Semantic Graph — Visual Examination Report

{
  "nodes": [
    {
      "id": "f",
      "type": "scalar",
      "latex": "f",
      "subexpr": "f",
      "chartScript": {
        "script": "f",
        "variables": [
          "f"
        ]
      }
    },
    {
      "id": "x",
      "type": "scalar",
      "latex": "x",
      "subexpr": "x",
      "chartScript": {
        "script": "x",
        "variables": [
          "x"
        ]
      }
    },
    {
      "id": "__deriv_1",
      "type": "operator",
      "op": "partial_derivative",
      "with_respect_to": "x",
      "subexpr": "\\frac{\\partial f}{\\partial x}",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    }
  ],
  "edges": [
    {
      "from": "f",
      "to": "__deriv_1"
    },
    {
      "from": "x",
      "to": "__deriv_1",
      "role": "wrt"
    }
  ],
  "classification": {
    "kind": "ODE",
    "order": 1,
    "dependent_variables": [
      "f"
    ],
    "independent_variables": [
      "x"
    ],
    "sympy_hints": [
      "nth_algebraic",
      "separable",
      "1st_exact",
      "1st_linear",
      "Bernoulli",
      "1st_homogeneous_coeff_best",
      "1st_homogeneous_coeff_subs_indep_div_dep",
      "1st_homogeneous_coeff_subs_dep_div_indep",
      "1st_power_series",
      "lie_group",
      "nth_linear_constant_coeff_homogeneous",
      "nth_linear_euler_eq_homogeneous"
    ],
    "linear": true,
    "homogeneous": true,
    "constant_coefficients": true
  }
}

flowchart RL
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  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
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  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  f["$f$"]:::scalar
  x["$x$"]:::scalar
  __deriv_1(("$\dfrac{\partial}{\partial x}$")):::operator
  f --> __deriv_1
  x -->|wrt| __deriv_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "\\frac{\\partial u}{\\partial t} = k \\frac{\\partial^2 u}{\\partial x^2}",
      "chartScript": {
        "script": "-k*partial*u/pow(x, 2)",
        "variables": [
          "k",
          "partial",
          "u",
          "x"
        ]
      }
    },
    {
      "id": "u",
      "type": "scalar",
      "latex": "u",
      "subexpr": "u",
      "chartScript": {
        "script": "u",
        "variables": [
          "u"
        ]
      }
    },
    {
      "id": "t",
      "type": "scalar",
      "latex": "t",
      "subexpr": "t",
      "chartScript": {
        "script": "t",
        "variables": [
          "t"
        ]
      }
    },
    {
      "id": "__deriv_2",
      "type": "operator",
      "op": "partial_derivative",
      "with_respect_to": "t",
      "subexpr": "\\frac{\\partial u}{\\partial t}",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    },
    {
      "id": "__multiply_3",
      "type": "operator",
      "op": "multiply",
      "subexpr": "k \\frac{\\partial^{2} u}{\\partial x^{2}}",
      "chartScript": {
        "script": "k*partial*u/pow(x, 2)",
        "variables": [
          "k",
          "partial",
          "u",
          "x"
        ]
      }
    },
    {
      "id": "k",
      "type": "scalar",
      "latex": "k",
      "subexpr": "k",
      "chartScript": {
        "script": "k",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "x",
      "type": "scalar",
      "latex": "x",
      "subexpr": "x",
      "chartScript": {
        "script": "x",
        "variables": [
          "x"
        ]
      }
    },
    {
      "id": "__deriv_4",
      "type": "operator",
      "op": "partial_derivative",
      "with_respect_to": "x",
      "subexpr": "\\frac{\\partial^{2} u}{\\partial x^{2}}",
      "chartScript": {
        "script": "partial*u/pow(x, 2)",
        "variables": [
          "partial",
          "u",
          "x"
        ]
      }
    }
  ],
  "edges": [
    {
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      "to": "__deriv_2"
    },
    {
      "from": "t",
      "to": "__deriv_2",
      "role": "wrt"
    },
    {
      "from": "__deriv_2",
      "to": "__equals_1"
    },
    {
      "from": "k",
      "to": "__multiply_3",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "u",
      "to": "__deriv_4"
    },
    {
      "from": "x",
      "to": "__deriv_4",
      "role": "wrt"
    },
    {
      "from": "__deriv_4",
      "to": "__multiply_3",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_3",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "PDE",
    "order": 2,
    "dependent_variables": [
      "u"
    ],
    "independent_variables": [
      "t",
      "x"
    ]
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
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  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
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  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  u["$u$"]:::scalar
  t["$t$"]:::scalar
  __deriv_2(("$\dfrac{\partial}{\partial t}$")):::operator
  __multiply_3(("$\times$")):::operator
  k["$k$"]:::scalar
  x["$x$"]:::scalar
