{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:ac724144-5a08-407a-a27a-796eb367a161", "labels": { "zh-CN": "度量张量", "en": "metric tensor" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "1b04c1f2d9bc76da010e7fb261e89c5fc80a9eca107f562a2ca3039b59481de0", "occurrences": [ { "concept_id": "urn:uuid:ac724144-5a08-407a-a27a-796eb367a161", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-07/02-multilinear-maps-and-tensors", "anchor": "first-52084e4e3766521c7bd5-zh", "chapter": "7", "section": "7.2", "offset": 11751, "range": [ 18882, 19120 ], "source_revision": "d4ec71fdf87fd3fc0add2e6bd217ee8e6ca96a929e9a58e94cc0ba3e89229d59", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "度量张量(metric tensor)", "context": { "range": [ 18504, 19423 ], "quote": "> [!example] 例 7.5 \n>\n> 设 $( V , \\cdot )$ 为实内积空间，如 第 5.1 节所述。由下式定义的映射 $g:V\\times V\\to\\mathbb R$\n>\n> $$\n> g (u, v) = u \\cdot v\n> $$\n>\n> 显然是双线性的，并且是一个2阶协变张量，称为内积的度量张量(metric tensor)。内积的分量$g _ { i j } = e _ { i } \\cdot e _ { j } = g ( e _ { i } , e _ { j } )$是度量张量关于基$\\{ e _ { i } \\}$的分量\n>\n> $$\n> g = g _ {i j} \\varepsilon^ {i} \\otimes \\varepsilon^ {j},\n> $$\n>\n> 而两个向量的内积是\n>\n> $$\n> u \\cdot v = g (u, v) = g _ {i j} u ^ {i} v ^ {j}.\n> $$\n>\n> 度量张量是对称的，\n>\n> $$\n> g (u, v) = g (v, u).\n> $$", "quote_sha256": "390b76f145489b6e26df36c46a5d918e843e2a88fbc4d101259299edd838f115", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-07/02-multilinear-maps-and-tensors", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-19b9a48c75ad99e4e015558e" }, { "concept_id": "urn:uuid:ac724144-5a08-407a-a27a-796eb367a161", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-07/02-multilinear-maps-and-tensors", "anchor": "first-52084e4e3766521c7bd5-en", "chapter": "7", "section": "7.2", "offset": 14061, "range": [ 20717, 20949 ], "source_revision": "40721e8f3f83073f4f97c691a9d2f801b5f414736ca94d691967c2992680ac85", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "metric tensor", "context": { "range": [ 20243, 21415 ], "quote": "> [!example] Example 7.5 \n>\n> Let $( V , \\cdot )$ be a real inner product space, as in Section 5.1. The map $g:V\\times V\\to\\mathbb R$ defined by\n>\n> $$\n> g (u, v) = u \\cdot v\n> $$\n>\n> is obviously bilinear, and is a covariant tensor of degree 2 called the metric tensor of the inner product. The components $g _ { i j } = e _ { i } \\cdot e _ { j } = g ( e _ { i } , e _ { j } )$ of the inner product are the components of the metric tensor with respect to the basis $\\{ e _ { i } \\}$\n>\n> $$\n> g = g _ {i j} \\varepsilon^ {i} \\otimes \\varepsilon^ {j},\n> $$\n>\n> while the inner product of two vectors is\n>\n> $$\n> u \\cdot v = g (u, v) = g _ {i j} u ^ {i} v ^ {j}.\n> $$\n>\n> The metric tensor is symmetric,\n>\n> $$\n> g (u, v) = g (v, u).\n> $$", "quote_sha256": "e53e9759dd9bd7c07275a203ba63361d968c413b30d95c63d773103fbb7efbdb", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "zh/chapter-07/02-multilinear-maps-and-tensors", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-0c9af9619926d3a4a3436bf8" }, { "concept_id": "urn:uuid:ac724144-5a08-407a-a27a-796eb367a161", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-18/04-pseudo-riemannian-manifolds", "anchor": "first-ee5625c59c4d344be166-zh", "chapter": "18", "section": "18.4", "offset": 744, "range": [ 7282, 7485 ], "source_revision": "1bbb27130bf7d8d0ec56ab51f9406b00c7ca2b06919edbe1a20361105c5aba6d", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "度量张量(metric tensor)", "context": { "range": [ 6869, 7694 ], "quote": "流形 $M$ 上类型为 (0, 2) 的张量场 $g$ 称为非奇异的，如果 $g _ { p } \\in T ^ { ( 0 , 2 ) }$ 在每一点 $p \\in M$ 处都是非奇异张量。一个 伪黎曼流形(pseudo-Riemannian manifold) $( M , g )$ 由一个可微流形 $M$ 连同类型为 (0, 2) 的对称非奇异张量场 $g$ 组成，该张量场称为 度量张量(metric tensor)。这等价于在每一点 $p \\in M$ 处的切空间 $T _ { p } ( M )$ 上定义一个内积 $X _ { p } \\cdot Y _ { p } = g _ { p } ( X _ { p } , Y _ { p } )$（见第 5 章和第 7 章）。