Differential Forms - Core Definitions

Differential forms provide the natural language for integration on manifolds. They generalize the integr ands from multivariable calculus to curved spaces and form the foundation of de Rham cohomology.

DefinitionDifferential k-Form

A differential $k$ -form on $M$ is a smooth section $\omega: M \to \Lambda^k(T^*M)$ of the $k$ -th exterior power of the cotangent bundle. At each point $p \in M$ , $\omega_p$ is an alternating multilinear map

\omega_p: T_pM \times \cdots \times T_pM \to \mathbb{R}

taking $k$ tangent vectors and returning a real number, with $\omega_p(v_1, \ldots, v_k) = 0$ whenever $v_i = v_j$ for some $i \neq j$ .

The alternating property means forms are antisymmetric under permutations: $\omega(v_{\sigma(1)}, \ldots, v_{\sigma(k)}) = \text{sgn}(\sigma) \omega(v_1, \ldots, v_k)$ for any permutation $\sigma$ .

DefinitionWedge Product

The wedge product $\omega \wedge \eta$ of a $k$ -form $\omega$ and an $\ell$ -form $\eta$ is a $(k+\ell)$ -form defined by

(\omega \wedge \eta)(v_1, \ldots, v_{k+\ell}) = \frac{1}{k!\ell!} \sum_{\sigma \in S_{k+\ell}} \text{sgn}(\sigma) \omega(v_{\sigma(1)}, \ldots, v_{\sigma(k)}) \eta(v_{\sigma(k+1)}, \ldots, v_{\sigma(k+\ell)})

The wedge product is associative and anticommutative: $\omega \wedge \eta = (-1)^{k\ell} \eta \wedge \omega$ .

ExampleForms in $\mathbb{R}^3$

On $\mathbb{R}^3$ with coordinates $(x, y, z)$ :

0-forms are functions: $f$
1-forms are expressions: $\omega = f_1 dx + f_2 dy + f_3 dz$
2-forms: $\eta = g_1 dy \wedge dz + g_2 dz \wedge dx + g_3 dx \wedge dy$
3-forms: $\alpha = h \, dx \wedge dy \wedge dz$ (volume forms)

Note $dx \wedge dy = -dy \wedge dx$ and $dx \wedge dx = 0$ .

DefinitionExterior Derivative

The exterior derivative is a linear operator $d: \Omega^k(M) \to \Omega^{k+1}(M)$ satisfying:

$d^2 = 0$ (nilpotency)
$d(f) = df$ for 0-forms (functions)
$d(\omega \wedge \eta) = d\omega \wedge \eta + (-1)^k \omega \wedge d\eta$ (Leibniz rule)
$d$ is natural with respect to pullbacks: $f^*(d\omega) = d(f^*\omega)$

In local coordinates $(x^1, \ldots, x^n)$ , if $\omega = \sum_I f_I dx^I$ where $I = (i_1 < \cdots < i_k)$ and $dx^I = dx^{i_1} \wedge \cdots \wedge dx^{i_k}$ , then

d\omega = \sum_I df_I \wedge dx^I = \sum_{I,j} \frac{\partial f_I}{\partial x^j} dx^j \wedge dx^I

ExampleExact and Closed Forms

A form $\omega$ is closed if $d\omega = 0$ , and exact if $\omega = d\eta$ for some form $\eta$ . Every exact form is closed (since $d^2 = 0$ ), but the converse depends on the topology of $M$ . On $\mathbb{R}^n$ , the Poincaré lemma states every closed form is exact.

Remark

The space $\Omega^k(M)$ of $k$ -forms forms a module over $C^\infty(M)$ . The exterior derivative $d$ makes $(\Omega^*(M), d)$ into a cochain complex, whose cohomology is the de Rham cohomology of $M$ .

Differential forms unify vector calculus operators: $d$ generalizes gradient, curl, and divergence, while the wedge product encodes cross products and determinants.