  __deriv_4(("$\dfrac{\partial^{2}}{\partial x^{2}}$")):::operator
  u --> __deriv_2
  t -->|wrt| __deriv_2
  __deriv_2 --> __equals_1
  k --> __multiply_3
  u --> __deriv_4
  x -->|wrt| __deriv_4
  __deriv_4 --> __multiply_3
  __multiply_3 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#ef5350,stroke-width:3px
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  linkStyle 5 stroke:#aaa,stroke-width:2px
  linkStyle 6 stroke:#ef5350,stroke-width:3px
  linkStyle 7 stroke:#aaa,stroke-width:2px

{
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      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "\\frac{\\partial^2 u}{\\partial t^2} = c^2 \\frac{\\partial^2 u}{\\partial x^2}",
      "chartScript": {
        "script": "-pow(c, 2)*partial*u/pow(x, 2) + partial*u/pow(t, 2)",
        "variables": [
          "c",
          "partial",
          "t",
          "u",
          "x"
        ]
      }
    },
    {
      "id": "u",
      "type": "scalar",
      "latex": "u",
      "subexpr": "u",
      "chartScript": {
        "script": "u",
        "variables": [
          "u"
        ]
      }
    },
    {
      "id": "t",
      "type": "scalar",
      "latex": "t",
      "subexpr": "t",
      "chartScript": {
        "script": "t",
        "variables": [
          "t"
        ]
      }
    },
    {
      "id": "__deriv_2",
      "type": "operator",
      "op": "partial_derivative",
      "with_respect_to": "t",
      "subexpr": "\\frac{\\partial^{2} u}{\\partial t^{2}}",
      "chartScript": {
        "script": "partial*u/pow(t, 2)",
        "variables": [
          "partial",
          "t",
          "u"
        ]
      }
    },
    {
      "id": "__multiply_3",
      "type": "operator",
      "op": "multiply",
      "subexpr": "c^{2} \\frac{\\partial^{2} u}{\\partial x^{2}}",
      "chartScript": {
        "script": "pow(c, 2)*partial*u/pow(x, 2)",
        "variables": [
          "c",
          "partial",
          "u",
          "x"
        ]
      }
    },
    {
      "id": "__power_4",
      "type": "operator",
      "op": "power",
      "exponent": "2",
      "subexpr": "c^{2}",
      "chartScript": {
        "script": "pow(c, 2)",
        "variables": [
          "c"
        ]
      }
    },
    {
      "id": "c",
      "type": "scalar",
      "latex": "c",
      "subexpr": "c",
      "chartScript": {
        "script": "c",
        "variables": [
          "c"
        ]
      }
    },
    {
      "id": "x",
      "type": "scalar",
      "latex": "x",
      "subexpr": "x",
      "chartScript": {
        "script": "x",
        "variables": [
          "x"
        ]
      }
    },
    {
      "id": "__deriv_5",
      "type": "operator",
      "op": "partial_derivative",
      "with_respect_to": "x",
      "subexpr": "\\frac{\\partial^{2} u}{\\partial x^{2}}",
      "chartScript": {
        "script": "partial*u/pow(x, 2)",
        "variables": [
          "partial",
          "u",
          "x"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "u",
      "to": "__deriv_2"
    },
    {
      "from": "t",
      "to": "__deriv_2",
      "role": "wrt"
    },
    {
      "from": "__deriv_2",
      "to": "__equals_1"
    },
    {
      "from": "c",
      "to": "__power_4"
    },
    {
      "from": "__power_4",
      "to": "__multiply_3",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "u",
      "to": "__deriv_5"
    },
    {
      "from": "x",
      "to": "__deriv_5",
      "role": "wrt"
    },
    {
      "from": "__deriv_5",
      "to": "__multiply_3",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_3",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "PDE",
    "order": 2,
    "dependent_variables": [
      "u"
    ],
    "independent_variables": [
      "t",
      "x"
    ]
  }
}

flowchart RL
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  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
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  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
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  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  u["$u$"]:::scalar
  t["$t$"]:::scalar
  __deriv_2(("$\dfrac{\partial^{2}}{\partial t^{2}}$")):::operator
  __multiply_3(("$\times$")):::operator
  __power_4(("${(\cdot)}^{2}$")):::operator
  c["$c$"]:::scalar
  x["$x$"]:::scalar
  __deriv_5(("$\dfrac{\partial^{2}}{\partial x^{2}}$")):::operator
  u --> __deriv_2
  t -->|wrt| __deriv_2
  __deriv_2 --> __equals_1
  c --> __power_4
  __power_4 --> __multiply_3
  u --> __deriv_5
  x -->|wrt| __deriv_5
  __deriv_5 --> __multiply_3
  __multiply_3 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#aaa,stroke-width:2px
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  linkStyle 5 stroke:#aaa,stroke-width:2px
  linkStyle 6 stroke:#aaa,stroke-width:2px
  linkStyle 7 stroke:#ef5350,stroke-width:3px
  linkStyle 8 stroke:#aaa,stroke-width:2px

{
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    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "\\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} = 0",
      "chartScript": {
