{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:1435ed7d-50c4-4a83-adfe-50f270d61d40", "labels": { "zh-CN": "核心指标记法", "en": "kernel index notation" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "09867675e4885fc1b51f4e2eea1cf0748d95ef0fbb16733e8ee21b03f18c3795", "occurrences": [ { "concept_id": "urn:uuid:1435ed7d-50c4-4a83-adfe-50f270d61d40", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-07/03-basis-representation-of-tensors", "anchor": "first-9682eeb03d3d57ac469d-zh", "chapter": "7", "section": "7.3", "offset": 7024, "range": [ 13430, 13678 ], "source_revision": "7e2e87e1c4bae49558bc101a3cbf92c7d730a5190f2d651e6c2822ad3c5a6f80", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "核心指标记法(kernel index notation)", "context": { "range": [ 13406, 14189 ], "quote": "在给出一般张量分量的变换律之前，先建立一种称为 核心指标记法(kernel index notation) 的约定是有益的。在这种记法中，我们用带撇的指标表示变换后的基和对偶基上的指标，即 $\\{ e _ { i ^ { \\prime } } ^ { \\prime } | i ^ { \\prime } = 1 , \\ldots , n \\}$ 和 $\\{ \\varepsilon ^ { \\prime j ^ { \\prime } } | j ^ { \\prime } = 1 , \\dots , n \\}$。在“核”字母 $e$ 和 $\\varepsilon$ 上加撇本质上是多余的，去掉它们几乎不会损失什么含义，只需将变换后的基写作 $e _ { i ^ { \\prime } }$ 和 $\\varepsilon ^ { j ^ { \\prime } }$。该约定甚至可以更进一步，要求带撇的指标在一个自然数的指标集 $i ^ { \\prime } = 1 ^ { \\prime } , \\dots , n ^ { \\prime }$ 上取值。这些做法可能显得有些古怪，也可能造成混淆。因此，我们只采用一种“半成熟”的核心指标记法，其关键要求是对变换后的量使用带撇的指标。核心指标记法的主要优点是使张量的变换律更容易记忆。", "quote_sha256": "a614dc791cd0274070f1d7b322af8269105be8e55050ca9780cd55ce789a6f89", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-07/03-basis-representation-of-tensors", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-4771c4b0f3720558152ac899" }, { "concept_id": "urn:uuid:1435ed7d-50c4-4a83-adfe-50f270d61d40", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-07/03-basis-representation-of-tensors", "anchor": "first-9682eeb03d3d57ac469d-en", "chapter": "7", "section": "7.3", "offset": 8899, "range": [ 14833, 15073 ], "source_revision": "19441a269ef9d2bb760a075f939505a17a733a91bf1b5e8bea10ab1d75088c9b", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "kernel index notation", "context": { "range": [ 14710, 16095 ], "quote": "Before giving the transformation law of components of a general tensor it is useful to establish a convention known as the kernel index notation. In this notation we denote the indices on the transformed bases and dual bases by a primed index, $\\{ e _ { i ^ { \\prime } } ^ { \\prime } | i ^ { \\prime } = 1 , \\ldots , n \\}$ and $\\{ \\varepsilon ^ { \\prime j ^ { \\prime } } | j ^ { \\prime } = 1 , \\dots , n \\}$ . The primes on the ‘kernel’ letters $e$ and $\\varepsilon$ are essentially superfluous and little meaning is lost in dropping them, simply writing $e _ { i ^ { \\prime } }$ and $\\varepsilon ^ { j ^ { \\prime } }$ for the transformed bases. The convention may go even further and require that the primed indices range over an indexed set of natural numbers $i ^ { \\prime } = 1 ^ { \\prime } , \\dots , n ^ { \\prime }$ . These practices may seem a little bizarre and possibly confusing. Accordingly, we will only follow a ‘half-blown’ kernel index notation, with the key requirement that primed indices be used on transformed quantities. 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