"codifferential" meaning in English

{
  "etymology_templates": [
    {
      "args": {
        "1": "en",
        "2": "co",
        "3": "differential"
      },
      "expansion": "co- + differential",
      "name": "prefix"
    }
  ],
  "etymology_text": "From co- + differential.",
  "forms": [
    {
      "form": "codifferentials",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "codifferential (plural codifferentials)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Mathematics",
          "orig": "en:Mathematics",
          "parents": [
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "ref": "2016, Terence Tao, “Finite time blowup for Lagrangian modifications of the three-dimensional Euler equation”, in arXiv:",
          "text": "In the language of differential geometry, the incompressible inviscid Euler equations can be written in vorticity-vector potential form as #x5C;begin#x7B;matrix#x7D;#x5C;partial#x5F;t#x5C;omega#x2B;#x7B;#x5C;mathcalL#x7D;#x5F;u#x5C;omega#x26;#x3D;0#x5C;#x5C;u#x26;#x3D;#x5C;delta#x5C;tilde#x5C;eta#x7B;-1#x7D;#x5C;Delta#x7B;-1#x7D;#x5C;omega#x5C;end#x7B;matrix#x7D; where #x5C;omega is the vorticity 2-form, #x7B;#x5C;mathcalL#x7D;#x5F;u denotes the Lie derivative with respect to the velocity field u, #x5C;Delta is the Hodge Laplacian, #x5C;delta is the codifferential (the negative of the divergence operator), and #x5C;tilde#x5C;eta#x7B;-1#x7D; is the canonical map from 2-forms to 2-vector fields induced by the Euclidean metric #x5C;eta.",
          "type": "quote"
        }
      ],
      "glosses": [
        "The projected differential of an extensor field."
      ],
      "id": "en-codifferential-en-noun-s9GGSNJj",
      "links": [
        [
          "mathematics",
          "mathematics"
        ]
      ],
      "raw_glosses": [
        "(mathematics) The projected differential of an extensor field."
      ],
      "topics": [
        "mathematics",
        "sciences"
      ]
    },
    {
      "categories": [
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Differential geometry",
          "orig": "en:Differential geometry",
          "parents": [
            "Geometry",
            "Mathematical analysis",
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        },
        {
          "_dis": "22 78",
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w+disamb"
        },
        {
          "_dis": "34 66",
          "kind": "other",
          "name": "English terms prefixed with co-",
          "parents": [],
          "source": "w+disamb"
        },
        {
          "_dis": "15 85",
          "kind": "other",
          "name": "Pages with 1 entry",
          "parents": [],
          "source": "w+disamb"
        },
        {
          "_dis": "20 80",
          "kind": "other",
          "name": "Pages with entries",
          "parents": [],
          "source": "w+disamb"
        }
      ],
      "examples": [
        {
          "ref": "2015, 1:19:34 from the start, in 2 Tangent Space, Differential Forms, Metric (Mathematics for Theoretical Physicists - Hirosi Ooguri), spoken by Hiroshi Ooguri (Hiroshi Ooguri), K Raviteja (YouTube):",
          "text": "One important concept is the introduction of codifferential operator",
          "type": "quote"
        }
      ],
      "glosses": [
        "the formal adjoint of the exterior derivative; a differential-geometric version of the divergence operator; the exterior derivative sandwiched between two Hodge star operators with some additional factor(s) that take(s) care of the sign; the Hermitian conjugate of the exterior derivative under the inner product for k-form fields over some manifold M: (α,β)=∫_Mα∧⋆β, so that (α,dβ)=(δα,β)."
      ],
      "id": "en-codifferential-en-noun-kpnRXw7u",
      "links": [
        [
          "differential geometry",
          "differential geometry"
        ],
        [
          "exterior derivative",
          "exterior derivative"
        ],
        [
          "divergence",
          "divergence"
        ],
        [
          "Hodge star",
          "Hodge star"
        ],
        [
          "Hermitian conjugate",
          "Hermitian conjugate"
        ],
        [
          "inner product",
          "inner product"
        ]
      ],
      "qualifier": "differential geometry",
      "raw_glosses": [
        "(differential geometry) the formal adjoint of the exterior derivative; a differential-geometric version of the divergence operator; the exterior derivative sandwiched between two Hodge star operators with some additional factor(s) that take(s) care of the sign; the Hermitian conjugate of the exterior derivative under the inner product for k-form fields over some manifold M: (α,β)=∫_Mα∧⋆β, so that (α,dβ)=(δα,β)."