        "script": "partial*u/pow(y, 2) + partial*u/pow(x, 2)",
        "variables": [
          "partial",
          "u",
          "x",
          "y"
        ]
      }
    },
    {
      "id": "__add_2",
      "type": "operator",
      "op": "add",
      "subexpr": "\\frac{\\partial^{2} u}{\\partial x^{2}} + \\frac{\\partial^{2} u}{\\partial y^{2}}",
      "chartScript": {
        "script": "partial*u/pow(y, 2) + partial*u/pow(x, 2)",
        "variables": [
          "partial",
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          "x",
          "y"
        ]
      }
    },
    {
      "id": "u",
      "type": "scalar",
      "latex": "u",
      "subexpr": "u",
      "chartScript": {
        "script": "u",
        "variables": [
          "u"
        ]
      }
    },
    {
      "id": "x",
      "type": "scalar",
      "latex": "x",
      "subexpr": "x",
      "chartScript": {
        "script": "x",
        "variables": [
          "x"
        ]
      }
    },
    {
      "id": "__deriv_3",
      "type": "operator",
      "op": "partial_derivative",
      "with_respect_to": "x",
      "subexpr": "\\frac{\\partial^{2} u}{\\partial x^{2}}",
      "chartScript": {
        "script": "partial*u/pow(x, 2)",
        "variables": [
          "partial",
          "u",
          "x"
        ]
      }
    },
    {
      "id": "y",
      "type": "scalar",
      "latex": "y",
      "subexpr": "y",
      "chartScript": {
        "script": "y",
        "variables": [
          "y"
        ]
      }
    },
    {
      "id": "__deriv_4",
      "type": "operator",
      "op": "partial_derivative",
      "with_respect_to": "y",
      "subexpr": "\\frac{\\partial^{2} u}{\\partial y^{2}}",
      "chartScript": {
        "script": "partial*u/pow(y, 2)",
        "variables": [
          "partial",
          "u",
          "y"
        ]
      }
    },
    {
      "id": "__num_5",
      "type": "number",
      "label": "0",
      "subexpr": "0",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    }
  ],
  "edges": [
    {
      "from": "u",
      "to": "__deriv_3"
    },
    {
      "from": "x",
      "to": "__deriv_3",
      "role": "wrt"
    },
    {
      "from": "__deriv_3",
      "to": "__add_2"
    },
    {
      "from": "u",
      "to": "__deriv_4"
    },
    {
      "from": "y",
      "to": "__deriv_4",
      "role": "wrt"
    },
    {
      "from": "__deriv_4",
      "to": "__add_2"
    },
    {
      "from": "__add_2",
      "to": "__equals_1"
    },
    {
      "from": "__num_5",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "PDE",
    "order": 2,
    "dependent_variables": [
      "u"
    ],
    "independent_variables": [
      "x",
      "y"
    ]
  }
}

flowchart RL
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  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
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  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
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  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  __add_2(("$+$")):::operator
  u["$u$"]:::scalar
  x["$x$"]:::scalar
  __deriv_3(("$\dfrac{\partial^{2}}{\partial x^{2}}$")):::operator
  y["$y$"]:::scalar
  __deriv_4(("$\dfrac{\partial^{2}}{\partial y^{2}}$")):::operator
  __num_5["$0$"]:::number
  u --> __deriv_3
  x -->|wrt| __deriv_3
  __deriv_3 --> __add_2
  u --> __deriv_4
  y -->|wrt| __deriv_4
  __deriv_4 --> __add_2
  __add_2 --> __equals_1
  __num_5 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
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  linkStyle 5 stroke:#aaa,stroke-width:2px
  linkStyle 6 stroke:#aaa,stroke-width:2px
  linkStyle 7 stroke:#aaa,stroke-width:2px

{
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    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "\\frac{\\partial u}{\\partial t} + c \\frac{\\partial u}{\\partial x} = 0",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    },
    {
      "id": "__add_2",
      "type": "operator",
      "op": "add",
      "subexpr": "c \\frac{\\partial u}{\\partial x} + \\frac{\\partial u}{\\partial t}",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    },
    {
      "id": "u",
      "type": "scalar",
      "latex": "u",
      "subexpr": "u",
      "chartScript": {
        "script": "u",
        "variables": [
          "u"
        ]
      }
    },
    {
      "id": "t",
      "type": "scalar",
      "latex": "t",
      "subexpr": "t",
      "chartScript": {
        "script": "t",
        "variables": [
          "t"
        ]
      }
    },
    {
      "id": "__deriv_3",
      "type": "operator",
      "op": "partial_derivative",
      "with_respect_to": "t",
      "subexpr": "\\frac{\\partial u}{\\partial t}",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    },
    {
      "id": "__multiply_4",
      "type": "operator",
      "op": "multiply",
      "subexpr": "c \\frac{\\partial u}{\\partial x}",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    },
    {
      "id": "c",
      "type": "scalar",
      "latex": "c",
