From the wedge product to one differentiating operator
In the previous guide you assembled the exterior algebra and learned to multiply forms with the wedge product, producing a differential k-form omega that eats k tangent vectors and spits out a number, antisymmetrically. That was algebra at a single point. Now we let the point move. The exterior derivative d is the one operator that differentiates a k-form into a (k+1)-form, d omega, and the miracle is that it needs no metric, no connection, no coordinates to be well defined — only the smooth structure of M. It is the most coordinate-free derivative in all of geometry.
Start where you already live: a 0-form is just a smooth function f. Its exterior derivative is the familiar differential, d f = (partial f / partial x^i) dx^i, the 1-form whose value on a vector X is the directional derivative X(f). Climbing up, on a general k-form written in coordinates as omega = a_I dx^I (summing over increasing multi-indices I), you differentiate the coefficient and wedge: d omega = (d a_I) ^ dx^I. So d acts on the only piece that is not already a constant form — the coefficient functions — and the wedge bumps the degree up by one. That single rule computes everything.
The four axioms that pin d down uniquely
You should not memorize the coordinate formula; instead remember the four properties that characterize d completely. (1) Linearity: d is R-linear. (2) On a function, d f is the ordinary differential. (3) A graded Leibniz rule: d(alpha ^ beta) = (d alpha) ^ beta + (-1)^p alpha ^ (d beta) when alpha is a p-form — the sign (-1)^p is the cost of sliding d past p one-forms, each of which anticommutes. (4) The headline: d^2 = 0, applying d twice gives zero. Any operator satisfying these is forced to be the exterior derivative; the coordinate formula above is just the unique solution.
Where does d^2 = 0 come from? Trace it to one fact you have known since multivariable calculus: mixed partials commute. For a function f, d f = (partial_i f) dx^i, and applying d again gives d(d f) = (partial_j partial_i f) dx^j ^ dx^i. The coefficient partial_j partial_i f is symmetric in i and j (Clairaut), but dx^j ^ dx^i is antisymmetric — and a symmetric thing summed against an antisymmetric thing is zero, term by term. So the equality of mixed partials is exactly the statement d^2 = 0 dressed in the language of forms. Every deeper appearance of d^2 = 0 traces back to this cancellation.
d secretly is grad, curl, and div
On R^3 the exterior derivative is not a new operation at all — it is the three vector-calculus operators wearing one coat. A 0-form is a function f, and d f packages exactly the gradient. A 1-form corresponds to a vector field, and its d packages the curl. A 2-form also corresponds to a vector field, and its d packages the divergence (times the volume form). The whole tower grad, curl, div is the single sequence f -> d f -> d(d f) -> ... read across degrees 0, 1, 2, 3.
degree: 0 --d--> 1 --d--> 2 --d--> 3 object: f A (1-form) B (2-form) C (3-form) d is: grad curl div d^2 = 0: curl(grad f) = 0 and div(curl A) = 0
Now read d^2 = 0 in this dictionary and it stops being abstract: curl(grad f) = 0 and div(curl A) = 0 are precisely the two identities d(d f) = 0 and d(d A) = 0. You proved these in vector calculus by grinding through components; here they fall out of one structural fact about forms, uniformly and in every dimension. This is the first real payoff of the formalism — three operators, two pet identities, and the awkward dependence on R^3 all collapse into a single d that lives on any manifold of any dimension.
Closed versus exact: the gap that becomes cohomology
Two words organize everything. A form omega is closed if d omega = 0, and exact if omega = d eta for some form eta one degree lower. Because d^2 = 0, every exact form is automatically closed: if omega = d eta then d omega = d(d eta) = 0. The deep question runs the other way — is every closed form exact? These are the closed and exact forms, and the gap between 'closed' and 'exact' is not a defect; it is information. It will become de Rham cohomology, the quotient H^k(M) = (closed k-forms) / (exact k-forms), and that quotient measures the holes of M.
The cleanest example lives on the punctured plane R^2 minus the origin. Consider the angle form omega = (x dy - y dx)/(x^2 + y^2). A direct computation gives d omega = 0, so it is closed. Yet it is not exact: it would have to be d(theta) for a global angle function theta, but theta jumps by 2*pi as you loop around the origin, so no single-valued theta exists. Integrate omega once around the puncture and you get 2*pi, not 0 — and an exact form always integrates to 0 over a loop. That stubborn 2*pi is the hole in the plane, detected by a closed-but-not-exact form.
The Poincaré lemma: locally, closed does mean exact
The angle form failed to be exact only because the plane had a hole. Remove the hole and the obstruction vanishes. The Poincaré lemma states it crisply: on a region that is star-shaped (or more generally smoothly contractible), every closed form of positive degree is exact. So on a ball, a cube, or all of R^n, closed and exact coincide for k >= 1, and H^k of such a region is zero. Locally there are no holes, so locally every closed form is the d of something — the failure of exactness is a purely global, topological phenomenon.
The proof is constructive and worth seeing in outline, because it actually hands you the antiderivative eta. On a star-shaped region one builds a homotopy operator K, a map lowering degree by one, satisfying the identity d K + K d = identity (on forms of positive degree). Read that identity on a closed form omega, where d omega = 0: the second term dies, leaving omega = d(K omega). So eta = K omega is your explicit primitive, manufactured by integrating omega radially along the straight lines from the star center. The operator K is built by pulling back along the contracting homotopy and integrating out the radial variable.
Two honest caveats. First, 'star-shaped' is a real hypothesis, not decoration: the lemma is false on the punctured plane, as the angle form proved, so never quote it without checking the domain has no holes. Second, the same homotopy identity d K + K d = identity reappears, with a different K, as the heart of the Cartan magic formula for Lie derivatives — the algebraic shape 'd of something plus something of d equals identity (or a Lie derivative)' is a recurring motif you will meet again when forms start to flow. The Poincaré lemma is your first and friendliest instance of it.