# Differential form

A differential form is a smooth, totally antisymmetric tensor field on a differentiable manifold: a *p*-form assigns, at each point, an alternating multilinear function of *p* tangent vectors, and the collection of all degrees forms an algebra under the wedge product and a derivative called the exterior derivative<sup>[1](https://www.damtp.cam.ac.uk/user/hsr1000/lecturenotes_2012.pdf)</sup>. Differential forms are the natural objects for integration over curved spaces, and their derivative encodes the classical operators of vector calculus<sup>[2](https://math-cs-compass.com/Mathematics/Calculus/differential_forms.html)</sup>.

| Key fact | Detail |
|---|---|
| Definition | A *p*-form on a differentiable manifold \( M \) is an antisymmetric \((0,p)\) tensor field; a 0-form is a function and a 1-form is a covector field<sup>[1](https://www.damtp.cam.ac.uk/user/hsr1000/lecturenotes_2012.pdf)</sup> |
| Dimension count | In an \( n \)-dimensional space there are \( {n \choose k} \) independent rank-\( k \) elements of the exterior algebra, and \( \Omega \equiv 0 \) for \( p > \dim M \)<sup>[3](https://books.physics.oregonstate.edu/GELG/forms.html)</sup><sup> • </sup><sup>[4](https://edu.itp.phys.ethz.ch/hs14/GR/gr.pdf)</sup> |
| Wedge product | Bilinear, associative, and graded-antisymmetric: \( X \wedge Y = (-1)^{pq} Y \wedge X \), so \( X \wedge X = 0 \) for odd \( p \)<sup>[1](https://www.damtp.cam.ac.uk/user/hsr1000/lecturenotes_2012.pdf)</sup> |
| Exterior derivative | A linear map \( d : \Omega^p \to \Omega^{p+1} \) with \( d^2 = 0 \) and \( d(\Omega_1 \wedge \Omega_2) = d\Omega_1 \wedge \Omega_2 + (-1)^{p_1} \Omega_1 \wedge d\Omega_2 \)<sup>[4](https://edu.itp.phys.ethz.ch/hs14/GR/gr.pdf)</sup> |
| Stokes' theorem | \( \int_{\partial M} \omega = \int_M d\omega \) for a compact oriented \((k+1)\)-dimensional submanifold-with-boundary and a smooth \( k \)-form \( \omega \)<sup>[5](https://math.libretexts.org/Bookshelves/Analysis/Tasty_Bits_of_Several_Complex_Variables_%28Lebl%29/07%3A_Appendix/7.03%3A_C-_Differential_Forms_and_Stokes%27_Theorem)</sup> |
| De Rham cohomology | \( H^\ell(M;\mathbb{R}) \) is the quotient of closed \( \ell \)-forms by exact \( \ell \)-forms; its dimension is the \( \ell \)-th Betti number<sup>[6](https://math.mit.edu/~dyatlov/18.155/155-notes.pdf)</sup> |
| Physics use | Maxwell's equations become \( dH = J \) and \( d\mathcal{F} = 0 \), with the field strength \( \mathcal{F} \) a 2-form and charge \( J \) a 3-form<sup>[7](https://m.fuw.edu.pl/event-reader/event4999.html)</sup> |

## Definition and low degrees

Let \( M \) be a differentiable manifold. A \( p \)-form on \( M \) is an antisymmetric \((0,p)\) tensor field on \( M \)<sup>[1](https://www.damtp.cam.ac.uk/user/hsr1000/lecturenotes_2012.pdf)</sup>. Equivalently, a \( k \)-form assigns to each point \( x \) a real value to \( k \) tangent vectors \( v_1, \ldots, v_k \in T_x X \), linear and antisymmetric in the arguments, varying smoothly with \( x \)<sup>[8](https://dept.math.lsa.umich.edu/~speyer/632Old/jan-11.pdf)</sup>. The space of \( k \)-forms is written \( \Omega^k(M) \). In local coordinates \( x^1, \ldots, x^n \), a typical \( k \)-form is a sum \( \sum_I f_{i_1 \ldots i_k}(x)\, dx^{i_1} \wedge \cdots \wedge dx^{i_k} \) with smooth coefficient functions<sup>[8](https://dept.math.lsa.umich.edu/~speyer/632Old/jan-11.pdf)</sup>.