      "subexpr": "c",
      "chartScript": {
        "script": "c",
        "variables": [
          "c"
        ]
      }
    },
    {
      "id": "x",
      "type": "scalar",
      "latex": "x",
      "subexpr": "x",
      "chartScript": {
        "script": "x",
        "variables": [
          "x"
        ]
      }
    },
    {
      "id": "__deriv_5",
      "type": "operator",
      "op": "partial_derivative",
      "with_respect_to": "x",
      "subexpr": "\\frac{\\partial u}{\\partial x}",
      "chartScript": {
        "script": "0",
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      "type": "number",
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      "subexpr": "0",
      "chartScript": {
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      }
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  "edges": [
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      "to": "__deriv_3",
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    },
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      "semantic": "direct",
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    },
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    },
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  "classification": {
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    "order": 1,
    "dependent_variables": [
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    "independent_variables": [
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}

flowchart RL
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  t["$t$"]:::scalar
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  c["$c$"]:::scalar
  x["$x$"]:::scalar
  __deriv_5(("$\dfrac{\partial}{\partial x}$")):::operator
  __num_6["$0$"]:::number
  u --> __deriv_3
  t -->|wrt| __deriv_3
  __deriv_3 --> __add_2
  c --> __multiply_4
  u --> __deriv_5
  x -->|wrt| __deriv_5
  __deriv_5 --> __multiply_4
  __multiply_4 --> __add_2
  __add_2 --> __equals_1
  __num_6 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
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{
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      "type": "relation",
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      "subexpr": "\\frac{\\partial C}{\\partial t} = D \\frac{\\partial^2 C}{\\partial x^2}",
      "chartScript": {
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    {
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      "subexpr": "\\frac{\\partial C}{\\partial t}",
      "chartScript": {
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      "with_respect_to": "x",
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      "chartScript": {
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  "edges": [
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      "weight": 1.0
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  "classification": {
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flowchart RL
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  x["$x$"]:::scalar
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  C --> __deriv_2
  t -->|wrt| __deriv_2
  __deriv_2 --> __equals_1
  D --> __multiply_3
  C --> __deriv_4
  x -->|wrt| __deriv_4
  __deriv_4 --> __multiply_3
  __multiply_3 --> __equals_1
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      "subexpr": "\\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} = f(x,y)",
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      "chartScript": {
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      "chartScript": {
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      "subexpr": "f{\\left(x,y \\right)}",
      "chartScript": {
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        "variables": [
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      }
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  "edges": [
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flowchart RL
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  x["$x$"]:::scalar
  __deriv_3(("$\dfrac{\partial^{2}}{\partial x^{2}}$")):::operator
  y["$y$"]:::scalar
  __deriv_4(("$\dfrac{\partial^{2}}{\partial y^{2}}$")):::operator
  __f_5{{"$f(\cdot, \cdot)$"}}:::function
  u --> __deriv_3
  x -->|wrt| __deriv_3
  __deriv_3 --> __add_2
  u --> __deriv_4
  y -->|wrt| __deriv_4
  __deriv_4 --> __add_2
  __add_2 --> __equals_1
  x --> __f_5
  y --> __f_5
  __f_5 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
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{
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      "type": "relation",
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      "subexpr": "\\frac{\\partial^2 \\phi}{\\partial x^2} + \\frac{\\partial^2 \\phi}{\\partial y^2} = 0",
      "chartScript": {
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    },
    {
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      }
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    {
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      }
