{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:c1d46382-437d-4926-a7fd-d9d0cd36a531", "labels": { "zh-CN": "切空间", "en": "tangent space" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "7d2f642442c19d4a79576bce4563b6a2fc2722151288033031f0410b99e0cbd5", "occurrences": [ { "concept_id": "urn:uuid:c1d46382-437d-4926-a7fd-d9d0cd36a531", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-15/03-tangent-cotangent-and-tensor-spaces", "anchor": "concept-15-tangent-space", "chapter": "15", "section": "15.3", "offset": 3497, "range": [ 10340, 10572 ], "source_revision": "42521000e4d5202ad14e779d03d39e10558dce1e3f583880e73256da77dce50d", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "切空间(tangent space)", "context": { "range": [ 10019, 10627 ], "quote": "是 $p$ 处的一个切向量，因为它满足式 (15.3) 和式 (15.4)。它称为 $p .$ 处的 切空间(tangent space)。若 $( U , \\phi )$ 是 $p$ 处的任一图卡，其坐标函数为 $x ^ { i }$，则定义算符", "quote_sha256": "f913ed5354337dc099c0fc98487c8125aff2108b651aa7efe8104e518ad817dc", "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-b124643a63bd4c79963041b4" }, { "concept_id": "urn:uuid:c1d46382-437d-4926-a7fd-d9d0cd36a531", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-15/03-tangent-cotangent-and-tensor-spaces", "anchor": "first-1fd2c442e5431ea73af8-en", "chapter": "15", "section": "15.3", "offset": 4949, "range": [ 11277, 11509 ], "source_revision": "8b0278d73f794378de0c8f3aa32c9c14db7ce60b01ab8b11faed63d6d00fda5f", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "tangent space", "context": { "range": [ 10917, 11619 ], "quote": "is a tangent vector at $p$ since it satisfies Eq. (15.3) and Eq. (15.4). It is called the tangent space at $p .$ . If $( U , \\phi )$ is any chart at $p$ with coordinate functions $x ^ { i }$ , define the operators", "quote_sha256": "11d356b768a7970ea3666f2358be0041de2ec918b98b54e4258b4042e69abdb2", "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-06d42b38b281ac3d4a5a6880" }, { "concept_id": "urn:uuid:c1d46382-437d-4926-a7fd-d9d0cd36a531", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-03/02-commutation-relations-for-pauli-matrices", "anchor": "first-c1cbeb5580362deafa51-zh", "chapter": "3", "section": "3.2", "offset": 3472, "range": [ 12130, 12398 ], "source_revision": "59260a454f4e5877261b20806d461df669f06f442a99b7596955f4ec200babfb", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "切空间(tangent space)", "context": { "range": [ 12038, 12447 ], "quote": "$X_j$ 及其对易关系可以看作完整的群及其群乘法则在单位元附近的无穷小形式。从流形几何来看，回忆 $SU(2)$ 作为空间就是 $S^3$。$X_j$ 构成 $SU(2)$ 单位元处切空间(tangent space) $\\mathbf R^3$ 的一组基，正如 $i$ 构成 $U(1)$ 单位元处切空间的一组基。", "quote_sha256": "aa3d7b7edaf492750ec62d4ddd2ea43a711f51c18366d59b77ec9ff266a7d4bd", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-03/02-commutation-relations-for-pauli-matrices", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-fd9ef01a114a25e0f1d25677" }, { "concept_id": "urn:uuid:c1d46382-437d-4926-a7fd-d9d0cd36a531", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-03/02-commutation-relations-for-pauli-matrices", "anchor": "first-c1cbeb5580362deafa51-en", "chapter": "3", "section": "3.2", "offset": 4594, "range": [ 11833, 12138 ], "source_revision": "d94860156429173a676b4ffcdc432f70d9a5648fad559409281d5b9cb75de2ec", "source_category": "original_transcription", "review_status": "ai-transcription-compared", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "tangent space", "context": { "range": [ 11511, 12253 ], "quote": "The $X_j$ and their commutation relations can be thought of as an infinitesimal version of the full group and its group multiplication law near the identity. 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