Cartan's magic formula
/ kar-TAHN /
Cartan's magic formula is a single short equation tying together the three fundamental operations on differential forms: the exterior derivative d (which raises degree), the interior product iota_X (which lowers degree by contracting with a vector field X), and the Lie derivative L_X (which measures how a form changes as you flow along X). The 'magic' is that the Lie derivative — a dynamical, flow-based notion — is rebuilt purely from the algebraic operations d and iota_X.
The formula reads L_X = d iota_X + iota_X d, often written L_X = d circ iota_X + iota_X circ d, valid on forms of every degree. Read it as: to see how omega changes along the flow of X, contract then differentiate, plus differentiate then contract. Here L_X omega is defined as the derivative at time zero of the pullback of omega by the flow of X, capturing the infinitesimal effect of dragging omega along X. The formula lets you compute Lie derivatives of forms with pure algebra, never needing to construct the flow explicitly. As an immediate corollary, L_X commutes with d (since d L_X = d iota_X d = L_X d), and on functions f it reduces to L_X f = iota_X df = df(X) = X(f), the directional derivative.
Its power is conceptual and computational. It is the engine behind the Poincaré lemma (the homotopy operator that proves closed implies locally exact is built from iota_X), behind Moser's trick in symplectic geometry (deforming forms along a flow), and behind countless invariance arguments: if L_X omega = 0 then omega is preserved by the flow of X, a conservation law. A frequent confusion: the formula is exact and needs no approximation — it is an identity, not a first-order expansion. And it holds for the Lie derivative of FORMS; the Lie derivative of a vector field Y is the bracket [X, Y], a different though related story.
Take omega = x dy on R^2 and X = partial_x. Then iota_X omega = iota_X(x dy) = 0 (no dx to delete), so d iota_X omega = 0. And d omega = dx ^ dy, so iota_X d omega = iota_{partial_x}(dx^dy) = dy. Adding: L_X omega = 0 + dy = dy. Check directly: the flow of partial_x is (x, y) -> (x + t, y), pulling back x dy to (x + t) dy, whose t-derivative at t = 0 is dy. The algebra and the flow agree.
Cartan's formula computes a Lie derivative of a form with two algebraic steps, matching the flow computation exactly.
Sometimes called the Cartan homotopy formula because d iota_X + iota_X d is the prototype of a chain homotopy. Do not confuse the Lie derivative L_X of a form with the covariant derivative nabla_X: L_X needs no connection (it uses the flow), while nabla_X needs one. They generally differ.