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    {
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    {
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      "label": "0",
      "subexpr": "0",
      "chartScript": {
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    }
  ],
  "edges": [
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flowchart RL
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  x["$x$"]:::scalar
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  y["$y$"]:::scalar
  __deriv_4(("$\dfrac{\partial^{2}}{\partial y^{2}}$")):::operator
  __num_5["$0$"]:::number
  phi --> __deriv_3
  x -->|wrt| __deriv_3
  __deriv_3 --> __add_2
  phi --> __deriv_4
  y -->|wrt| __deriv_4
  __deriv_4 --> __add_2
  __add_2 --> __equals_1
  __num_5 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
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{
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      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "i \\hbar \\frac{\\partial \\psi}{\\partial t} = H \\psi",
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      }
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      "chartScript": {
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    {
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    },
    {
      "id": "hbar",
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      "latex": "\\hbar",
      "subexpr": "\\hbar",
      "chartScript": {
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        "variables": [
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    {
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      "chartScript": {
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        "variables": []
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    },
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      "type": "operator",
      "op": "multiply",
      "subexpr": "\\psi H",
      "chartScript": {
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        "variables": [
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          "psi"
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    },
    {
      "id": "H",
      "type": "scalar",
      "latex": "H",
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      "chartScript": {
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        "variables": [
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    {
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  "classification": {
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    "order": 1,
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    "independent_variables": [
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    "sympy_hints": [
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      "1st_linear",
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      "nth_linear_constant_coeff_homogeneous"
    ],
    "linear": true,
    "homogeneous": true,
    "constant_coefficients": true
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}

flowchart RL
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  __multiply_3(("$\times$")):::operator
  hbar["$\hbar$"]:::scalar
  psi["$\psi$"]:::scalar
  t["$t$"]:::scalar
  __deriv_4(("$\dfrac{\partial}{\partial t}$")):::operator
  __multiply_5(("$\times$")):::operator
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  i --> __multiply_2
  hbar --> __multiply_3
  psi --> __deriv_4
  t -->|wrt| __deriv_4
  __deriv_4 --> __multiply_3
  __multiply_3 --> __multiply_2
  __multiply_2 --> __equals_1
  H --> __multiply_5
  psi --> __multiply_5
  __multiply_5 --> __equals_1
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{
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    "linear": true,
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}

flowchart RL
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  t["$t$"]:::scalar
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  __multiply_4(("$\times$")):::operator
  nabla["$\nabla$"]:::scalar
  __multiply_5(("$\times$")):::operator
  v["$v$"]:::scalar
  __num_6["$0$"]:::number
  rho --> __deriv_3
  t -->|wrt| __deriv_3
  __deriv_3 --> __add_2
  nabla --> __multiply_4
  rho --> __multiply_5
  v --> __multiply_5
  __multiply_5 --> __multiply_4
  __multiply_4 --> __add_2
  __add_2 --> __equals_1
  __num_6 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
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{
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flowchart RL
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  t["$t$"]:::scalar
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  __multiply_4(("$\times$")):::operator
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  __multiply_6(("$\times$")):::operator
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  t -->|wrt| __deriv_3
  __deriv_3 --> __add_2
  u --> __multiply_4
  u --> __deriv_5
  x -->|wrt| __deriv_5
  __deriv_5 --> __multiply_4
  __multiply_4 --> __add_2
  __add_2 --> __equals_1
  nu --> __multiply_6
  u --> __deriv_7
