Exterior Derivative
Definition. The unique map $d: \Omega^k(M) \to \Omega^{k+1}(M)$ that: agrees with the differential on functions, satisfies the graded Leibniz rule $d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^{|\alpha|}\alpha \wedge d\beta$, and obeys $d^2 = 0$. In coordinates: $d\omega = \partial_j \omega_{i_1 \dots i_k}\, dx^j \wedge dx^{i_1} \wedge \cdots$.
Intuition. Grad, curl, and div unified into one coordinate-free operator. $d^2 = 0$ is $\nabla \times \nabla f = 0$ and $\nabla \cdot (\nabla \times \vec{v}) = 0$, both at once.
Vocabulary. $\omega$ is closed if $d\omega = 0$, exact if $\omega = d\eta$. Exact ⇒ closed (by $d^2=0$); the converse fails globally — measured by de Rham Cohomology.
Physics anchor. $F = dA$ (Landau §23: $F_{ik} = \partial_i A_k - \partial_k A_i$). Then $dF = d^2A = 0$ is the first pair of Maxwell equations (§26) — automatically true, needing no physical input. This is the first big geometric dividend of the whole study.
Related: Differential Form · Stokes Theorem · de Rham Cohomology · Curvature 2-Form · Module 02 - Differential Forms Refresher
Sources: Sternberg Ch. III; Landau §23, §26.