore accumulates work along a directed path. In Euclidean coordinates its components can be displayed as arrows, but identifying a one-form with a vector field uses an inner product; the covector field itself does not require a Riemannian metric.

Let $W$ consist of real-linear maps $\alpha$ from smooth vector fields to smooth functions such that $X_p=0$ implies $\alpha (X)(p)=0$. These are exactly evaluations by smooth differential one-forms. Define ${\omega }_p(v)=\alpha (X)(p)$ using any global field with $X_p=v$; the vanishing condition makes it well-defined. Bump-extended local frame vectors make its coefficients smooth. Also $\alpha (fX)(p)=f(p)\alpha (X)(p)$, so real linearity and pointwise vanishing automatically give $C^{\infty }$-linearity. The evaluation correspondence is a natural module isomorphism.

The exterior derivative is the unique graded derivation $d:{\Omega }^p(M)\to {\Omega }^{p+1}(M)$ that agrees with the differential on functions and satisfies $d^2=0$.

On a smooth planar one-form $\alpha =Pdx+Qdy$, the product rule and antisymmetry of the wedge product of differential forms giveFor $\alpha =xdy$, the coefficient is $1$: there is a nonzero integral around the boundary of every positively oriented rectangle. For $\beta =d(xy)$, the coefficient is ${\partial }_{x}x-{\partial }_{y}y=0$. More generally, the identity $d^2=0$ makes every exact differential form a closed differential form. It expresses cancellation between mixed partial derivatives locally; it does not imply that every closed differential form has a global potential.

For a smooth one-form $A$, the components ${\partial }_a{A}_b-{\partial }_b{A}_a$ transform as a covariant rank-two tensor. The non-tensorial second derivatives in the ordinary covector derivative under nonlinear coordinate changes are symmetric in $a,b$ and cancel. For a torsion-free connection the same components are ${\nabla }_a{A}_b-{\nabla }_b{A}_a$. This is the coordinate expression of $dA$, without choosing a metric.

An $\mathbb{R}$-linear degree-one graded derivation of the exterior algebra of smooth differential forms that equals the differential on functions and squares to zero is unique. A smooth bump function and the graded Leibniz rule first establish locality. Then $d(d{x}^i)=0$, and applying the graded Leibniz rule to $\alpha ={\sum }_I{a}_{I}d{x}^I$ gives the displayed coordinate formula. The chain rule makes that formula coordinate-independent.

For a smooth one-form $\alpha$ and smooth vector fields $X,Y$,In coordinates the derivatives of the components of $X,Y$ cancel against the Lie bracket of vector fields, leaving $({\partial }_i{\alpha }_j-{\partial }_j{\alpha }_i)X^i{Y}^j$. This formula extends the coordinate definition of the exterior derivative to arbitrary frames.

If $X$ is an oriented $n$-manifold with boundary, $F:\partial X\hookrightarrow X$ is the inclusion, and $\omega$ is a compactly supported $(n-1)$-form, thenwhere the boundary has the outward-normal-first boundary orientation. A subordinate partition of unity reduces the theorem to the fundamental theorem of calculus in oriented coordinate half-spaces; local finiteness and compact support ensure that only finitely many terms contribute.

For the counterclockwise unit-square boundary and $\omega =xdy$, the right-hand side is $1$, while the left-hand side is ${\int }_{[0,1{]}^2}dx\wedge dy=1$. The equation says that the boundary integral measures the total local circulation in the region. Reversing the orientation of a smooth manifold reverses both sides together.

In dimension one, the boundary of an interval consists of its endpoint with a plus sign and its starting point with a minus sign; the identity becomes ${\int }_a^{b}df=f(b)-f(a)$. In the plane it gives Green theorem. Applied to appropriate differential forms in three dimensions it gives both Stokes theorem and the divergence theorem. The boundary and the differential being integrated must belong to the stated domain: removing a singular point can make a spanning disc unavailable.

