{
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    "publishableCanonical": false,
    "generatedAt": "2026-09-12T23:35:12Z",
    "providers": [
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    "pass": 2,
    "wave": 3,
    "engine": "claude",
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      "producedAt": "2026-10-06T10:00:00Z",
      "ingestedAt": "2026-10-06T06:58:41+00:00",
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    "registryId": "vr.tr.differential-operator",
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      "XCT.QLT"
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  "model": {
    "registry_id": "vr.tr.differential-operator",
    "name": "differential operator",
    "purpose": "Let an agent explain differential operators and their kinds, relay their properties and role in differential equations and physics, help apply common operators, and separate the general concept from specialised operators listed as aliases.",
    "definition": "An operator that maps a function to another function by differentiation, such as the derivative, gradient, divergence, curl, Laplacian, exterior derivative and material derivative, and more general linear combinations of partial derivatives with coefficients; differential operators are central to calculus, differential equations, physics and geometry, are classified by order and properties such as ellipticity and hypoellipticity, and extend to pseudodifferential and microdifferential operators and stochastic derivatives such as the Malliavin derivative.",
    "what_it_is_for": "Expressing rates of change and differential equations.",
    "affordances": [
      "explain the concept and kinds",
      "relay properties and classification",
      "help apply common operators",
      "relate to specialised operators"
    ],
    "distinguishing_features": [
      "Built from derivatives",
      "Linear in most cases",
      "Classified by order and type",
      "Central to differential equations"
    ],
    "appearance": "Not a visible object; symbols such as d/dx, nabla and the Laplacian.",
    "visual_identification": [
      "Operator built from derivatives",
      "Gradient, divergence, Laplacian and general forms",
      "Integral operators integrate; difference operators act on sequences"
    ],
    "physical_properties": [],
    "families_and_kinds": [
      "ordinary and partial differential operators",
      "vector calculus operators: gradient, divergence, curl and Laplacian",
      "the exterior derivative and covariant derivatives in geometry",
      "the material and convective derivatives in fluid mechanics",
      "elliptic, hyperbolic, parabolic and hypoelliptic operators",
      "pseudodifferential and microdifferential operators",
      "stochastic derivatives such as the Malliavin derivative"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "function",
        "why": "in registry terms, an operator on functions"
      },
      {
        "relation": "is a kind of",
        "target": "linear operator",
        "why": "in most cases"
      },
      {
        "relation": "defines",
        "target": "differential equation",
        "why": "when set equal to a function"
      },
      {
        "relation": "includes",
        "target": "Laplace operator",
        "why": "a central example"
      }
    ],
    "identifiers": [],
    "standards_and_regulation": [
      "ISO 80000-2 notation for derivatives and operators"
    ],
    "failure_modes_and_hazards": [
      "Confusing operators with the equations they define",
      "Domain and smoothness issues",
      "Notation clashes across fields"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "may": [
        "Explain differential operators such as gradient, divergence, curl and the Laplacian.",
        "Apply operators to functions supplied by a user, stating domains and smoothness assumptions.",
        "Check symbolic or numerical computations for sign and notation errors."
      ],
      "must_not": [
        "Apply an operator outside the domain or smoothness where it is defined without saying so.",
        "Mix notation from different fields without defining it.",
        "Present results of a numerical approximation as exact.",
        "Confuse an operator with the equation it appears in."
      ],
      "requires_human": [
        "Relying on operator calculations for safety-critical engineering models."
      ]
    },
    "ethics": {
      "considerations": [
        "Errors in models built on these operators can lead to unsafe designs.",
        "Clear exposition makes advanced mathematics accessible."
      ],
      "affected_parties": [
        "Students",
        "Engineers and scientists"
      ]
    },
    "owners": {
      "steward": "Nobody: a mathematical concept held in common.",
      "master_systems": []
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q1058681: differential operator",
      "url": "https://www.wikidata.org/wiki/Q1058681",
      "url_status": "live",
      "what_it_supports": "identity and sense of the item",
      "checked_by": "wikidata-api"
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      "id": "wikipedia-en",
      "title": "Wikipedia: Differential operator",
      "url": "https://en.wikipedia.org/wiki/Differential_operator",
      "url_status": "live",
      "what_it_supports": "general description of the item",
      "checked_by": "wikidata-api sitelink"
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        "id": "define",
        "name": "Define",
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        "rationale": "Definition.",
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            "id": "definition",
            "name": "Definition",
            "description": "Definition and kinds.",
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                "id": "definition-finding",
                "name": "Definition",
                "description": "Definition.",
                "questions": [
                  {
                    "text": "What is a differential operator, and what common kinds exist?",
                    "kind": "definition"
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                  {
                    "text": "Is the object a differential operator, an integral operator or a difference operator?",
                    "kind": "boundary"
                  }
                ]
              }
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            "description": "Classification.",
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                "id": "classification-finding",
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                    "kind": "definition"
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                    "text": "Which entry fits the specific operator?",
                    "kind": "action"
                  }
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              }
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            "description": "Computing with operators.",
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                "description": "Compute.",
                "questions": [
                  {
                    "text": "How are gradient, divergence, curl and Laplacian applied in different coordinate systems?",
                    "kind": "action"
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                  {
                    "text": "What is the result for the function in question?",
                    "kind": "measurement"
                  }
                ]
              }
            ]
          },
          {
            "id": "equations",
            "name": "Equations",
            "description": "Differential equations.",
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              {
                "id": "equations-finding",
                "name": "Equations",
                "description": "Equations.",
                "questions": [
                  {
                    "text": "How do operators define and solve differential equations?",
                    "kind": "action"
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                  {
                    "text": "Which entry fits partial differential equation?",
                    "kind": "action"
                  }
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              }
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            "name": "Properties",
            "description": "Properties.",
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                "name": "Properties",
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                "questions": [
                  {
                    "text": "What do ellipticity, hypoellipticity and regularity theory say about operators?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which references are standard?",
                    "kind": "provenance"
                  }
                ]
              }
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            "id": "generalisations",
            "name": "Generalisations",
            "description": "Generalisations.",
            "findings": [
              {
                "id": "generalisations-finding",
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                "description": "Generalisations.",
                "questions": [
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                    "text": "What are pseudodifferential, microdifferential and stochastic differential operators such as the Malliavin derivative?",
                    "kind": "provenance"
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                    "text": "Which entry fits the specific generalisation?",
                    "kind": "action"
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      {
        "id": "fields",
        "name": "Fields",
        "description": "Uses in physics and geometry.",
        "rationale": "Application.",
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            "id": "physics",
            "name": "Physics",
            "description": "Physics and fluids.",
            "findings": [
              {
                "id": "physics-finding",
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                "questions": [
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                    "text": "How are the material derivative and other operators used in fluid mechanics and field theory?",
                    "kind": "provenance"
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                  {
                    "text": "Which entry fits material derivative?",
                    "kind": "action"
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            "id": "geometry",
            "name": "Geometry",
            "description": "Geometry.",
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              {
                "id": "geometry-finding",
                "name": "Geometry",
                "description": "Geometry.",
                "questions": [
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                    "text": "How do the exterior derivative and covariant derivative work in differential geometry?",
                    "kind": "provenance"
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                  {
                    "text": "Which sources are cited?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should each named operator be a separate entry?",
    "How should reference texts be linked?",
    "The registry entry has merged aliases naming specialised operators; should they be split off?"
  ],
  "statistics": {
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    "layers": 8,
    "findings": 8,
    "questions": 16
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}