de Rham Cohomology

Maxwell's equations, Stokes's theorem, Green's theorem - all are special cases of a single formula: ∫_M dω = ∫_{∂M} ω. de Rham cohomology is the algebra measuring global obstructions to solving ω = dη. Put another way: differential equations on manifolds, governed by topology.

• **Electromagnetism:** Maxwell equations = dF=0, d*F=J; the potential A exists iff H¹=0 (no holes in spacetime)
• **Computer graphics:** exterior calculus on surfaces powers discrete differential geometry: fluid simulation, mesh smoothing, UV parametrization
• **Gauge theory:** particle physics = differential forms on fiber bundles; H¹ is the space of gauge fields

Предварительные знания

• Manifolds

Differential Forms

A **0-form** on a manifold M is a smooth function f: M → R. A **1-form** integrates along curves: at each point p it is a linear functional on the tangent space T_p M. In coordinates: ω = f₁dx₁ + f₂dx₂ + ··· + fₙdxₙ, where the fᵢ are smooth functions.

A **k-form** is a totally antisymmetric k-linear map on T_p M: it takes k tangent vectors and returns a number. In coordinates: ω = Σ_{i₁<···<iₖ} f_{i₁···iₖ} dxᵢ₁ ∧ ··· ∧ dxᵢₖ. The number of independent k-forms on an n-dimensional space is C(n,k).

Forms are the 'correct' objects to integrate: they transform properly under coordinate changes. The integral of a k-form over a k-dimensional surface is coordinate-independent. This is the foundation for defining integrals on manifolds without reference to specific coordinates.

The Exterior Derivative

The **exterior derivative** d: Ω^k(M) → Ω^{k+1}(M) is the unique operator satisfying: 1. on 0-forms, d is the differential df 2. d(ω ∧ η) = dω ∧ η + (−1)^k ω ∧ dη (Leibniz rule) 3. **d ∘ d = 0**. The last property - the boundary of a boundary is zero - is fundamental.

**Maxwell's equations as forms:** In 4D spacetime, all Maxwell equations reduce to dF = 0 (Faraday + Gauss laws) and d*F = J (Ampere + Gauss laws), where F is the electromagnetic 2-form and *F is its Hodge dual. One symbol d replaces all four Maxwell equations.

de Rham Cohomology

Since d ∘ d = 0, every exact form is closed: im(d: Ω^{k-1} → Ω^k) ⊆ ker(d: Ω^k → Ω^{k+1}). A **closed** k-form satisfies dω = 0. An **exact** k-form satisfies ω = dη for some (k−1)-form η. The **de Rham cohomology** is H^k_dR(M) = closed k-forms / exact k-forms.

The de Rham Theorem

**de Rham's theorem:** For a smooth manifold M, the de Rham cohomology is isomorphic to singular cohomology with real coefficients: H^k_dR(M) ≅ H^k(M; R). In particular, dim H^k_dR(M) = βₖ (the k-th Betti number). This links differential geometry to algebraic topology.

**Physical applications:** 1. Maxwell's equations: dF=0 (Faraday + Gauss for B) and d*F=J (Ampere + Gauss for E). 2. Electromagnetic potential A: F=dA, so dF=d²A=0 automatically. 3. Generalized Stokes theorem: ∫_M dω = ∫_{∂M} ω - unifies Green's, Gauss's, and Stokes's theorems.

Key Ideas

• **k-forms:** antisymmetric k-linear functions on tangent spaces; the correct objects to integrate
• **Exterior derivative d:** d∘d=0; generalizes grad, curl, div; Stokes: ∫dω = ∫_{boundary} ω
• **de Rham cohomology:** H^k_dR = closed / exact; measures k-dimensional holes via forms
• **de Rham theorem:** H^k_dR(M) ≅ H^k(M;R); dim = βₖ; bridge between geometry and topology

Related Topics

de Rham cohomology completes algebraic topology and opens the door to differential geometry:

• Manifolds — Differential forms are only defined on smooth manifolds; the smooth structure is essential
• Topological Data Analysis — Discrete exterior calculus (the discrete analogue of de Rham) is used in computational geometry and TDA

Вопросы для размышления

• Maxwell's equation dF=0 says F is closed. Under what topological condition on spacetime does a potential A with F=dA exist? What happens when H²≠0?
• The generalized Stokes theorem ∫_M dω = ∫_{∂M} ω unifies Green's theorem, the divergence theorem, and classical Stokes. Write each as a special case.
• The Hodge star operator *: Ω^k → Ω^{n-k} links H^k_dR and H^{n-k}_dR. What is Poincare duality, and why does it require orientability?

Связанные уроки

• calc-24-line-integrals