{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:7731c967-eca6-43a2-a75d-2070d953614c", "labels": { "zh-CN": "分布", "en": "distribution" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "1a6d00d39336067469be91e6e51e40328f290ec841986cfb94f8274aac3f64ae", "occurrences": [ { "concept_id": "urn:uuid:7731c967-eca6-43a2-a75d-2070d953614c", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-12/01-test-functions-and-distributions", "anchor": "first-89a4a784510541d6735a-zh", "chapter": "12", "section": "12.1", "offset": 6350, "range": [ 10605, 10840 ], "source_revision": "1313c447e739f007a893e8a36396d845ca81d5070f59e96e03fc03f2c0d3bce9", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "分布(distribution)", "context": { "range": [ 10272, 11158 ], "quote": "其连续到 $m$ 阶；也就是说，若 $\\varphi _ { k } \\to \\varphi$ 是 ${ \\mathcal { D } } ( \\mathbb { R } ^ { n } )$ 中任一按 $m$ 阶收敛的函数序列，则 $T ( \\varphi _ { k } ) \\to T ( \\varphi )$ 。${ \\mathcal { D } } ( \\mathbb { R } ^ { n } )$ 上关于 ${ \\mathcal { D } } ( \\mathbb { R } ^ { n } )$ 中按所有阶收敛的序列 $\\varphi _ { i }$ 连续的线性泛函 $T$ ，将简称为 $\\mathbb { R } ^ { n }$ 上的 分布(distribution) 。在这种连续性意义下，$\\mathbb { R } ^ { n }$ 上 $m$ 阶分布的空间是 ${ \\mathcal { D } } ^ { m } ( \\mathbb { R } ^ { n } )$ 的对偶空间（见 10.9 节），而分布的空间是 ${ \\mathcal { D } } ( \\mathbb { R } ^ { n } )$ 的对偶空间。因此，它们分别记作 ${ \\mathcal { D } } ^ { \\prime m } ( \\mathbb { R } ^ { n } )$ ）和 ${ \\mathcal { D } } ^ { \\prime } ( \\mathbb { R } ^ { n } )$ 。", "quote_sha256": "f2687ced78502d1c12ada91a7ff61e1ee054a748da307959781fe82fb4085084", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-12/01-test-functions-and-distributions", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-f8c48276f7d227d8db64ede3" }, { "concept_id": "urn:uuid:7731c967-eca6-43a2-a75d-2070d953614c", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-15/06-distributions-and-frobenius-theorem", "anchor": "first-89a4a784510541d6735a-en", "chapter": "15", "section": "15.6", "offset": 349, "range": [ 5314, 5545 ], "source_revision": "9dc392eaf382187323a5ca1c4e44f8254de3aca53722f1a557d9dcc93bf1b795", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "distribution", "context": { "range": [ 5296, 6262 ], "quote": "A $k$-dimensional distribution $D ^ { k }$ on a manifold $M$ is an assignment of a $k$-dimensional subspace $D ^ { k } ( p )$ of the tangent space $T _ { p } ( M )$ at every point $p \\in M$ . The distribution is said to be $C ^ { \\infty }$ or smooth if for all $p \\in M$ there is an open neighbourhood $U$ and $k$ smooth vector fields $X _ { 1 } , \\ldots , X _ { k }$ on $U$ that span $D ^ { k } ( q )$ at each point $q \\in U$ . A vector field $X$ on an open domain $A$ is said to lie in or belong to the distribution $D ^ { k }$ if $X _ { p } \\in D ^ { k } ( p )$ ) at each point $p \\in A$ . A one-dimensional distribution is equivalent to a vector field up to an arbitrary scalar factor at every point, and is sometimes called a direction field.", "quote_sha256": "079e87f9d0d2a2ab1a854c3f5c450402d6515092bdc5ba2ae04625b6d5c5a323", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "zh/chapter-15/06-distributions-and-frobenius-theorem", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-b8b8d4553ce996faa4b820f4" }, { "concept_id": "urn:uuid:7731c967-eca6-43a2-a75d-2070d953614c", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-15/06-distributions-and-frobenius-theorem", "anchor": "concept-15-distribution", "chapter": "15", "section": "15.6", "offset": 349, "range": [ 5828, 6057 ], "source_revision": "225aab8a17b4738fe4c2c90a0fbd4d1303e79044e70ca9d93a75b916d5200722", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "分布(distribution)", "context": { "range": [ 5810, 6426 ], "quote": "流形 $M$ 上的一个 $k$ 维 分布(distribution) $D ^ { k }$ 是在每一点 $p \\in M$ 处对切空间 $T _ { p } ( M )$ 的一个 $k$ 维子空间 $D ^ { k } ( p )$ 的指定。若对所有的 $p \\in M$，都存在一个开邻域 $U$ 以及 $U$ 上的 $k$ 个光滑向量场 $X _ { 1 } , \\ldots , X _ { k }$，使得它们在每一点 $q \\in U$ 处张成 $D ^ { k } ( q )$，则称该分布是 $C ^ { \\infty }$ 或光滑的。若在每一点 $p \\in A$ 处都有 $X _ { p } \\in D ^ { k } ( p )$，则称开定义域 $A$ 上的向量场 $X$ 位于或属于分布 $D ^ { k }$。一维分布等价于在每一点处相差一个任意标量因子的向量场，有时称为方向场。", "quote_sha256": "8bb320983af6fe8a64a46b30e6224ed9a84e87a66e67d048ee04906d4f3fcada", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-15/06-distributions-and-frobenius-theorem", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-65f881c14eb188f55b6d0004" } ], "registered_positions": [ { "route": "zh/chapter-12/01-test-functions-and-distributions", "lang": "zh-CN", "section": "12.1", "signal": "term-mark", "label": "分布", "english": "distribution", "offset": 5395 }, { "route": "en/chapter-15/06-distributions-and-frobenius-theorem", "lang": "en", "section": "15.6", "signal": "term-mark", "label": "distribution", "english": "distribution", "offset": 349 }, { "route": "zh/chapter-15/06-distributions-and-frobenius-theorem", "lang": "zh-CN", "section": "15.6", "signal": "term-mark", "label": "分布", "english": "distribution", "offset": 349 } ], "statements": [], "resolved_references": [], "reading_order": [ "occ-f8c48276f7d227d8db64ede3", "occ-b8b8d4553ce996faa4b820f4", "occ-65f881c14eb188f55b6d0004" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 3, "located_occurrences": 3, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "a7c1f16783d5893d5dca50e1cbc3db1e289dafbda5744ac871af3d6606ba5a42", "optional_enrichments": [] }