  x -->|wrt| __deriv_7
  __deriv_7 --> __multiply_6
  __multiply_6 --> __equals_1
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      "type": "relation",
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      "latex": "c",
      "subexpr": "c",
      "chartScript": {
        "script": "c",
        "variables": [
          "c"
        ]
      }
    },
    {
      "id": "x",
      "type": "scalar",
      "latex": "x",
      "subexpr": "x",
      "chartScript": {
        "script": "x",
        "variables": [
          "x"
        ]
      }
    },
    {
      "id": "__deriv_8",
      "type": "operator",
      "op": "partial_derivative",
      "with_respect_to": "x",
      "subexpr": "\\frac{\\partial^{2} \\phi}{\\partial x^{2}}",
      "chartScript": {
        "script": "partial*phi/pow(x, 2)",
        "variables": [
          "partial",
          "phi",
          "x"
        ]
      }
    },
    {
      "id": "__multiply_9",
      "type": "operator",
      "op": "multiply",
      "subexpr": "\\phi m^{2}",
      "chartScript": {
        "script": "pow(m, 2)*phi",
        "variables": [
          "m",
          "phi"
        ]
      }
    },
    {
      "id": "__power_10",
      "type": "operator",
      "op": "power",
      "exponent": "2",
      "subexpr": "m^{2}",
      "chartScript": {
        "script": "pow(m, 2)",
        "variables": [
          "m"
        ]
      }
    },
    {
      "id": "m",
      "type": "scalar",
      "latex": "m",
      "subexpr": "m",
      "chartScript": {
        "script": "m",
        "variables": [
          "m"
        ]
      }
    },
    {
      "id": "__num_11",
      "type": "number",
      "label": "0",
      "subexpr": "0",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    }
  ],
  "edges": [
    {
      "from": "phi",
      "to": "__deriv_4"
    },
    {
      "from": "t",
      "to": "__deriv_4",
      "role": "wrt"
    },
    {
      "from": "__deriv_4",
      "to": "__add_3"
    },
    {
      "from": "c",
      "to": "__power_7"
    },
    {
      "from": "__power_7",
      "to": "__multiply_6",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "phi",
      "to": "__deriv_8"
    },
    {
      "from": "x",
      "to": "__deriv_8",
      "role": "wrt"
    },
    {
      "from": "__deriv_8",
      "to": "__multiply_6",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_6",
      "to": "__negation_5"
    },
    {
      "from": "__negation_5",
      "to": "__add_3"
    },
    {
      "from": "__add_3",
      "to": "__add_2"
    },
    {
      "from": "m",
      "to": "__power_10"
    },
    {
      "from": "__power_10",
      "to": "__multiply_9",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "phi",
      "to": "__multiply_9",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_9",
      "to": "__add_2"
    },
    {
      "from": "__add_2",
      "to": "__equals_1"
    },
    {
      "from": "__num_11",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "PDE",
    "order": 2,
    "dependent_variables": [
      "phi"
    ],
    "independent_variables": [
      "t",
      "x"
    ]
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  __add_2(("$+$")):::operator
  __add_3(("$+$")):::operator
  phi["$\phi$"]:::scalar
  t["$t$"]:::scalar
  __deriv_4(("$\dfrac{\partial^{2}}{\partial t^{2}}$")):::operator
  __negation_5@{ shape: "flip-tri", label: "$-$" }
  __multiply_6(("$\times$")):::operator
  __power_7(("${(\cdot)}^{2}$")):::operator
  c["$c$"]:::scalar
  x["$x$"]:::scalar
  __deriv_8(("$\dfrac{\partial^{2}}{\partial x^{2}}$")):::operator
  __multiply_9(("$\times$")):::operator
  __power_10(("${(\cdot)}^{2}$")):::operator
  m["$m$"]:::scalar
  __num_11["$0$"]:::number
  class __negation_5 operator
  phi --> __deriv_4
  t -->|wrt| __deriv_4
  __deriv_4 --> __add_3
  c --> __power_7
  __power_7 --> __multiply_6
  phi --> __deriv_8
  x -->|wrt| __deriv_8
  __deriv_8 --> __multiply_6
  __multiply_6 --> __negation_5
  __negation_5 --> __add_3
  __add_3 --> __add_2
  m --> __power_10
  __power_10 --> __multiply_9
  phi --> __multiply_9
  __multiply_9 --> __add_2
  __add_2 --> __equals_1
  __num_11 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#ef5350,stroke-width:3px
  linkStyle 5 stroke:#aaa,stroke-width:2px
  linkStyle 6 stroke:#aaa,stroke-width:2px
  linkStyle 7 stroke:#ef5350,stroke-width:3px
  linkStyle 8 stroke:#aaa,stroke-width:2px
  linkStyle 9 stroke:#aaa,stroke-width:2px
  linkStyle 10 stroke:#aaa,stroke-width:2px
  linkStyle 11 stroke:#aaa,stroke-width:2px
  linkStyle 12 stroke:#ef5350,stroke-width:3px
  linkStyle 13 stroke:#ef5350,stroke-width:3px
  linkStyle 14 stroke:#aaa,stroke-width:2px
  linkStyle 15 stroke:#aaa,stroke-width:2px
  linkStyle 16 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "\\frac{\\partial z}{\\partial x} = \\frac{\\partial z}{\\partial u} \\frac{\\partial u}{\\partial x} + \\frac{\\partial z}{\\partial v} \\frac{\\partial v}{\\partial x}",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    },
    {
      "id": "z",
      "type": "scalar",
      "latex": "z",
