{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:2c8de24d-b302-4e88-9c0f-db37c70e2f7d", "labels": { "zh-CN": "外法向规则", "en": "outward normal rule" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "f0a87cdb3215192de54ba0c27a26a8e6a2811dae2bef3ac77fbd4ce5e202c110", "occurrences": [ { "concept_id": "urn:uuid:2c8de24d-b302-4e88-9c0f-db37c70e2f7d", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-17/03-stokes-theorem", "anchor": "first-85d5bb855bb031797b63-zh", "chapter": "17", "section": "17.3", "offset": 2890, "range": [ 7155, 7400 ], "source_revision": "8106886fa25dc64b6ef3cd88e17b299a5dc48af6933e24a846386f26821d7ae2", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "外法向规则(outward normal rule)", "context": { "range": [ 6427, 7463 ], "quote": "这些映射可分别视为 $\\Gamma _ { k }$ 的 ($i$, 0)-面与 ($i$, 1)-面。若标准 $k$-立方体 $\\Gamma _ { k }$ 的内部按自然方式定向，即指定 $k$-形式 $\\mathrm { d } x ^ { 1 } \\wedge \\cdots \\wedge \\mathrm { d } x ^ { k }$ 在 $\\Gamma _ { k }$ 上为正定向，则对每个面映射 $x ^ { i } = 0$ 或 1，$\\Gamma _ { k - 1 }$ 上的定向按如下规则指定：令 $k$-形式 $\\mathrm { d } x ^ { i } \\wedge \\mathrm { d } x ^ { 1 } \\wedge \\dots \\wedge \\mathrm { d } x ^ { i - 1 } \\wedge \\mathrm { d } x ^ { i + 1 } \\wedge \\dots \\wedge \\mathrm { d } x ^ { k }$ 为正定向，若 ${ \\mathrm { i f } } x ^ { i }$ 在该面处向外递增，否则为负定向。按此规则，$(k-1)$-形式 $\\mathrm { d } \\boldsymbol { y } ^ { 1 } \\wedge \\mathrm { d } y ^ { 2 } \\wedge \\dots \\wedge \\mathrm { d } y ^ { k - 1 }$ 在 ($i$, 0)-面上具有定向 $( - 1 ) ^ { i }$，而在 ($i$, 1)-面上具有定向 $( - 1 ) ^ { i + 1 }$。这有时称为 外法向规则(outward normal rule)——边界曲面上的定向必须如此选取，使得在每一点处都存在局部正定向坐标，使第一个坐标 $x ^ { 1 }$ 从该曲面指向外侧。", "quote_sha256": "d5ea7cf0b566276b9b48ac2fd8a71ce313ee8dd71a8c9607a5af5818e9156d82", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-17/03-stokes-theorem", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-47b1fac5320b2d2bc8adbb57" }, { "concept_id": "urn:uuid:2c8de24d-b302-4e88-9c0f-db37c70e2f7d", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-17/03-stokes-theorem", "anchor": "first-85d5bb855bb031797b63-en", "chapter": "17", "section": "17.3", "offset": 3731, "range": [ 7524, 7762 ], "source_revision": "89e328aa625ac31a097a870b7d570f879bf72c427e5a34ed8eac8460efde855a", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "outward normal rule", "context": { "range": [ 6412, 7971 ], "quote": "These maps can be thought of as the ($i$, 0)-face and ($i$, 1)-face respectively of $\\Gamma _ { k }$ . If the interior of the standard $k$-cube $\\Gamma _ { k }$ is oriented in the natural way, by assigning the $k$-form $\\mathrm { d } x ^ { 1 } \\wedge \\cdots \\wedge \\mathrm { d } x ^ { k }$ to be positively oriented over $\\Gamma _ { k }$ , then for each face map $x ^ { i } = 0$ or 1 the orientation on $\\Gamma _ { k - 1 }$ is assigned according to the following rule: set the $k$-form $\\mathrm { d } x ^ { i } \\wedge \\mathrm { d } x ^ { 1 } \\wedge \\dots \\wedge \\mathrm { d } x ^ { i - 1 } \\wedge \\mathrm { d } x ^ { i + 1 } \\wedge \\dots \\wedge \\mathrm { d } x ^ { k }$ to be positively oriented ${ \\mathrm { i f } } x ^ { i }$ is increasing outwards at the face, else it is negatively oriented. According to this rule the $(k-1)$-form $\\mathrm { d } \\boldsymbol { y } ^ { 1 } \\wedge \\mathrm { d } y ^ { 2 } \\wedge \\dots \\wedge \\mathrm { d } y ^ { k - 1 }$ has orientation $( - 1 ) ^ { i }$ on the ($i$, 0)-face, while on the ($i$, 1)-face it has orientation $( - 1 ) ^ { i + 1 }$ . 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