A closed differential form satisfies $d\alpha =0$; an exact differential form has $\alpha =d\beta$. The identity $d^2=0$ makes every exact form closed. Global failure of the converse is measured by de Rham cohomology; the Poincaré lemma gives the local converse in positive degrees.

A differential form $\alpha$ is closed when $d\alpha =0$.

On the punctured plane, this one-form is closed and locally the differential of the polar angle. Its line integral around a closed curve is $2\pi$ times the winding number. A once-winding circle therefore has nonzero integral, proving that no global single-valued angle potential exists and the form is not an exact differential form.

For a counterclockwise circle of radius $R>0$, substitute $x=R\cos t$, $y=R\sin t$. The pullback of a differential form isThe radius cancels. In coordinates, $P=-y/(x^2+y^2)$ and $Q=x/(x^2+y^2)$ satisfy ${\partial }_{x}Q={\partial }_{y}P$ away from the origin, so $d\omega =0$ there. The nonzero integral is a global obstruction to being an exact differential form, not a nonzero local exterior derivative.

Compare this with the smooth one-form $\eta =(-ydx+xdy)/2$ on the whole plane. Here $d\eta =dx\wedge dy$, and its integral around a radius-$R$ circle is $\pi R^2$, the enclosed area. The Generalized Stokes theorem applies to the full disc for $\eta$. For $\omega$, that disc crosses the excluded origin, where the one-form is undefined. On an annulus the theorem applies, but its induced inner boundary is clockwise: the two integrals $2\pi$ and $-2\pi$ cancel. This explains exactly why closedness and nonzero circulation can coexist.

For real $a,b\ne 0$, this differential one-form on the punctured plane is the pullback of a differential form of the nonexact angular one-form under the invertible linear transformation $(X,Y)=(ax,by)$. It is a closed differential form for every such pair. On a region admitting a continuous complex argument, it has the local potential $f=\mathrm{Arg}(ax+iby)$. In a chart with $x\ne 0$, this can be written $f=\arctan (by/(ax))+C$.

There is no globally single-valued potential on the whole punctured plane. On the closed path $x=\cos t/a$, $y=\sin t/b$, $0\le t\le 2\pi$, substitution gives ${\omega }_{a,b}=dt$, hence its line integral is $2\pi \ne 0$. This contradicts the zero integral around every closed path required of an exact differential form. Around a positively oriented once-winding path the integral is $2\pi \mathrm{sgn}(ab)$. The local equation ${\omega }_{a,b}=0$ gives radial rays; away from a vertical ray its solutions as graphs are $y=Cx$ on intervals not crossing the excluded origin.

A differential one-form is closed when its exterior derivative vanishes. In coordinates, $\eta ={\eta }_{i}d{x}^i$ is closed exactly when ${\partial }_i{\eta }_j={\partial }_j{\eta }_i$. The Poincaré lemma makes it locally an exact differential form, but global exactness can fail through nonzero integrals around closed curves.

Remove the north pole and south pole separately. Each resulting chart is diffeomorphic to the plane, so the Poincaré lemma supplies local primitives of a closed one-form. Their difference has zero derivative on the connected overlap and is constant. Adjust one primitive by that constant and glue. This proves exactness without assuming a prior computation of de Rham cohomology.

Average a closed one-form over fibre rotations. The average differs from the original by an exact form because each rotation is isotopic to the identity; the Cartan formula for the Lie derivative supplies an explicit homotopy integral. Its vertical value $a$ is constant, since closedness and rotational invariance give $da=0$. Subtracting $a\omega$ leaves a basic form ${\pi }^{\ast }\xi$, and closedness gives $d\xi =aK{\Omega }_a$. For a zero-flux magnetic field, the Liouville mean of $\zeta (X+fV)$ is zero: the directional mean of $\xi (v)$ vanishes, the vertical mean is $2\pi a\int f{\Omega }_a$, and the coboundary has zero mean by volume preservation.