**Low degrees.** A 0-form is a smooth function and a 1-form is a covector field<sup>[1](https://www.damtp.cam.ac.uk/user/hsr1000/lecturenotes_2012.pdf)</sup>. The differential \( df \) of a smooth function \( f \) is the basic example of a 1-form: it acts on a vector field \( X \) by the directional derivative, \( df(X) = X(f) \), and in a coordinate basis this is partial differentiation<sup>[3](https://books.physics.oregonstate.edu/GELG/forms.html)</sup>. A \( p \)-form vanishes identically when \( p \) exceeds the dimension of \( M \)<sup>[4](https://edu.itp.phys.ethz.ch/hs14/GR/gr.pdf)</sup>.

## The wedge product

The wedge (exterior) product of a \( p \)-form \( X \) and a \( q \)-form \( Y \) is the \((p+q)\)-form defined by antisymmetrizing the tensor product,

\[
(X \wedge Y)_{a_1 \ldots a_p b_1 \ldots b_q} = \frac{(p+q)!}{p!\,q!}\, X_{[a_1 \ldots a_p} Y_{b_1 \ldots b_q]} .
\]

It is bilinear and associative, and graded-antisymmetric: \( X \wedge Y = (-1)^{pq} Y \wedge X \), so the wedge of a form of odd degree with itself is zero<sup>[1](https://www.damtp.cam.ac.uk/user/hsr1000/lecturenotes_2012.pdf)</sup>. For two 1-forms \( A, B \) the components are \( (A \wedge B)_{ik} = A_i B_k - A_k B_i \)<sup>[4](https://edu.itp.phys.ethz.ch/hs14/GR/gr.pdf)</sup>. In an \( n \)-dimensional space the wedge products of \( k \) basis covectors span a space of dimension \( {n \choose k} \), and the forms of all degrees assemble into the exterior algebra \( \bigwedge(M) \)<sup>[3](https://books.physics.oregonstate.edu/GELG/forms.html)</sup>.

## The exterior derivative

The exterior derivative \( d \) sends a \( p \)-form to a \((p+1)\)-form. In a coordinate basis it is given by

\[
(dX)_{\mu_1 \ldots \mu_{p+1}} = (p+1)\, \partial_{[\mu_1} X_{\mu_2 \ldots \mu_{p+1}]} ,
\]

where the brackets denote antisymmetrization over the indices<sup>[1](https://www.damtp.cam.ac.uk/user/hsr1000/lecturenotes_2012.pdf)</sup>. On a 0-form \( f \), \( df \) is the 1-form \( df(X) = Xf \); on a 1-form \( \Omega \), \( d\Omega(X_1, X_2) = X_1\Omega(X_2) - X_2\Omega(X_1) - \Omega([X_1, X_2]) \)<sup>[4](https://edu.itp.phys.ethz.ch/hs14/GR/gr.pdf)</sup>. The operator satisfies four defining properties: linearity from \( p \)-forms to \((p+1)\)-forms, the graded product rule \( d(\Omega_1 \wedge \Omega_2) = d\Omega_1 \wedge \Omega_2 + (-1)^{p_1}\Omega_1 \wedge d\Omega_2 \), the identity \( d^2 = 0 \), and \( df(X) = Xf \)<sup>[4](https://edu.itp.phys.ethz.ch/hs14/GR/gr.pdf)</sup>.

**Closed and exact forms.** A form \( \omega \) with \( d\omega = 0 \) is *closed*; a form \( \omega = d\eta \) is *exact*. Every exact form is closed, but the converse generally fails. The local converse is the Poincaré lemma: on a star-shaped open domain, any closed \( p \)-form \( \omega \) can be written \( \omega = d\eta \) for some \((p-1)\)-form \( \eta \), and \( \eta \) is not unique since \( \eta \to \eta + d\rho \) leaves \( d\eta \) unchanged<sup>[4](https://edu.itp.phys.ethz.ch/hs14/GR/gr.pdf)</sup>.

