"Jacobi identity" meaning in English

{
  "etymology_text": "After German mathematician Carl Gustav Jakob Jacobi (1804-1851).",
  "forms": [
    {
      "form": "Jacobi identities",
      "tags": [
        "plural"
      ]
    }
  ],
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      "expansion": "Jacobi identity (countable and uncountable, plural Jacobi identities)",
      "name": "en-noun"
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  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
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          "kind": "other",
          "name": "English entries with incorrect language header",
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          "kind": "topical",
          "langcode": "en",
          "name": "Mathematics",
          "orig": "en:Mathematics",
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        }
      ],
      "derived": [
        {
          "word": "super Jacobi identity"
        }
      ],
      "examples": [
        {
          "ref": "1995, Stephen L. Adler, Quaternionic Quantum Mechanics and Quantum Fields, Oxford University Press, page 535:",
          "text": "We give here two proofs of the Jacobi identity for the generalized Poisson bracket defined in Eq. (13.69a).",
          "type": "quote"
        },
        {
          "text": "2004, ames Lepowsky, Haisheng Li, Introduction to Vertex Operator Algebras and Their Representations, Springer (Birkhäuser), page 12,\nAs we have already mentioned, the Jacobi identity is actually the generating function of an infinite list of generally highly nontrivial identities, and one needs many of these individual componenent identities in working with the theory."
        },
        {
          "ref": "2005, Martin Kröger, Models for Polymeric and Anisotropic Liquids, Springer, page 113:",
          "text": "It is sufficient to test the Jacobi identity against three linear functions [354] (this reference also provides a code for evaluating Jacobi identities).",
          "type": "quote"
        }
      ],
      "glosses": [
        "Given a binary operation × defined on a set S which also has additive operation + and additive identity 0, the property that a × (b×c) + b × (c×a) + c × (a×b) = 0 for all a, b, c in S."
      ],
      "id": "en-Jacobi_identity-en-noun-wCVVwAVy",
      "links": [
        [
          "mathematics",
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      "raw_glosses": [
        "(mathematics) Given a binary operation × defined on a set S which also has additive operation + and additive identity 0, the property that a × (b×c) + b × (c×a) + c × (a×b) = 0 for all a, b, c in S."
      ],
      "tags": [
        "countable",
        "uncountable"
      ],
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        "mathematics",
        "sciences"
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      "translations": [
        {
          "code": "it",
          "lang": "Italian",
          "sense": "property of a binary operation",
          "tags": [
            "feminine"
          ],
          "word": "identità di Jacobi"
        }
      ],
      "wikipedia": [
        "Carl Gustav Jakob Jacobi",
        "Jacobi identity"
      ]
    }
  ],
  "word": "Jacobi identity"
}

{
  "derived": [
    {
      "word": "super Jacobi identity"
    }
  ],
  "etymology_text": "After German mathematician Carl Gustav Jakob Jacobi (1804-1851).",
  "forms": [
    {
      "form": "Jacobi identities",
      "tags": [
        "plural"
      ]
    }
  ],
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
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        "English countable nouns",
        "English entries with incorrect language header",
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      "examples": [
        {
          "ref": "1995, Stephen L. Adler, Quaternionic Quantum Mechanics and Quantum Fields, Oxford University Press, page 535:",
          "text": "We give here two proofs of the Jacobi identity for the generalized Poisson bracket defined in Eq. (13.69a).",
          "type": "quote"
        },
        {
          "text": "2004, ames Lepowsky, Haisheng Li, Introduction to Vertex Operator Algebras and Their Representations, Springer (Birkhäuser), page 12,\nAs we have already mentioned, the Jacobi identity is actually the generating function of an infinite list of generally highly nontrivial identities, and one needs many of these individual componenent identities in working with the theory."
        },
        {
          "ref": "2005, Martin Kröger, Models for Polymeric and Anisotropic Liquids, Springer, page 113:",
          "text": "It is sufficient to test the Jacobi identity against three linear functions [354] (this reference also provides a code for evaluating Jacobi identities).",
          "type": "quote"
        }
      ],
      "glosses": [
        "Given a binary operation × defined on a set S which also has additive operation + and additive identity 0, the property that a × (b×c) + b × (c×a) + c × (a×b) = 0 for all a, b, c in S."
      ],
      "links": [
        [
          "mathematics",
          "mathematics"
        ]
      ],
      "raw_glosses": [
        "(mathematics) Given a binary operation × defined on a set S which also has additive operation + and additive identity 0, the property that a × (b×c) + b × (c×a) + c × (a×b) = 0 for all a, b, c in S."
      ],
      "tags": [
        "countable",
        "uncountable"
      ],
      "topics": [
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "Carl Gustav Jakob Jacobi",
        "Jacobi identity"
      ]
    }
  ],
  "translations": [
    {
      "code": "it",
      "lang": "Italian",
      "sense": "property of a binary operation",
      "tags": [
        "feminine"
      ],
      "word": "identità di Jacobi"
    }
  ],
  "word": "Jacobi identity"
}