Every smooth closed differential form of positive degree is locally an exact differential form. On a coordinate ball centered at zero, put $F_t(x)=tx$ and let $R=x^i{\partial }_i$ be the radial vector field. Define the degree-lowering operatorFor a $p$-form with $p>0$, the integrand has a factor $t^{p-1}$ in its coordinate coefficients and is smooth at zero. Cartan's magic formula and differentiation of the pullback of a differential form give ${\partial }_t(F_t^{\ast }\alpha )=t^{-1}{F}_t^{\ast }(d{\iota }_R\alpha +{\iota }_{R}d\alpha )$. Integrating from zero to one gives $dK\alpha +Kd\alpha =\alpha$, since $F_0^{\ast }\alpha =0$ in positive degree. If $d\alpha =0$, take $\beta =K\alpha$. The local potential of a closed differential one-form gives the explicit degree-one construction. Global exactness can fail through nonzero periods, measured by de Rham cohomology.

Let a neighborhood of a submanifold retract onto it by fiberwise radial contraction in a tubular neighborhood. If a closed differential form $\beta$ pulls back to zero on the submanifold, the radial homotopy formula gives $\beta =d\alpha$, with $\alpha$ vanishing along the zero section. Applying the homotopy operator to $\beta$ gives the primitive directly. This is the relative ingredient in the relative Moser theorem.

On a star-shaped domain, let ${\rho }_t(x)=tx$ and $R=\sum x_j{\partial }_{x_j}$. On positive-degree differential forms, $H\alpha ={\int }_0^1{\rho }_t^{\ast }({\iota }_R\alpha )dt/t$ satisfies $dH+Hd=\mathrm{id}-{\rho }_0^{\ast }$. It gives a primitive of every closed positive-degree form. On a star-shaped domain in a complex vector space, contraction with the two type components of $R$ shows that $H$ sends type $(p,q)$ into $(p-1,q)\oplus (p,q-1)$. There is no primitive assertion for a nonzero constant function.

For radial contraction on a star-shaped domain, the homotopy formula is the identity in positive form degrees because pullback by the constant map vanishes there. On functions it is $h_{1}df=f-f(0)$. A nonzero constant is a counterexample to the uncorrected degree-zero identity. The Poincaré lemma is therefore an exactness assertion in positive degrees, while closed degree-zero forms are locally constant.

On a fiberwise star-shaped neighborhood, let $H_s$ scale fibers by $s$ and let $E$ be the radial vector field. For a closed differential form $\delta$ with zero pullback to the zero section, the homotopy primitive $\eta ={\int }_0^1{s}^{-1}{H}_s^{\ast }({\iota }_E\delta )ds$ satisfies $d\eta =\delta$. For a two-form, it vanishes as an ambient covector along that section, providing the fixed-submanifold condition in the relative Moser theorem.

On a star-shaped planar domain containing zero, let $\eta$ be a smooth closed differential one-form and set $f(x)={\int }_{[0,x]}\eta$. Green theorem on the triangle with vertices $0,x,x+h$ gives $f(x+h)-f(x)={\int }_{[x,x+h]}\eta$. Dividing by a displacement and taking its limit proves $df=\eta$. Thus every such form is an exact differential form, a degree-one version of the Poincaré lemma which uses only the planar integral theorem.

For a smooth differential one-form $\alpha ={\alpha }_i(x)d{x}^i$ with $d\alpha =0$, take a manifold chart identifying a neighborhood with a ball centered at zero, and set $\Phi (x)={\int }_0^1{x}^i{\alpha }_i(sx)ds$. Closedness means ${\partial }_j{\alpha }_i={\partial }_i{\alpha }_j$. ConsequentlyThus $d\Phi =\alpha$ in any dimension. This is the one-form case of the Poincaré lemma, and applies to spacetime as well as three-dimensional vector calculus. A global potential need not exist on a domain with nonzero periods.

A differential form is exact when it is the exterior derivative of a form of one lower degree. Since $d^2=0$, every exact form is closed.

The $p$th de Rham cohomology group is the quotient of closed $p$-forms by exact $p$-forms.