## Integration and Stokes' theorem

A \( k \)-form is an object that can be integrated over a \( k \)-dimensional domain, taking orientation into account<sup>[5](https://math.libretexts.org/Bookshelves/Analysis/Tasty_Bits_of_Several_Complex_Variables_%28Lebl%29/07%3A_Appendix/7.03%3A_C-_Differential_Forms_and_Stokes%27_Theorem)</sup>. Integration of an \( n \)-form on an oriented \( n \)-manifold requires an orientation, an atlas of positively oriented charts; not every manifold is orientable, the [Möbius strip](https://www.edgepedia.org/mobius-strip) being a standard example<sup>[4](https://edu.itp.phys.ethz.ch/hs14/GR/gr.pdf)</sup>.

The generalized [Stokes' theorem](https://www.edgepedia.org/stokes-theorem) states that for \( M \) an embedded compact smooth oriented \((k+1)\)-dimensional submanifold-with-boundary, with \( \partial M \) carrying the induced orientation, and \( \omega \) a smooth \( k \)-form on \( M \),

\[
\int_{\partial M} \omega = \int_M d\omega .
\]

The classical theorems of vector calculus are special cases: the fundamental theorem of calculus for line integrals, [Green's theorem](https://www.edgepedia.org/greens-theorem), the classical Stokes' theorem (the case \( n = 3 \), \( k = 2 \)), and the divergence theorem<sup>[5](https://math.libretexts.org/Bookshelves/Analysis/Tasty_Bits_of_Several_Complex_Variables_%28Lebl%29/07%3A_Appendix/7.03%3A_C-_Differential_Forms_and_Stokes%27_Theorem)</sup><sup> • </sup><sup>[6](https://math.mit.edu/~dyatlov/18.155/155-notes.pdf)</sup>. In the classical language the line integral is an integral of a vector field rather than a 1-form, and the derivative \( d\omega \) corresponds to the curl operator<sup>[5](https://math.libretexts.org/Bookshelves/Analysis/Tasty_Bits_of_Several_Complex_Variables_%28Lebl%29/07%3A_Appendix/7.03%3A_C-_Differential_Forms_and_Stokes%27_Theorem)</sup>.

## De Rham cohomology

The order \( \ell \) [de Rham cohomology](https://www.edgepedia.org/de-rham-cohomology) space is the quotient of the space of closed \( \ell \)-forms by the space of exact \( \ell \)-forms,

\[
H^\ell(M; \mathbb{R}) := \ker d_\ell \, / \, d_{\ell-1}(C^\infty(M; \Omega^{\ell-1})) ,
\]

a real vector space whose dimension is the \( \ell \)-th [Betti number](https://www.edgepedia.org/betti-number) \( b_\ell(M) \)<sup>[6](https://math.mit.edu/~dyatlov/18.155/155-notes.pdf)</sup>. The wedge product descends to cohomology: the wedge of two closed forms is closed, and the wedge of a closed and an exact form is exact, so \( [u] \in H^\ell \) and \( [v] \in H^m \) give \( [u \wedge v] \in H^{\ell+m} \)<sup>[6](https://math.mit.edu/~dyatlov/18.155/155-notes.pdf)</sup>.

**A topological example.** On \( M = \mathbb{R}^2 \setminus \{0\} \), the 1-form

\[
u = \frac{x_1\, dx_2 - x_2\, dx_1}{x_1^2 + x_2^2}
\]

is closed but not exact: \( b_1(M) = 1 \) and \( [u] \) spans \( H^1(M; \mathbb{R}) \). Its integral over the unit circle is \( 2\pi \), and for a general immersed closed curve the integral \( \frac{1}{2\pi} \oint_\gamma u \) is the signed winding number of the curve about the origin<sup>[6](https://math.mit.edu/~dyatlov/18.155/155-notes.pdf)</sup>. For the sphere \( S^n \), \( b_\ell = 1 \) for \( \ell = 0 \) or \( \ell = n \) and 0 otherwise; for a compact connected oriented surface of genus \( g \), \( b_0 = b_2 = 1 \) and \( b_1 = 2g \)<sup>[6](https://math.mit.edu/~dyatlov/18.155/155-notes.pdf)</sup>.