      ]
    }
  ],
  "word": "codifferential"
}

{
  "categories": [
    "English countable nouns",
    "English entries with incorrect language header",
    "English lemmas",
    "English nouns",
    "English terms prefixed with co-",
    "Pages with 1 entry",
    "Pages with entries"
  ],
  "etymology_templates": [
    {
      "args": {
        "1": "en",
        "2": "co",
        "3": "differential"
      },
      "expansion": "co- + differential",
      "name": "prefix"
    }
  ],
  "etymology_text": "From co- + differential.",
  "forms": [
    {
      "form": "codifferentials",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "codifferential (plural codifferentials)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        "English terms with quotations",
        "en:Mathematics"
      ],
      "examples": [
        {
          "ref": "2016, Terence Tao, “Finite time blowup for Lagrangian modifications of the three-dimensional Euler equation”, in arXiv:",
          "text": "In the language of differential geometry, the incompressible inviscid Euler equations can be written in vorticity-vector potential form as #x5C;begin#x7B;matrix#x7D;#x5C;partial#x5F;t#x5C;omega#x2B;#x7B;#x5C;mathcalL#x7D;#x5F;u#x5C;omega#x26;#x3D;0#x5C;#x5C;u#x26;#x3D;#x5C;delta#x5C;tilde#x5C;eta#x7B;-1#x7D;#x5C;Delta#x7B;-1#x7D;#x5C;omega#x5C;end#x7B;matrix#x7D; where #x5C;omega is the vorticity 2-form, #x7B;#x5C;mathcalL#x7D;#x5F;u denotes the Lie derivative with respect to the velocity field u, #x5C;Delta is the Hodge Laplacian, #x5C;delta is the codifferential (the negative of the divergence operator), and #x5C;tilde#x5C;eta#x7B;-1#x7D; is the canonical map from 2-forms to 2-vector fields induced by the Euclidean metric #x5C;eta.",
          "type": "quote"
        }
      ],
      "glosses": [
        "The projected differential of an extensor field."
      ],
      "links": [
        [
          "mathematics",
          "mathematics"
        ]
      ],
      "raw_glosses": [
        "(mathematics) The projected differential of an extensor field."
      ],
      "topics": [
        "mathematics",
        "sciences"
      ]
    },
    {
      "categories": [
        "English terms with quotations",
        "en:Differential geometry"
      ],
      "examples": [
        {
          "ref": "2015, 1:19:34 from the start, in 2 Tangent Space, Differential Forms, Metric (Mathematics for Theoretical Physicists - Hirosi Ooguri), spoken by Hiroshi Ooguri (Hiroshi Ooguri), K Raviteja (YouTube):",
          "text": "One important concept is the introduction of codifferential operator",
          "type": "quote"
        }
      ],
      "glosses": [
        "the formal adjoint of the exterior derivative; a differential-geometric version of the divergence operator; the exterior derivative sandwiched between two Hodge star operators with some additional factor(s) that take(s) care of the sign; the Hermitian conjugate of the exterior derivative under the inner product for k-form fields over some manifold M: (α,β)=∫_Mα∧⋆β, so that (α,dβ)=(δα,β)."
      ],
      "links": [
        [
          "differential geometry",
          "differential geometry"
        ],
        [
          "exterior derivative",
          "exterior derivative"
        ],
        [
          "divergence",
          "divergence"
        ],
        [
          "Hodge star",
          "Hodge star"
        ],
        [
          "Hermitian conjugate",
          "Hermitian conjugate"
        ],
        [
          "inner product",
          "inner product"
        ]
      ],
      "qualifier": "differential geometry",
      "raw_glosses": [
        "(differential geometry) the formal adjoint of the exterior derivative; a differential-geometric version of the divergence operator; the exterior derivative sandwiched between two Hodge star operators with some additional factor(s) that take(s) care of the sign; the Hermitian conjugate of the exterior derivative under the inner product for k-form fields over some manifold M: (α,β)=∫_Mα∧⋆β, so that (α,dβ)=(δα,β)."
      ]
    }
  ],
  "word": "codifferential"
}