For a holomorphic submersion, relative differential forms are forms modulo those containing a base differential. The relative exterior derivative gives a complex whose fibrewise hypercohomology is complex cohomology for a smooth proper family. Its filtration by degree defines the Hodge filtration; retaining exactly one base differential in the absolute complex constructs the Gauss–Manin connection.

There are no exact forms in degree zero. A smooth function has zero exterior derivative precisely when it is locally constant, so $H_{\mathrm{dR}}^0(M)$ is the space of locally constant functions. On a connected nonempty smooth manifold, it is $\mathbb{R}$. An overlapping cover by two connected open subsets therefore does not give the direct-sum formula for de Rham cohomology of a finite disjoint union: restriction sends a constant to equal constants on both pieces.

Restriction of differential forms to the finitely many open and closed components gives an isomorphism of cochain complexes ${\Omega }^{\ast }(⨆M_j)\cong {⨁}_j{\Omega }^{\ast }(M_j)$. Piecewise forms glue because there are no overlaps, and the exterior derivative acts componentwise. Taking the quotient of closed differential forms by exact differential forms proves the displayed de Rham cohomology isomorphism. For an infinite disjoint union, forms and cohomology give products instead.

The complex of smooth differential forms resolves the constant complex sheaf by the Poincaré lemma and consists of fine sheaves. Its global cohomology therefore agrees with sheaf cohomology of that constant sheaf. A good-cover Čech-de Rham double complex gives an explicit comparison, used in Čech-de Rham curvature descent.

For a compact connected oriented boundaryless $n$-dimensional smooth manifold, choose a Riemannian metric. Harmonic functions are constant because $⟨f,\Delta f⟩=\Vert df{\Vert }^2$, and Hodge star commutes with the Hodge Laplacian. Thus the Hodge star operator identifies the constant functions with the harmonic differential forms of top degree, exactly $\mathbb{R}{\omega }_g$. The Hodge decomposition theorem identifies these with the top de Rham cohomology. The metric volume form has nonzero class by Stokes theorem. The harmonic-representative theorem is stated in Denis Auroux's Hodge theory lecture.

A closed differential form of degree one on the two-sphere has primitives on its north-punctured and south-punctured charts by the Poincaré lemma. The overlap is connected, so their difference is constant. Adjust one constant and glue the primitives. Thus every closed one-form is an exact differential form, proving the first de Rham cohomology vanishes.

For a free smooth action of a finite group $G$ on a smooth manifold $M$, the quotient projection inducesInvariant differential forms descend uniquely through local inverse branches of the covering map. Averaging by $|G{|}^{-1}{\sum }_{g\in G}{g}^{\ast }$ commutes with the exterior derivative, produces invariant representatives of invariant classes, and produces an invariant primitive for every invariant exact form. Hence taking invariant forms and taking invariant cohomology give the same result over the real numbers.

On a compact connected oriented $n$-dimensional smooth manifold without boundary, integration is a linear isomorphism $H_{\mathrm{dR}}^n(M)\to \mathbb{R}$. Thus two top-degree differential forms have the same de Rham cohomology class exactly when their integrals agree.

A compact Lie group has finitely many connected components, each an oriented compact manifold of dimension $n$. Componentwise integration identifies its top-degree de Rham cohomology with one real coordinate per component. For a connected group, if top-degree classes are represented by invariant forms, the invariant volume form on a Lie group spans the cohomology: invariant top forms are one-dimensional, and its positive integral proves that its class is nonzero by the Generalized Stokes theorem. A disconnected group cannot have an invariant representative of every top-degree class, since left translations permute its components. For example, $S^1\times \mathbb{Z}/2$ has two independent degree-one classes but only one independent invariant top form.

Let $S^1$ act on $M\times S^1$ by rotation of the second factor. Averaging a differential form over this action is a cochain map. Every rotation is homotopic to the identity through smooth rotations, and integrating Cartan's magic formula along them gives a cochain homotopy between the averaging map and the identity. Consequently every de Rham cohomology class has a rotation-invariant representative, and an invariant exact form has an invariant primitive obtained by averaging any primitive.

