On a smooth manifold $M$, a $k$-form is a smooth section of $\Lambda^k T^* M$ — locally, an antisymmetric multilinear function on $k$ tangent vectors. Familiar special cases: 0-forms = functions, 1-forms = covector fields, 3-forms in $\mathbb R^3$ = scalar densities times $dV$.

Key operations:

• Wedge product $\alpha \wedge \beta$: associative, distributive, $\alpha \wedge \beta = (-1)^{kl} \beta \wedge \alpha$ (graded-commutative).
• Exterior derivative $d$: $d^2 = 0$; on a 0-form $f$, $df = \sum (\partial_i f)\, dx^i$.
• Pullback $\phi^* \omega$ along a smooth map $\phi$: natural with $d$ ($\phi^* d = d \phi^*$).

In $\mathbb R^3$, $d$ unifies grad, curl, div via the identifications $1$-form $\leftrightarrow$ vector field, $2$-form $\leftrightarrow$ vector field (via Hodge star). The identities $d^2 = 0$ become $\nabla \times \nabla f = 0$ and $\nabla \cdot (\nabla \times \mathbf F) = 0$.

Stokes' theorem: for a $k$-form $\omega$ on a compact oriented $(k+1)$-manifold $M$ with boundary $\partial M$,

Specializations: fundamental theorem of calculus ($k=0$), Green's theorem ($k=1, 2D$), classical Stokes ($k=1, 3D$), divergence theorem ($k=2, 3D$). All of vector calculus is one identity.