我们将假设 $g$ 是可微张量场，从而对于每一对光滑向量场 $X$、$Y \\in { \\mathcal { T } } ( M )$，其内积都是可微函数，", "quote_sha256": "6e7986246e8b7d2e9734e73d3015a1df56840055901b652b8ad555ab255f5627", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-18/04-pseudo-riemannian-manifolds", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-4b8b8f9baab9e57f2ba0ee85" }, { "concept_id": "urn:uuid:ac724144-5a08-407a-a27a-796eb367a161", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-18/04-pseudo-riemannian-manifolds", "anchor": "first-ee5625c59c4d344be166-en", "chapter": "18", "section": "18.4", "offset": 882, "range": [ 6948, 7145 ], "source_revision": "abe4ca4f42fe5272187e66105b690e8b9f1e858c2725731a33be8d30fbcf1939", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "metric tensor", "context": { "range": [ 6397, 7533 ], "quote": "A tensor field $g$ of type (0, 2) on a manifold $M$ is said to be non-singular if $g _ { p } \\in T ^ { ( 0 , 2 ) }$ is a non-singular tensor at every point $p \\in M$ . A pseudo-Riemannian manifold $( M , g )$ consists of a differentiable manifold $M$ together with a symmetric non-singular tensor field $g$ of type (0, 2), called a metric tensor. This is equivalent to defining an inner produc $X _ { p } \\cdot Y _ { p } = g _ { p } ( X _ { p } , Y _ { p } )$ on the tangent space $T _ { p } ( M )$ at every point $p \\in M$ (see Chapters 5 and 7). We will assume $g$ is a differentiable tensor field, so that for every pair of smooth vector fields $X$, $Y \\in { \\mathcal { T } } ( M )$ the inner product is a differentiable function,", "quote_sha256": "91f8f7700274958dd51739d2152dacba0f1a69b39378fc9ae848ba0cf6575fad", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "zh/chapter-18/04-pseudo-riemannian-manifolds", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-46d0631af9b7e6c0e54da2fc" } ], "registered_positions": [ { "route": "en/chapter-07/02-multilinear-maps-and-tensors", "lang": "en", "section": "7.2", "signal": "term-mark", "label": "metric tensor", "english": "metric tensor", "offset": 13226 }, { "route": "zh/chapter-07/02-multilinear-maps-and-tensors", "lang": "zh-CN", "section": "7.2", "signal": "term-mark", "label": "度量张量", "english": "metric tensor", "offset": 10916 }, { "route": "en/chapter-18/04-pseudo-riemannian-manifolds", "lang": "en", "section": "18.4", "signal": "term-mark", "label": "metric tensor", "english": "metric tensor", "offset": 691 }, { "route": "zh/chapter-18/04-pseudo-riemannian-manifolds", "lang": "zh-CN", "section": "18.4", "signal": "term-mark", "label": "度量张量", "english": "metric tensor", "offset": 553 } ], "statements": [], "resolved_references": [ { "raw": "href=\"#example-7-5\"", "href": "#example-7-5", "range": [ 18545, 18564 ], "relation": "references", "target_route": "zh/chapter-07/02-multilinear-maps-and-tensors", "target_anchor": "example-7-5", "resolved": true, "reason": null, "source_revision": "d4ec71fdf87fd3fc0add2e6bd217ee8e6ca96a929e9a58e94cc0ba3e89229d59", "source_route": "zh/chapter-07/02-multilinear-maps-and-tensors", "occurrence_id": "occ-19b9a48c75ad99e4e015558e" }, { "raw": "href=\"../chapter-05/01-real-inner-product-spaces#section-5-1\"", "href": "../chapter-05/01-real-inner-product-spaces#section-5-1", "range": [ 18610, 18671 ], "relation": "references", "target_route": "zh/chapter-05/01-real-inner-product-spaces", "target_anchor": "section-5-1", "resolved": true, "reason": null, "source_revision": "d4ec71fdf87fd3fc0add2e6bd217ee8e6ca96a929e9a58e94cc0ba3e89229d59", "source_route": "zh/chapter-07/02-multilinear-maps-and-tensors", "occurrence_id": "occ-19b9a48c75ad99e4e015558e" }, { "raw": "href=\"#example-7-5\"", "href": "#example-7-5", "range": [ 20284, 20303 ], "relation": "references", "target_route": "en/chapter-07/02-multilinear-maps-and-tensors", "target_anchor": "example-7-5", "resolved": true, "reason": null, "source_revision": "40721e8f3f83073f4f97c691a9d2f801b5f414736ca94d691967c2992680ac85", "source_route": "en/chapter-07/02-multilinear-maps-and-tensors", "occurrence_id": "occ-0c9af9619926d3a4a3436bf8" }, { "raw": "href=\"../chapter-05/01-real-inner-product-spaces#section-5-1\"", "href": "../chapter-05/01-real-inner-product-spaces#section-5-1", "range": [ 20385, 20446 ], "relation": "references", "target_route": "en/chapter-05/01-real-inner-product-spaces", "target_anchor": "section-5-1", "resolved": true, "reason": null, "source_revision": "40721e8f3f83073f4f97c691a9d2f801b5f414736ca94d691967c2992680ac85", "source_route": "en/chapter-07/02-multilinear-maps-and-tensors", "occurrence_id": "occ-0c9af9619926d3a4a3436bf8" } ], "reading_order": [ "occ-19b9a48c75ad99e4e015558e", "occ-0c9af9619926d3a4a3436bf8", "occ-4b8b8f9baab9e57f2ba0ee85", "occ-46d0631af9b7e6c0e54da2fc" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 4, "located_occurrences": 4, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "79bff036389ece79f49dee93b7c96c3614f9a7c69f974b70e85a273d6b052d86", "optional_enrichments": [] }