Choose a closed one-form $\nu$ on $S^1$ with integral one and let $p:M\times S^1\to M$ be projection. Every rotation-invariant $k$-form is uniquelyThe exterior derivative acts componentwise. Averaging therefore proves thatis the displayed isomorphism.

Smoothly homotopic maps induce the same map on de Rham cohomology. Integrating Cartan's magic formula along a homotopy gives a cochain homotopy

For an open cover $X=U\cup V$, restriction of forms gives a short exact sequence of cochain complexesand hence a long exact sequence in de Rham cohomology.

Cover the sphere by complements of its two poles. Each open set is contractible, and their intersection deformation retracts to the equatorial sphere. The Poincaré lemma and homotopy invariance of de Rham cohomology make the two neighboring groups in the Mayer--Vietoris sequence for de Rham cohomology zero, so its connecting map is an isomorphism. Its inverse takes a top-degree form to the difference of its local primitives on the equator, up to the connecting-map sign convention.

On an oriented Riemannian $n$-manifold, the Hodge star is determined byIt maps $p$-forms to $(n-p)$-forms and satisfies ${\ast }^2=(-1{)}^{p(n-p)}$ on $p$-forms.

For a Hermitian manifold with the pointwise Hermitian inner product linear in the first variable, define the conjugate-linear star by $\alpha \wedge \ast \beta =⟨\alpha ,\beta ⟩{\omega }^n/n!$. It is the usual real Hodge star operator extended complex-linearly and then composed with complex conjugation. It has the displayed bidegree and satisfies ${\ast }^2=(-1{)}^{p+q}$ on $(p,q)$-forms.

For inner products linear in the first argument, define ${\ast }_E$ by contraction in $\psi \wedge {\ast }_E\eta =(\psi ,\eta )dV$. It is conjugate-linear and takes $(p,q)$-forms with values in $E$ to $(n-p,n-q)$-forms with values in $E^{\ast }$. With induced dual metrics, ${\ast }_{E^{\ast }}{\ast }_E=(-1{)}^{p+q}$.

On an oriented Euclidean metric four-space, the Hodge star operator on two-forms is an orthogonal involution. Its two eigenspaces each have dimension three. A self-dual two-form has eigenvalue $+1$, while an anti-self-dual two-form has eigenvalue $-1$. The splitting uses the metric-normalized Riemannian volume form; an arbitrarily rescaled volume in the defining wedge identity would rescale the operator itself.

The Hodge star operator is self-adjoint on real two-forms in four Euclidean dimensions, so its opposite eigenspaces are orthogonal. Since $H\wedge G=⟨H,\ast G⟩{\mathrm{vol}}_g$, a self-dual two-form and an anti-self-dual two-form also have zero wedge product of differential forms. Applied to Lie-algebra components with an invariant pairing, this identity eliminates mixed terms from the Yang-Mills action.

On $m$-forms in an oriented Riemannian manifold of dimension $2m$, the Hodge star operator is orthogonal and satisfies ${\ast }^2=(-1{)}^{m^2}$. HenceIt is self-adjoint when $m$ is even and skew-adjoint when $m$ is odd.

If a metric in dimension $n$ is rescaled by $\tilde{g}=e^{2\omega }g$, then its Hodge star on $p$-forms satisfies $\tilde{\ast }=e^{(n-2p)\omega }\ast$. The exponent vanishes for middle-degree forms with $n=2p$, so their self-dual and anti-self-dual subspaces depend only on the conformal class.

The Hodge star extends complex-linearly to complex differential forms. With the Hermitian pointwise inner product, it is characterized byOn a Hermitian manifold it maps forms of type $(p,q)$ to forms of type $(n-q,n-p)$.

Use the complex-linear Hodge star operator on the direct sum of all form degrees. Its square on degree $r$ is $(-1{)}^{r(n-r)}$. In positive odd dimension this is always one; in positive even dimension its sign is positive on even-degree forms and negative on odd-degree forms. Complementary basis wedges provide both roots of each sign, so the displayed eigenvalues all occur. On a single degree the Hodge star operator is an endomorphism only in middle degree; on middle degree $m$ in dimension $2m$ its possible eigenvalues are $\pm 1$ for even $m$ and $\pm i$ for odd $m$.

