"Green's theorem" meaning in All languages combined

{
  "etymology_text": "Named after the mathematician George Green.",
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
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      "examples": [
        {
          "text": "∬_R(∂Q/∂x-∂P/∂y)dx,dy=∮_(∂R)P,dx+Q,dy."
        }
      ],
      "glosses": [
        "A generalization of the fundamental theorem of calculus to the two-dimensional plane, which states that given two scalar fields P and Q and a simply connected region R, the area integral of derivatives of the fields equals the line integral of the fields, or"
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      "raw_glosses": [
        "(calculus) A generalization of the fundamental theorem of calculus to the two-dimensional plane, which states that given two scalar fields P and Q and a simply connected region R, the area integral of derivatives of the fields equals the line integral of the fields, or"
      ],
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        "uncountable"
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      "examples": [
        {
          "text": "∬_R∇∧⃑Gdx,dy=∮_(∂R)⃑G·d⃑l"
        },
        {
          "text": "where ∧ is the wedge product, or equivalently, as\n∬_R∇·⃑Gdx,dy=∮_(∂R)⃑G∧d⃑l,"
        },
        {
          "text": "with the earlier formula resembling Stokes' theorem, and the latter resembling the divergence theorem."
        }
      ],
      "glosses": [
        "Letting ⃑G=(P,Q) be a vector field, and d⃑l=(dx,dy) this can be restated as"
      ],
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        "(calculus) Letting ⃑G=(P,Q) be a vector field, and d⃑l=(dx,dy) this can be restated as"
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  "wikipedia": [
    "George Green (mathematician)",
    "Green's theorem"
  ],
  "word": "Green's theorem"
}

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  "etymology_text": "Named after the mathematician George Green.",
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        {
          "text": "∬_R(∂Q/∂x-∂P/∂y)dx,dy=∮_(∂R)P,dx+Q,dy."
        }
      ],
      "glosses": [
        "A generalization of the fundamental theorem of calculus to the two-dimensional plane, which states that given two scalar fields P and Q and a simply connected region R, the area integral of derivatives of the fields equals the line integral of the fields, or"
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        "(calculus) A generalization of the fundamental theorem of calculus to the two-dimensional plane, which states that given two scalar fields P and Q and a simply connected region R, the area integral of derivatives of the fields equals the line integral of the fields, or"
      ],
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      ],
      "examples": [
        {
          "text": "∬_R∇∧⃑Gdx,dy=∮_(∂R)⃑G·d⃑l"
        },
        {
          "text": "where ∧ is the wedge product, or equivalently, as\n∬_R∇·⃑Gdx,dy=∮_(∂R)⃑G∧d⃑l,"
        },
        {
          "text": "with the earlier formula resembling Stokes' theorem, and the latter resembling the divergence theorem."
        }
      ],
      "glosses": [
        "Letting ⃑G=(P,Q) be a vector field, and d⃑l=(dx,dy) this can be restated as"
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        "(calculus) Letting ⃑G=(P,Q) be a vector field, and d⃑l=(dx,dy) this can be restated as"
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  "word": "Green's theorem"
}