{
"schema_version": 1,
"kind": "ConceptView",
"producer": "concept-view-1",
"concept_id": "urn:uuid:168b04d1-9b02-428a-b637-0ce8914e7384",
"labels": {
"zh-CN": "微分形式",
"en": "differential forms"
},
"alternative_labels": [],
"scope_note": "",
"identity_status": "active",
"identity_revision": "019ebd532d88baa786fde177f62e9a878a314fa12fe9fc199b4557b00dad537d",
"occurrences": [
{
"concept_id": "urn:uuid:168b04d1-9b02-428a-b637-0ce8914e7384",
"book_id": "woit-qtgr",
"edition_id": "2025-10-20",
"lang": "zh-CN",
"route": "books/woit-qtgr/2025-10-20/zh/chapter-30/01-the-grassmann-algebra-of-polynomials-on-anticommuting-generators",
"anchor": "first-370d8aecf498723703b7-zh",
"chapter": "30",
"section": "30.1",
"offset": 1632,
"range": [
8703,
9020
],
"source_revision": "4a1079350069bab75a696253558e58679b40e35740fe65e8ee7c3f6ea1446698",
"source_category": "machine_draft",
"review_status": "machine-draft-not-reviewed",
"role": "occurrence",
"role_basis": {
"kind": "explicit-source-uuid-marker"
},
"label": "微分形式(differential forms)",
"context": {
"range": [
8650,
9417
],
"quote": "\n> [!theorem] 拓展 · 微分形式\n>\n> *读者也许已在 $\\mathbb R^n$ 上的[微分形式(differential forms)](../../../../../concepts/differential-geometry/differential-form)中见过Grassmann代数。物理学家把微分形式称为“反对称张量场”：取外代数 $\\Lambda^*(\\mathbb R^n)$ 的元素，使系数由常数变为 $\\mathbb R^n$ 上的函数，就得到这一构造。它在流形理论中很重要。流形 $M$ 的点 $x$ 处有切空间 $T_xM$ 及其对偶 $(T_xM)^*$；局部坐标 $x_j$ 给出 $(T_xM)^*$ 的基元素 $dx_j$。微分形式在局部可写为以下各项之和：*\n>\n> $$\n> f(x_1,x_2,\\ldots,x_n)\\,dx_j\\wedge\\cdots\\wedge dx_k\\wedge\\cdots\\wedge dx_l\n> $$\n>\n> *其中 $1\\le j [!theorem] Digression · Differential forms\n>\n> *Readers may have seen the Grassmann algebra in the context of [differential forms](../../../../../concepts/differential-geometry/differential-form) on $\\mathbb R^n$. Physicists call these “antisymmetric tensor fields”. They are elements of $\\Lambda^*(\\mathbb R^n)$ with coefficients that are functions on $\\mathbb R^n$, rather than constants. In manifold theory, a point $x$ of a manifold $M$ has a tangent space $T_xM$ and its dual $(T_xM)^*$. Local coordinates $x_j$ on $M$ give basis elements $dx_j$ of $(T_xM)^*$, and differential forms are locally sums of terms of the form*\n>\n> $$\n> f(x_1,x_2,\\ldots,x_n)\\,dx_j\\wedge\\cdots\\wedge dx_k\\wedge\\cdots\\wedge dx_l\n> $$\n>\n> *where $1\\le j