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<!DOCTYPE html>
<html lang="en">
<head>
    <meta charset="UTF-8">
    <meta name="viewport" content="width=device-width, initial-scale=1.0">
    <title>Differential Forms - Mathematics Process</title>
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</head>
<body>
    <div class="container">
        <div class="header">
            <h1>Differential Forms</h1>
            <div class="header-meta">
                <span class="meta-item">Differential Geometry</span>
                <span class="meta-item">Mathematics</span>
            </div>
        </div>
        <div class="nav-links">
            <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>
        </div>
        <div class="content">
            <div class="description">
                <h2>Description</h2>
                <p>Exterior algebra, differential forms, exterior derivative, and integration. Stokes theorem and de Rham cohomology.</p>
            </div>
            <div class="flowchart-container">
                <h2>Dependency Flowchart</h2>
                <div class="mermaid">
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
                </div>
            </div>
            <div class="color-legend">
                <h3>Color Scheme</h3>
                <div class="color-grid">
                    <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>
                </div>
            </div>
            <div class="info-card">
                <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>
                </ul>
            </div>
        </div>
    </div>
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