Manifolds 49 | Cartan Derivatives

Exterior (Cartan) derivatives are the unique way to differentiate differential forms on a smooth manifold so that three core rules hold: they extend the usual differential of functions, they satisfy the “complex” property d∘d = 0, and they obey a graded product rule with the wedge product. This matters because those properties are exactly what later power generalized Stokes-type integration theorems on manifolds.

On an n-dimensional manifold M, differential forms come in degrees: 0-forms are smooth functions, 1-forms act on tangent vectors, and k-forms are alternating multilinear objects. The construction starts with the familiar map f ↦ df for functions f, where df is a 1-form. From there, the goal is to extend the operator d so it maps k-forms to (k+1)-forms for every k up to n. The key theorem asserted in the transcript guarantees existence and uniqueness of such an extension: there is one and only one sequence of linear maps d_k: Ω^k(M) → Ω^{k+1}(M) that (1) agrees with the ordinary differential on 0-forms, (2) satisfies d_{k+1}∘d_k = 0 for all k (the “complex property”), and (3) follows the graded Leibniz rule for wedge products.

The graded Leibniz rule takes the form d(ω ∧ η) = dω ∧ η + (−1)^r ω ∧ dη, where ω is an r-form. The sign is not cosmetic: it reflects the alternating nature of forms and ensures consistency across degrees. The transcript also notes that the resulting chain of spaces and maps is known as the de Rham complex of M, and that the complex property is the structural reason Stokes’s theorem can be formulated in this generalized setting.

To justify the construction, the transcript sketches how the exterior derivative can be defined locally in a coordinate chart. In a chart (U, h) with h: U → ℝ^n and inverse φ, any k-form ω can be written as a finite sum of component functions times wedge products of basic 1-forms. Those basic 1-forms are tied to the chart: if dx^j is paired with the tangent vector ∂/∂x^k (represented via the chart’s differential), it returns the Kronecker delta δ^j_k. Equivalently, dx^j corresponds to the j-th component of d(h).

The local definition is then engineered to satisfy d∘d = 0 and the graded product rule. By applying linearity and the product rule repeatedly, the calculation reduces to differentiating the component functions and wedging the remaining 1-forms. Using the chart, each component function becomes a composition of maps into ℝ, so its differential is computed via the chain rule and expressed through the Jacobian. Concretely, the derivative of a component function produces a sum of partial derivatives with respect to the ℝ^n coordinates, multiplied by the corresponding basic 1-forms dx^p.

The upshot is a practical formula: for any k-form written in local coordinates, d_k ω is obtained by replacing the differential of each component function with its coordinate expression (involving partial derivatives) and then summing over all wedge-product placements. The transcript closes by emphasizing that this local coordinate formula is the starting point for deeper development of exterior derivatives in later installments, which ultimately feed into integration theorems.