{
  "schema_version": 1,
  "kind": "ConceptView",
  "producer": "concept-view-1",
  "concept_id": "urn:uuid:f871eb88-c0a5-4b52-8d60-f5daef5ddef1",
  "labels": {
    "zh-CN": "余切空间",
    "en": "cotangent space"
  },
  "alternative_labels": [],
  "scope_note": "",
  "identity_status": "active",
  "identity_revision": "2b8437f4656cf2fe2b34f1b7006faaf4f8d0749ccefaeb52307564d34b7920f4",
  "occurrences": [
    {
      "concept_id": "urn:uuid:f871eb88-c0a5-4b52-8d60-f5daef5ddef1",
      "book_id": "szekeres",
      "edition_id": "source-snapshot-bc3404156f6e",
      "lang": "zh-CN",
      "route": "zh/chapter-15/03-tangent-cotangent-and-tensor-spaces",
      "anchor": "concept-15-cotangent-space",
      "chapter": "15",
      "section": "15.3",
      "offset": 11971,
      "range": [
        18814,
        19051
      ],
      "source_revision": "42521000e4d5202ad14e779d03d39e10558dce1e3f583880e73256da77dce50d",
      "source_category": "translation",
      "review_status": "unreviewed",
      "role": "occurrence",
      "role_basis": {
        "kind": "explicit-source-uuid-marker"
      },
      "label": "余切空间(cotangent space)",
      "context": {
        "range": [
          18750,
          19522
        ],
        "quote": "与 $p \\in M$ 处的切空间相关联的对偶空间 $T _ { p } ^ { * } ( M )$ 称为 $p .$ 处的 <u id=\"concept-15-cotangent-space\" class=\"concept-term\" data-concept-identity=\"urn:uuid:f871eb88-c0a5-4b52-8d60-f5daef5ddef1\" data-concept-route=\"/concept-identities#concept-f871eb88-c0a5-4b52-8d60-f5daef5ddef1\">余切空间(cotangent space)</u>。它由 $T _ { p } ( M )$ 上的所有线性泛函组成，也称为 $p .$ 处的余向量或 1-形式。$p$ 处的余向量 $\\omega _ { p }$ 对切向量 $X _ { p }$ 的作用将记为 $\\omega _ { p } ( X _ { p } ) \\langle \\omega _ { p } , X _ { p } \\rangle$ 或 $\\langle X _ { p } , \\omega _ { p } \\rangle$。由<a href=\"../chapter-03/07-dual-spaces#section-3-7\" class=\"cross-reference\" id=\"xref-03-tangent-cotangent-and-tensor-spaces-11\" data-reference=\"section-3-7\">第 3.7 节</a>可知，$\\dim$ $T _ { p } ^ { * } ( M ) = n = \\dim T _ { p } ( M ) =$ $\\dim$ $M$。",
        "quote_sha256": "6683c3936efe7fe5bda36c601d1646792e4e3fce5cf4d2b9eca555a8ae5fa6be",
        "environment": null,
        "excerpt_omitted": false
      },
      "parallel_readings": [
        {
          "route": "en/chapter-15/03-tangent-cotangent-and-tensor-spaces",
          "lang": "en",
          "source_category": "original_transcription",
          "basis": "registered-book-edition-section-language-context; not-concept-equivalence"
        }
      ],
      "id": "occ-76d047e97883449367702610"
    },
    {
      "concept_id": "urn:uuid:f871eb88-c0a5-4b52-8d60-f5daef5ddef1",
      "book_id": "szekeres",
      "edition_id": "source-snapshot-bc3404156f6e",
      "lang": "en",
      "route": "en/chapter-15/03-tangent-cotangent-and-tensor-spaces",
      "anchor": "first-469b2fbb724b75494d6c-en",
      "chapter": "15",
      "section": "15.3",
      "offset": 15045,
      "range": [
        21373,
        21607
      ],
      "source_revision": "8b0278d73f794378de0c8f3aa32c9c14db7ce60b01ab8b11faed63d6d00fda5f",
      "source_category": "original_transcription",
      "review_status": "unreviewed",
      "role": "occurrence",
      "role_basis": {
        "kind": "explicit-source-uuid-marker"
      },
      "label": "cotangent space",
      "context": {
        "range": [
          21271,
          22211
        ],
