the exterior derivative
The exterior derivative d is the one true derivative of differential forms. It is a single operation that, depending on degree, reproduces the gradient, the curl, and the divergence of vector calculus all at once — and it does so without any choice of metric or coordinates. Apply d to a function and you get its differential (a 1-form recording how the function changes); apply d again and you always get zero. That last fact, d^2 = 0, is the seed of all of cohomology.
Precisely, d: Omega^k(M) -> Omega^{k+1}(M) is the unique linear operator satisfying: (i) on 0-forms (functions) f, df is the ordinary differential, df = sum (partial f / partial x^i) dx^i; (ii) the graded Leibniz (antiderivation) rule d(alpha ^ beta) = d alpha ^ beta + (-1)^{deg alpha} alpha ^ d beta; and (iii) d(d omega) = 0 for every omega. In coordinates, for omega = f_I dx^I you just take d omega = df_I ^ dx^I, differentiating only the coefficient functions and wedging on their differential. It commutes with pullback (d f^* = f^* d), which is what makes it geometric rather than coordinate-bound.
Why d^2 = 0 matters: it says exact forms (those of the form d alpha) are automatically closed (d of them is zero), and the gap between closed and exact is precisely de Rham cohomology. The mixed-partials symmetry of second derivatives is exactly what forces d^2 = 0. A classic translation: in R^3, d on functions is grad, d on 1-forms is curl, d on 2-forms is div; then d^2 = 0 encodes the two identities curl(grad f) = 0 and div(curl F) = 0. So vector calculus's two famous vanishing identities are a single statement, d^2 = 0.
On R^3 with omega = P dx + Q dy + R dz, compute d omega = dP^dx + dQ^dy + dR^dz, expand and collect: d omega = (Q_x - P_y) dx^dy + (R_y - Q_z) dy^dz + (P_z - R_x) dz^dx. The coefficients are exactly the components of curl(P, Q, R). And for a function f, d(df) = sum f_{ij} dx^i^dx^j = 0 because f_{ij} = f_{ji} is symmetric while dx^i^dx^j = -dx^j^dx^i is antisymmetric — equal symmetric times antisymmetric sums to zero.
d on a 1-form in R^3 is the curl; d^2 = 0 is the symmetry of mixed partials, i.e. curl(grad) = 0.
The exterior derivative needs no connection, no metric, and no orientation — it is purely smooth-structure data, which is what makes de Rham cohomology a diffeomorphism invariant for free. Do not confuse d with the covariant derivative nabla, which DOES need a connection and acts on general tensors, not just alternating ones.