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        "quote": "设 $\\mathcal { C } ^ { r } ( M ) = \\Lambda _ { r } ( M )$ 为所有 $r$-形式在 $M$ 上构成的实向量空间。它的元素也称为 $M$ 上的 $r$-<u class=\"concept-term\" data-concept-identity=\"urn:uuid:04adfa20-e5d3-4dd7-910d-d1fc29b4820b\" data-concept-route=\"/concept-identities#concept-04adfa20-e5d3-4dd7-910d-d1fc29b4820b\" id=\"first-66320d0b2eb1c12f93aa-zh\">上链(cochain)</u>。外微分 $d$ 是一个线性算符 $d$：$\\begin{array} { r } { \\begin{array} { r } { \\mathcal { C } ^ { r } ( M ) \\to \\mathcal { C } ^ { r + 1 } ( M ) } \\end{array} } \\end{array}$，对每个 $r = 0 , 1 , \\ldots , n$ 成立，并具有性质 $\\mathrm { d } ^ { 2 } = 0$）。我们将其在 $\\mathcal { C } ^ { r } ( M )$ 上的限制记为 $d$。",
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        "quote": "Let $\\mathcal { C } ^ { r } ( M ) = \\Lambda _ { r } ( M )$ be the real vector space consisting of all differential $r$-forms on $M$. Its elements are also known as $r$-<u class=\"concept-term\" data-concept-identity=\"urn:uuid:04adfa20-e5d3-4dd7-910d-d1fc29b4820b\" data-concept-route=\"/concept-identities#concept-04adfa20-e5d3-4dd7-910d-d1fc29b4820b\" id=\"first-e2e602e31df1891319af-en\">cochains</u> on $M$. The exterior derivative $d$ is a linear operator $d$ : $\\begin{array} { r } { \\begin{array} { r } { \\mathcal { C } ^ { r } ( M ) \\to \\mathcal { C } ^ { r + 1 } ( M ) } \\end{array} } \\end{array}$ for each $r = 0 , 1 , \\ldots , n$ , with the property $\\mathrm { d } ^ { 2 } = 0$ ). We write its restriction to $\\mathcal { C } ^ { r } ( M )$ as $d$ .",
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