For the complex-linear Hodge star operator and complex orientation on a Hermitian manifold of complex dimension $n$, every $(n,0)$-form has the displayed eigenvalue. In an adapted orthonormal real coframe, each factor $d{x}_j+id{y}_j$ has two-dimensional star eigenvalue $-i$. Regrouping the $n$ factors introduces $(-1{)}^{n(n-1)/2}$, giving $(-1{)}^{n(n+1)/2}{i}^n$. This is a pointwise metric identity; the metric need not be Kähler.

On an oriented Riemannian four-manifold, a two-form $\alpha$ is self-dual when $\ast \alpha =\alpha$.

On an oriented Riemannian four-manifold, a two-form is self-dual when $\ast \alpha =\alpha$. Here $\ast$ is the Hodge star operator.

For the orientation $d{x}^1\wedge d{x}^2\wedge d{x}^3\wedge d{x}^4$, define ${\eta }_{ij}^a={ϵ}_{aij}$, ${\eta }_{i4}^a={\delta }_{ai}$, and extend antisymmetrically. The three forms $\frac{1}{2}{\eta }_{\mu \nu }^{a}d{x}^{\mu }\wedge d{x}^{\nu }$ are self-dual. Reversing the sign of the components involving index four gives the anti-self-dual symbols $\bar{\eta }$. They identify the two three-dimensional eigenspaces of the Hodge star on differential two-forms and give compact formulas for the BPST instanton.

For $w=x^1+i{x}^2$, $z=x^3+i{x}^4$ and the standard oriented Euclidean metric, the real and imaginary parts of $dw\wedge dz$, together with $i(dw\wedge d\bar{w}+dz\wedge d\bar{z})$, span the self-dual two-forms. The printed complex four-form $dw\wedge dz\wedge d\bar{w}\wedge d\bar{z}$ is four times the metric volume form and gives the same orientation.

On an oriented Riemannian four-manifold, a two-form $\alpha$ is anti-self-dual when $\ast \alpha =-\alpha$.

On an oriented Riemannian four-manifold, a two-form is anti-self-dual when $\ast \alpha =-\alpha$. Here $\ast$ is the Hodge star operator.

If an exact two-form $\beta =d\alpha$ on a compact oriented Riemannian four-manifold is anti-self-dual, thenso $\beta =0$.

The codifferential is the formal $L^2$ adjoint of the exterior derivative. On $p$-forms in dimension $n$, one convention is

On a closed oriented Riemannian manifold, let $\alpha$ have degree $r-1$ and $\beta$ degree $r$. The graded Leibniz rule and Stokes theorem give $\int d\alpha \wedge \ast \beta =(-1{)}^r\int \alpha \wedge d\ast \beta$. The codifferential convention $\delta \beta =(-1{)}^{n(r+1)+1}\ast d\ast \beta$, combined with the square of the Hodge star, gives $\ast \delta \beta =(-1{)}^{r}d\ast \beta$. This proves that the codifferential is the formal adjoint of the exterior derivative. A boundary requires the corresponding boundary term or specified boundary conditions.

On an oriented boundaryless Riemannian manifold, apply Stokes theorem to the compactly supported form $f\ast \alpha$. The graded Leibniz rule gives $\int df\wedge \ast \alpha =-\int fd\ast \alpha$. The Hodge star operator identifies these top forms with the scalar integrands displayed above. This proves that the codifferential on one-forms is $\delta =-\ast d\ast$, the formal adjoint of the exterior derivative. Compact support of $f$ suffices.

A differential form is coclosed when its codifferential vanishes. On a closed Riemannian manifold, a form is a harmonic differential form exactly when it is both a closed differential form and coclosed: the identity $⟨\Delta \alpha ,\alpha ⟩=\Vert d\alpha {\Vert }^2+\Vert \delta \alpha {\Vert }^2$ proves the converse as well as necessity.