        "quote": "The dual space $T _ { p } ^ { * } ( M )$ associated with the tangent space at $p \\in M$ is called the <u class=\"concept-term\" data-concept-identity=\"urn:uuid:f871eb88-c0a5-4b52-8d60-f5daef5ddef1\" data-concept-route=\"/concept-identities#concept-f871eb88-c0a5-4b52-8d60-f5daef5ddef1\" id=\"first-469b2fbb724b75494d6c-en\">cotangent space</u> at $p .$ . It consists of all linear functionals on $T _ { p } ( M )$ , also called covectors or 1-forms at $p .$ . The action of a covector $\\omega _ { p }$ at $p$ on a tangent vector $X _ { p }$ will be denoted by $\\omega _ { p } ( X _ { p } ) \\langle \\omega _ { p } , X _ { p } \\rangle$ or $\\langle X _ { p } , \\omega _ { p } \\rangle$ . From <a href=\"../chapter-03/07-dual-spaces#section-3-7\" class=\"cross-reference\" id=\"xref-03-tangent-cotangent-and-tensor-spaces-11\" data-reference=\"section-3-7\">Section 3.7</a> we have that $\\dim$ $T _ { p } ^ { * } ( M ) = n = \\dim T _ { p } ( M ) =$ $\\dim$ $M$.",
        "quote_sha256": "52426bf51f4c9f222dd960d188c03694bf64d2eec8e5a358e229d7b1a7dc1c67",
        "environment": null,
        "excerpt_omitted": false
      },
      "parallel_readings": [
        {
          "route": "zh/chapter-15/03-tangent-cotangent-and-tensor-spaces",
          "lang": "zh-CN",
          "source_category": "translation",
          "basis": "registered-book-edition-section-language-context; not-concept-equivalence"
        }
      ],
      "id": "occ-7a504bb014b18f1ddfc7fad5"
    }
  ],
  "registered_positions": [
    {
      "route": "en/chapter-15/03-tangent-cotangent-and-tensor-spaces",
      "lang": "en",
      "section": "15.3",
      "signal": "term-mark",
      "label": "cotangent space",
      "english": "cotangent space",
      "offset": 14592
    },
    {
      "route": "zh/chapter-15/03-tangent-cotangent-and-tensor-spaces",
      "lang": "zh-CN",
      "section": "15.3",
      "signal": "term-mark",
      "label": "余切空间",
      "english": "cotangent space",
      "offset": 11588
    }
  ],
  "statements": [],
  "resolved_references": [
    {
      "raw": "href=\"../chapter-03/07-dual-spaces#section-3-7\"",
      "href": "../chapter-03/07-dual-spaces#section-3-7",
      "range": [
        19282,
        19329
      ],
      "relation": "references",
      "target_route": "zh/chapter-03/07-dual-spaces",
      "target_anchor": "section-3-7",
      "resolved": true,
      "reason": null,
      "source_revision": "42521000e4d5202ad14e779d03d39e10558dce1e3f583880e73256da77dce50d",
      "source_route": "zh/chapter-15/03-tangent-cotangent-and-tensor-spaces",
      "occurrence_id": "occ-76d047e97883449367702610"
    },
    {
      "raw": "href=\"../chapter-03/07-dual-spaces#section-3-7\"",
      "href": "../chapter-03/07-dual-spaces#section-3-7",
      "range": [
        21956,
        22003
      ],
      "relation": "references",
      "target_route": "en/chapter-03/07-dual-spaces",
      "target_anchor": "section-3-7",
      "resolved": true,
      "reason": null,
      "source_revision": "8b0278d73f794378de0c8f3aa32c9c14db7ce60b01ab8b11faed63d6d00fda5f",
      "source_route": "en/chapter-15/03-tangent-cotangent-and-tensor-spaces",
      "occurrence_id": "occ-7a504bb014b18f1ddfc7fad5"
    }
  ],
  "reading_order": [
    "occ-76d047e97883449367702610",
    "occ-7a504bb014b18f1ddfc7fad5"
  ],
  "recorded_dependencies": [],
  "reading_paths": [],
  "coverage": {
    "registered_positions": 2,
    "located_occurrences": 2,
    "whole_book_search_complete": false,
    "registered_positions_truncated": false,
    "independent_human_review_implied": false
  },
  "unresolved": [],
  "base_revision": "bb33a9831f6bc4e3c8686ea995431b14701cf8b5178227699b31b18f2226fc37",
  "optional_enrichments": []
}
