Differential Forms

graph TD D1["D1 Exterior algebra Λ^k T*M\nAlternating k-forms on tangent space"] D2["D2 Exterior derivative d\nCartan formula: dω(X,Y) = X ω(Y) − Y ω(X) − ω([X,Y])"] D3["D3 Closed and exact forms\ndω = 0 vs ω = dη"] D4["D4 Integration of n-forms\nOriented manifold, ∫_M ω"] T1["T1 d² = 0\nExterior derivative squared is zero"] T2["T2 Stokes theorem\n∫_M dω = ∫_∂M ω"] T3["T3 Poincaré lemma\nContractible ⇒ closed = exact locally"] T4["T4 de Rham cohomology H^k\nClosed / exact forms; topological invariant"] T5["T5 Hodge decomposition\nΩ^k = im d ⊕ im d* ⊕ harmonic"] D1 --> D2 D2 --> D3 D1 --> D4 D2 --> T1 D2 --> T2 D4 --> T2 D3 --> T3 D3 --> T4 D2 --> T5 T1 --> T4 T2 --> T3 classDef definition fill:#3498db,color:#fff,stroke:#2980b9 classDef theorem fill:#1abc9c,color:#fff,stroke:#16a085 class D1,D2,D3,D4 definition class T1,T2,T3,T4,T5 theorem