      "subexpr": "z",
      "chartScript": {
        "script": "z",
        "variables": [
          "z"
        ]
      }
    },
    {
      "id": "x",
      "type": "scalar",
      "latex": "x",
      "subexpr": "x",
      "chartScript": {
        "script": "x",
        "variables": [
          "x"
        ]
      }
    },
    {
      "id": "__deriv_2",
      "type": "operator",
      "op": "partial_derivative",
      "with_respect_to": "x",
      "subexpr": "\\frac{\\partial z}{\\partial x}",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    },
    {
      "id": "__add_3",
      "type": "operator",
      "op": "add",
      "subexpr": "\\frac{\\partial z}{\\partial u} \\frac{\\partial u}{\\partial x} + \\frac{\\partial z}{\\partial v} \\frac{\\partial v}{\\partial x}",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    },
    {
      "id": "__multiply_4",
      "type": "operator",
      "op": "multiply",
      "subexpr": "\\frac{\\partial z}{\\partial u} \\frac{\\partial u}{\\partial x}",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    },
    {
      "id": "u",
      "type": "scalar",
      "latex": "u",
      "subexpr": "u",
      "chartScript": {
        "script": "u",
        "variables": [
          "u"
        ]
      }
    },
    {
      "id": "__deriv_5",
      "type": "operator",
      "op": "partial_derivative",
      "with_respect_to": "u",
      "subexpr": "\\frac{\\partial z}{\\partial u}",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    },
    {
      "id": "__deriv_6",
      "type": "operator",
      "op": "partial_derivative",
      "with_respect_to": "x",
      "subexpr": "\\frac{\\partial u}{\\partial x}",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    },
    {
      "id": "__multiply_7",
      "type": "operator",
      "op": "multiply",
      "subexpr": "\\frac{\\partial z}{\\partial v} \\frac{\\partial v}{\\partial x}",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    },
    {
      "id": "v",
      "type": "scalar",
      "latex": "v",
      "subexpr": "v",
      "chartScript": {
        "script": "v",
        "variables": [
          "v"
        ]
      }
    },
    {
      "id": "__deriv_8",
      "type": "operator",
      "op": "partial_derivative",
      "with_respect_to": "v",
      "subexpr": "\\frac{\\partial z}{\\partial v}",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    },
    {
      "id": "__deriv_9",
      "type": "operator",
      "op": "partial_derivative",
      "with_respect_to": "x",
      "subexpr": "\\frac{\\partial v}{\\partial x}",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    }
  ],
  "edges": [
    {
      "from": "z",
      "to": "__deriv_2"
    },
    {
      "from": "x",
      "to": "__deriv_2",
      "role": "wrt"
    },
    {
      "from": "__deriv_2",
      "to": "__equals_1"
    },
    {
      "from": "z",
      "to": "__deriv_5"
    },
    {
      "from": "u",
      "to": "__deriv_5",
      "role": "wrt"
    },
    {
      "from": "__deriv_5",
      "to": "__multiply_4",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "u",
      "to": "__deriv_6"
    },
    {
      "from": "x",
      "to": "__deriv_6",
      "role": "wrt"
    },
    {
      "from": "__deriv_6",
      "to": "__multiply_4",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_4",
      "to": "__add_3"
    },
    {
      "from": "z",
      "to": "__deriv_8"
    },
    {
      "from": "v",
      "to": "__deriv_8",
      "role": "wrt"
    },
    {
      "from": "__deriv_8",
      "to": "__multiply_7",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "v",
      "to": "__deriv_9"
    },
    {
      "from": "x",
      "to": "__deriv_9",
      "role": "wrt"
    },
    {
      "from": "__deriv_9",
      "to": "__multiply_7",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_7",
      "to": "__add_3"
    },
    {
      "from": "__add_3",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "PDE",
    "order": 1,
    "dependent_variables": [
      "u",
      "v",
      "z"
    ],
    "independent_variables": [
      "u",
      "v",
      "x"
    ]
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  z["$z$"]:::scalar
  x["$x$"]:::scalar
  __deriv_2(("$\dfrac{\partial}{\partial x}$")):::operator
  __add_3(("$+$")):::operator
  __multiply_4(("$\times$")):::operator
  u["$u$"]:::scalar
  __deriv_5(("$\dfrac{\partial}{\partial u}$")):::operator
  __deriv_6(("$\dfrac{\partial}{\partial x}$")):::operator
  __multiply_7(("$\times$")):::operator
  v["$v$"]:::scalar
  __deriv_8(("$\dfrac{\partial}{\partial v}$")):::operator
  __deriv_9(("$\dfrac{\partial}{\partial x}$")):::operator
  z --> __deriv_2
  x -->|wrt| __deriv_2
  __deriv_2 --> __equals_1
  z --> __deriv_5
  u -->|wrt| __deriv_5
  __deriv_5 --> __multiply_4
  u --> __deriv_6
  x -->|wrt| __deriv_6
  __deriv_6 --> __multiply_4
  __multiply_4 --> __add_3
  z --> __deriv_8
  v -->|wrt| __deriv_8
  __deriv_8 --> __multiply_7
  v --> __deriv_9
  x -->|wrt| __deriv_9
