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| <h1>Differential Forms</h1> | |
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| <span class="meta-item">Differential Geometry</span> | |
| <span class="meta-item">Mathematics</span> | |
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| <a href="../../mathematics_index.html">← Back to Mathematics Index</a> | |
| <a href="differential_geometry-index.html">Differential Geometry Index</a> | |
| <a href="https://arxiv.org/list/math.DG/recent" target="_blank">arXiv: Differential Geometry (math.DG)</a> | |
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| <h2>Description</h2> | |
| <p>Exterior algebra, differential forms, exterior derivative, and integration. Stokes theorem and de Rham cohomology.</p> | |
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| <h2>Dependency Flowchart</h2> | |
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| 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 | |
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| <h3>Color Scheme</h3> | |
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| <div class="color-item"><div class="color-box" style="background:#3498db"></div><div><strong>Blue</strong><br><small>Definitions (D1–D4)</small></div></div> | |
| <div class="color-item"><div class="color-box" style="background:#1abc9c"></div><div><strong>Teal</strong><br><small>Theorems (T1–T5)</small></div></div> | |
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| <h3>Info</h3> | |
| <ul> | |
| <li><strong>Subcategory:</strong> differential_geometry</li> | |
| <li><strong>Keywords:</strong> differential form, exterior derivative, Stokes, de Rham, Hodge</li> | |
| <li><strong>Research frontier:</strong> <a href="https://arxiv.org/list/math.DG/recent" target="_blank">arXiv math.DG</a></li> | |
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