Differential forms are alternating covariant tensors designed for integration. Their wedge product and exterior derivative form a coordinate-free calculus that contains gradients, curls, divergences, and the integrands of Stokes' theorem.
The exterior algebra
Definition. A differential \(k\)-form is a smooth section of \(\Lambda^kT^*M\). In coordinates, \[ \omega=\sum_{i_1<\cdots<i_k} \omega_{i_1\cdots i_k}\, \mathrm{d}x^{i_1}\wedge\cdots\wedge\mathrm{d}x^{i_k}. \] The wedge product satisfies \[ \alpha\wedge\beta=(-1)^{k\ell}\beta\wedge\alpha \] for \(\alpha\in\Omega^k\), \(\beta\in\Omega^\ell\).
Exterior differentiation
Theorem. There is a unique family of linear maps \(\mathrm{d}:\Omega^k(M)\to\Omega^{k+1}(M)\) such that \[ \mathrm{d}f \text{ is the ordinary differential},\qquad \mathrm{d}(\alpha\wedge\beta) =\mathrm{d}\alpha\wedge\beta+(-1)^k\alpha\wedge\mathrm{d}\beta, \qquad \mathrm{d}^2=0. \] In coordinates, \[ \mathrm{d}\!\left(\sum_I\omega_I\,\mathrm{d}x^I\right) =\sum_I\sum_j\partial_j\omega_I\, \mathrm{d}x^j\wedge\mathrm{d}x^I. \]
Proof of \(\mathrm{d}^2=0\). It suffices to calculate in coordinates: \[ \mathrm{d}^2\omega =\sum_{I,j,k}\partial_k\partial_j\omega_I\, \mathrm{d}x^k\wedge\mathrm{d}x^j\wedge\mathrm{d}x^I=0. \] The coefficient is symmetric in \(j,k\), while the wedge factor is antisymmetric.
Proposition (naturality). For every smooth \(F:M\to N\), \[ \mathrm{d}(F^*\omega)=F^*(\mathrm{d}\omega). \]
This identity can be checked in coordinates and then is intrinsic. It is the mechanism by which changes of variables preserve differential relations.
Closed and exact forms
Definition. A form is closed if \(\mathrm{d}\omega=0\), and exact if \(\omega=\mathrm{d}\eta\). Every exact form is closed, but the converse can fail globally.
Lemma (Poincar?). Every closed \(k\)-form, \(k\ge1\), on a star-shaped open set \(U\subset\mathbb R^n\) is exact.
Proof. Let \(H_t(x)=tx\) and let \(R=\sum_ix^i\partial_i\) be the radial field. Define \[ K\omega=\int_0^1t^{k-1}H_t^*(\iota_R\omega)\,\mathrm{d}t. \] Differentiating \(H_t^*\omega\) and using Cartan's formula gives the homotopy identity \[ \mathrm{d}K+K\mathrm{d}=\operatorname{id}-H_0^*. \] For \(k\ge1\), \(H_0^*\omega=0\); if \(\mathrm{d}\omega=0\), then \(\omega=\mathrm{d}(K\omega)\).
Example. On \(\mathbb R^2\setminus\{0\}\), \[ \omega=\frac{-y\,\mathrm{d}x+x\,\mathrm{d}y}{x^2+y^2} \] is closed, but \(\int_{S^1}\omega=2\pi\). It cannot be exact because the integral of an exact form over a closed curve is zero. This local-global failure is measured by de Rham cohomology.