  __deriv_9 --> __multiply_7
  __multiply_7 --> __add_3
  __add_3 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#aaa,stroke-width:2px
  linkStyle 5 stroke:#ef5350,stroke-width:3px
  linkStyle 6 stroke:#aaa,stroke-width:2px
  linkStyle 7 stroke:#aaa,stroke-width:2px
  linkStyle 8 stroke:#ef5350,stroke-width:3px
  linkStyle 9 stroke:#aaa,stroke-width:2px
  linkStyle 10 stroke:#aaa,stroke-width:2px
  linkStyle 11 stroke:#aaa,stroke-width:2px
  linkStyle 12 stroke:#ef5350,stroke-width:3px
  linkStyle 13 stroke:#aaa,stroke-width:2px
  linkStyle 14 stroke:#aaa,stroke-width:2px
  linkStyle 15 stroke:#ef5350,stroke-width:3px
  linkStyle 16 stroke:#aaa,stroke-width:2px
  linkStyle 17 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "dz = \\frac{\\partial z}{\\partial x} dx + \\frac{\\partial z}{\\partial y} dy",
      "chartScript": {
        "script": "dz",
        "variables": [
          "dz"
        ]
      }
    },
    {
      "id": "dz",
      "type": "scalar",
      "latex": "dz",
      "subexpr": "dz",
      "chartScript": {
        "script": "dz",
        "variables": [
          "dz"
        ]
      }
    },
    {
      "id": "__add_2",
      "type": "operator",
      "op": "add",
      "subexpr": "\\frac{\\partial z}{\\partial x} dx + \\frac{\\partial z}{\\partial y} dy",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    },
    {
      "id": "__multiply_3",
      "type": "operator",
      "op": "multiply",
      "subexpr": "\\frac{\\partial z}{\\partial x} dx",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    },
    {
      "id": "z",
      "type": "scalar",
      "latex": "z",
      "subexpr": "z",
      "chartScript": {
        "script": "z",
        "variables": [
          "z"
        ]
      }
    },
    {
      "id": "x",
      "type": "scalar",
      "latex": "x",
      "subexpr": "x",
      "chartScript": {
        "script": "x",
        "variables": [
          "x"
        ]
      }
    },
    {
      "id": "__deriv_4",
      "type": "operator",
      "op": "partial_derivative",
      "with_respect_to": "x",
      "subexpr": "\\frac{\\partial z}{\\partial x}",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    },
    {
      "id": "dx",
      "type": "scalar",
      "latex": "dx",
      "subexpr": "dx",
      "chartScript": {
        "script": "dx",
        "variables": [
          "dx"
        ]
      }
    },
    {
      "id": "__multiply_5",
      "type": "operator",
      "op": "multiply",
      "subexpr": "\\frac{\\partial z}{\\partial y} dy",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    },
    {
      "id": "y",
      "type": "scalar",
      "latex": "y",
      "subexpr": "y",
      "chartScript": {
        "script": "y",
        "variables": [
          "y"
        ]
      }
    },
    {
      "id": "__deriv_6",
      "type": "operator",
      "op": "partial_derivative",
      "with_respect_to": "y",
      "subexpr": "\\frac{\\partial z}{\\partial y}",
      "chartScript": {
        "script": "0",
        "variables": []
      }
    },
    {
      "id": "dy",
      "type": "scalar",
      "latex": "dy",
      "subexpr": "dy",
      "chartScript": {
        "script": "dy",
        "variables": [
          "dy"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "dz",
      "to": "__equals_1"
    },
    {
      "from": "z",
      "to": "__deriv_4"
    },
    {
      "from": "x",
      "to": "__deriv_4",
      "role": "wrt"
    },
    {
      "from": "__deriv_4",
      "to": "__multiply_3",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "dx",
      "to": "__multiply_3",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_3",
      "to": "__add_2"
    },
    {
      "from": "z",
      "to": "__deriv_6"
    },
    {
      "from": "y",
      "to": "__deriv_6",
      "role": "wrt"
    },
    {
      "from": "__deriv_6",
      "to": "__multiply_5",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "dy",
      "to": "__multiply_5",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_5",
      "to": "__add_2"
    },
    {
      "from": "__add_2",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "PDE",
    "order": 1,
    "dependent_variables": [
      "z"
    ],
    "independent_variables": [
      "x",
      "y"
    ]
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  dz["$dz$"]:::scalar
  __add_2(("$+$")):::operator
  __multiply_3(("$\times$")):::operator
  z["$z$"]:::scalar
  x["$x$"]:::scalar
  __deriv_4(("$\dfrac{\partial}{\partial x}$")):::operator
  dx["$dx$"]:::scalar
  __multiply_5(("$\times$")):::operator
  y["$y$"]:::scalar
  __deriv_6(("$\dfrac{\partial}{\partial y}$")):::operator
  dy["$dy$"]:::scalar
  dz --> __equals_1
  z --> __deriv_4
  x -->|wrt| __deriv_4
  __deriv_4 --> __multiply_3
  dx --> __multiply_3
  __multiply_3 --> __add_2
  z --> __deriv_6
  y -->|wrt| __deriv_6
  __deriv_6 --> __multiply_5
  dy --> __multiply_5
  __multiply_5 --> __add_2
  __add_2 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#ef5350,stroke-width:3px
  linkStyle 4 stroke:#ef5350,stroke-width:3px
  linkStyle 5 stroke:#aaa,stroke-width:2px
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