SYSTEMATIC MATHEMATICS

Differential Forms, de Rham Cohomology & Hodge Theory

Why does curl automatically have zero divergence, when does a closed form have a potential, how can integration detect a hole that no local coordinate chart sees, and why does every real cohomology class on a closed Riemannian manifold have one canonical harmonic representative? This course answers those questions in sixty-one visible bilingual chapters across eleven content-sized units of lengths 7, 5, 6, 4, 7, 5, 6, 5, 7, 4, and 5. It develops alternating tensors, wedge, pullback, contraction, exterior differentiation, Lie derivatives, integration and Stokes, homotopy operators and the Poincaré lemma, de Rham cohomology computations, Mayer–Vietoris and good covers, the de Rham comparison theorem, compact support and duality, Riemannian form inner products, Hodge star, codifferential, Hodge Laplacian, ellipticity, Green operators, Hodge decomposition, harmonic representatives, boundary conditions, noncompact failure modes, and complete circle, sphere, annulus, and flat-torus dossiers. Every manifold, dimension, form degree, orientation, metric, coefficient field, support, boundary condition, operator domain, compactness assumption, local/global distinction, and theorem boundary is named. The course proves the finite-dimensional differential-form and cohomological results it uses and gives a structured proof of the compact Hodge theorem; it does not claim to replace full courses in elliptic regularity, pseudodifferential operators, geometric measure theory, L² cohomology on singular or noncompact spaces, complex Hodge theory, Kähler geometry, index theory, gauge theory, or research geometric analysis.

Before this course: Completed Differential Geometry & Manifolds, Algebraic Topology, Differential Topology, Vector Bundles & Morse Theory, Partial Differential Equations, Functional Analysis, and Measure & Lebesgue Integration. Students should know smooth manifolds, tangent and cotangent bundles, orientations, partitions of unity, Stokes’ theorem, vector bundles, singular cohomology, exact sequences, Hilbert spaces, unbounded self-adjoint operators, weak derivatives, Sobolev-space motivation, and compactness arguments. No prior full de Rham theorem, Hodge star, codifferential, Hodge Laplacian, elliptic complex, Green operator, Hodge decomposition, or boundary Hodge theory is assumed.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

Alternating covariant tensors measure oriented k-dimensional input

Objective: Why is ω(v,w)=0 not enough to conclude that v and w are linearly dependent in R³?

A one-form eats one vector and returns a scalar. A k-form at one point eats k tangent vectors. Linearity in each slot makes the measurement compatible with vector addition and scaling; alternation forces the result to vanish when the inputs lose k-dimensional independence. Before manipulating symbols, name the manifold, its dimension, the degree of every form, the tangent or cotangent space where each input lives, and every orientation, metric, compactness, boundary, or support hypothesis. The exact conclusion is: If dim V=n, then dim Λ^k(V^*)=binomial(n,k) for 0≤k≤n and Λ^k(V^*)={0} for k>n. An alternating k-form vanishes on every linearly dependent k-tuple.

The new object means: For a finite-dimensional vector space V, an alternating covariant k-tensor is a multilinear map ω:V^k→R satisfying ω(v_{σ(1)},…,v_{σ(k)})=sign(σ)ω(v_1,…,v_k) for every permutation σ. The space of these maps is Λ^k(V^*). Set Λ^0(V^*)=R. A differential form is not a decorative string of dx symbols: it is an alternating multilinear measurement with a declared point, degree, and transformation law.

Rebuild the worked problem “On V=R³, let ω((a_1,a_2,a_3),(b_1,b_2,b_3))=a_1b_2−a_2b_1. Evaluate ω on v=(1,2,3), w=(4,5,6), verify alternation, and identify its kernel behavior.” from the definition, including every sign and degree. Keep this failure boundary visible: Alternating does not mean merely antisymmetric in one selected pair; it controls every permutation. A k-form is covariant: vectors are inputs and scalars are outputs. When k>dim V the only alternating form is zero, so a formal dx_1∧⋯∧dx_k cannot be nonzero in too small a space.

If dim V=n, then dim Λ^k(V^*)=binomial(n,k) for 0≤k≤n and Λ^k(V^*)={0} for k>n. An alternating k-form vanishes on every linearly dependent k-tuple.

Choose a basis e_1,…,e_n with dual basis e^1,…,e^n. Alternation says a value is determined by increasing index lists i_1<⋯<i_k.

There are binomial(n,k) such index lists, and the wedge basis e^{i_1}∧⋯∧e^{i_k} is linearly independent, proving the dimension formula.

If v_1,…,v_k are dependent, write one as a combination of the others. Multilinearity expands the value into terms with repeated inputs, and alternation makes each repeated-input term zero.

On V=R³, let ω((a_1,a_2,a_3),(b_1,b_2,b_3))=a_1b_2−a_2b_1. Evaluate ω on v=(1,2,3), w=(4,5,6), verify alternation, and identify its kernel behavior.

  1. Substitute directly: ω(v,w)=1·5−2·4=−3.
  2. Swapping inputs gives ω(w,v)=4·2−5·1=3=−ω(v,w), and ω(v,v)=1·2−2·1=0.
  3. The third coordinates never enter. Any pair whose projections to the first two coordinates are dependent gives zero; dependence in all of R³ also forces zero.

Result: ω(v,w)=−3, ω(w,v)=3, and repeated or projected-dependent inputs give zero; in dual notation ω=e^1∧e^2.