## Uses in physics

**Electromagnetism.** In the generally covariant, metric-free formulation of classical electrodynamics, Maxwell's field equations are \( dH = J \) and \( d\mathcal{F} = 0 \), where \( H \) (the field excitation) and \( \mathcal{F} \) (the field strength) are 2-forms and \( J \) (charge) is a 3-form, together with a constitutive law \( H = \kappa(\mathcal{F}) \) relating \( H \) and \( \mathcal{F} \)<sup>[7](https://m.fuw.edu.pl/event-reader/event4999.html)</sup>. The homogeneous equation \( d\mathcal{F} = 0 \) encodes Faraday's induction law and the sourcelessness of \( B \), and in vacuum the constitutive law is \( H = \lambda_0\, {^\star}\mathcal{F} \) with the metric-dependent Hodge star<sup>[9](https://arxiv.org/abs/0807.4249)</sup>. Setting the constitutive law to the Hodge star yields [Maxwell's equations](https://www.edgepedia.org/maxwells-equations) in vacuum, while more general choices describe inhomogeneities, anisotropies, dispersion, and nonlinear media such as birefringent media or Born–Infeld theory<sup>[7](https://m.fuw.edu.pl/event-reader/event4999.html)</sup>.

**General relativity.** In the tetrad (vielbein) formalism, the torsion and curvature of a connection are written as forms, giving the Cartan (Maurer–Cartan) structure equations

\[
T^a = de^a + \omega^a{}_b \wedge e^b , \qquad R^a{}_b = d\omega^a{}_b + \omega^a{}_c \wedge \omega^c{}_b ,
\]

where \( e^a \) is the coframe and \( \omega^a{}_b \) the spin connection; these are equivalent to the usual component definitions of torsion and curvature<sup>[10](https://www.physics.smu.edu/scalise/P3368sp17/CarrollGR.pdf)</sup>. The [Bianchi identities](https://www.edgepedia.org/bianchi-identities) take the form \( dR^a{}_b + \omega^a{}_c \wedge R^c{}_b - R^a{}_c \wedge \omega^c{}_b = 0 \)<sup>[10](https://www.physics.smu.edu/scalise/P3368sp17/CarrollGR.pdf)</sup>.

**Frobenius integrability.** For a 1-form \( \omega \), the condition \( \omega \wedge d\omega = 0 \) is equivalent to \( \omega \) being locally of the form \( \lambda\, df \), which characterizes when the hyperplanes annihilated by \( \omega \) can be tangent to a foliation<sup>[4](https://edu.itp.phys.ethz.ch/hs14/GR/gr.pdf)</sup>.

## References

1. [Differential forms, Chapter 9 of General Relativity lecture notes, DAMTP, University of Cambridge](https://www.damtp.cam.ac.uk/user/hsr1000/lecturenotes_2012.pdf)
2. [Differential Forms on Manifolds, MATH-CS COMPASS](https://math-cs-compass.com/Mathematics/Calculus/differential_forms.html)
3. [Differential Forms, Oregon State University physics texts](https://books.physics.oregonstate.edu/GELG/forms.html)
4. [General Relativity lecture notes, ETH Zürich](https://edu.itp.phys.ethz.ch/hs14/GR/gr.pdf)
5. [Differential Forms and Stokes' Theorem, Mathematics LibreTexts (Lebl)](https://math.libretexts.org/Bookshelves/Analysis/Tasty_Bits_of_Several_Complex_Variables_%28Lebl%29/07%3A_Appendix/7.03%3A_C-_Differential_Forms_and_Stokes%27_Theorem)
6. [Differential Forms and Hodge Theory, MIT 18.155 lecture notes](https://math.mit.edu/~dyatlov/18.155/155-notes.pdf)
7. [Classical Electrodynamics in Linear Media, seminar abstract, University of Warsaw](https://m.fuw.edu.pl/event-reader/event4999.html)
8. [Notes for January 11: Differential Forms, University of Michigan](https://dept.math.lsa.umich.edu/~speyer/632Old/jan-11.pdf)
9. [Hehl, F. W. (2008). Maxwell's equations in Minkowski's world: their premetric generalization and the electromagnetic energy-momentum tensor. arXiv:0807.4249](https://arxiv.org/abs/0807.4249)
10. [Carroll, S. Lecture Notes on General Relativity, Chapter 3: Curvature](https://www.physics.smu.edu/scalise/P3368sp17/CarrollGR.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Relativity and gravitation › General relativity and curved spacetime › Foundations and field equations › Mathematical structure of curved spacetime › Differential forms and variational geometry*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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