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  "title": "06 - Gradient, Divergence and Curl",
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    "xg": "x_0",
    "yg": "y_0",
    "gradV_at": "\\nabla V",
    "Egrad_at": "\\mathbf{E}",
    "xdiv": "x",
    "divA": "\\nabla\\cdot\\mathbf{A}",
    "curl_z": "(\\nabla\\times\\mathbf{A})_z",
    "aCube": "a",
    "div_volume": "\\int_V(\\nabla\\cdot\\mathbf{A})\\,\\mathrm{d}V",
    "surface_flux": "\\oint_S\\mathbf{A}\\cdot\\mathrm{d}\\mathbf{S}",
    "divergence_theorem_residual": "\\varepsilon_{\\mathrm{div}}",
    "Rcirc": "R",
    "circulation": "\\oint_C\\mathbf{A}\\cdot\\mathrm{d}\\mathbf{l}",
    "curl_flux": "\\int_S(\\nabla\\times\\mathbf{A})\\cdot\\mathrm{d}\\mathbf{S}",
    "stokes_residual": "\\varepsilon_{\\mathrm{Stokes}}",
    "lapV": "\\nabla^2 V",
    "hfd": "h",
    "xfd": "x_0",
    "fd_second": "V^{\\prime\\prime}_{\\mathrm{FD}}",
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      "content": "06 - Gradient, Divergence and Curl",
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      "content": "Interactive M²S companion to EM Course Document 06.\nLocal differential operators, integral-theorem checks, Laplacian identities and finite differences.\nEdit the input Math regions, then Calculate. M²S values are unitless, so SI units are stated explicitly in labels and notes.",
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      "content": "In Cartesian coordinates the course uses grad f = (∂f/∂x,∂f/∂y,∂f/∂z), div A = ∂Ax/∂x+∂Ay/∂y+∂Az/∂z, and curl A as the local circulation operator. Gradient maps scalar→vector; divergence maps vector→scalar; curl maps vector→vector.",
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      "id": "ch06-head-004",
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      "content": "1. Gradient: steepest increase and electrostatic field",
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    },
    {
      "id": "ch06-note-005",
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      "x": 65,
      "y": 511,
      "w": 990,
      "content": "Use V=x²+0.5y². Its gradient is (2x,y), and an electrostatic E field is -grad(V).",
      "heading": false
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      "content": "2. Divergence: local sources and sinks",
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    },
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      "id": "ch06-note-013",
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      "content": "Course checkpoint field A=(x²,2y,-z) has divergence 2x+2-1 = 2x+1.",
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      "expr": "0.75"
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      "id": "ch06-math-015",
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      "w": 990,
      "target": "divA",
      "expr": "2*xdiv+1"
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      "title": "Field A=(x²,2y,-z)",
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      "id": "ch06-head-017",
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      "content": "3. Curl: local circulation",
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    },
    {
      "id": "ch06-note-018",
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      "x": 65,
      "y": 3481,
      "w": 990,
      "content": "For A=(-y,x,0), curl(A)=2 zhat. Field lines circulate around z, and the local curl points along +z.",
      "heading": false
    },
    {
      "id": "ch06-math-019",
      "type": "math",
      "x": 65,
      "y": 3596,
      "w": 990,
      "target": "curl_z",
      "expr": "2"
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      "id": "ch06-plot-020",
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      "title": "Rotational field A=(-y,x)",
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      "content": "4. Divergence theorem: flux equals integrated divergence",
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      "id": "ch06-note-022",
      "type": "text",
      "x": 65,
      "y": 4441,
      "w": 990,
      "content": "For F=(x,y,z), div(F)=3. Over the cube [-a,a]^3 the volume integral is 3(2a)^3, and the outward flux is 6 faces × 4a³ = 24a³.",
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      "expr": "1.2"
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      "id": "ch06-math-024",
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      "w": 990,
      "target": "div_volume",
      "expr": "3*(2*aCube)^3"
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      "headersText": "Value",
      "rowLabels": "Volume integral of div(F)\nClosed-surface flux\nResidual",
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    },
    {
      "id": "ch06-head-028",
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      "y": 5586,
      "w": 990,
      "content": "5. Stokes theorem: circulation equals curl flux",
      "heading": true
    },
    {
      "id": "ch06-math-029",
      "type": "math",
      "x": 65,
      "y": 5691,
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      "target": "Rcirc",
      "expr": "1.3"
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    {
      "id": "ch06-math-030",
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      "y": 5856,
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      "expr": "2*pi*Rcirc^2"
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      "id": "ch06-math-031",
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      "y": 6021,
      "w": 990,
      "target": "curl_flux",
      "expr": "curl_z*pi*Rcirc^2"
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      "id": "ch06-math-032",
      "type": "math",
      "x": 65,
      "y": 6186,
      "w": 990,
      "target": "stokes_residual",
      "expr": "circulation-curl_flux"
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      "title": "Stokes theorem check",
      "headersText": "Value",
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      "id": "ch06-head-034",
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      "y": 6721,
      "w": 990,
      "content": "6. Laplacian and finite difference",
      "heading": true
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    {
      "id": "ch06-math-035",
      "type": "math",
      "x": 65,
      "y": 6826,
      "w": 990,
      "target": "lapV",
      "expr": "2+1"
    },
    {
      "id": "ch06-math-036",
      "type": "math",
      "x": 65,
      "y": 6991,
      "w": 475,
      "target": "hfd",
      "expr": "0.02"
    },
    {
      "id": "ch06-math-037",
      "type": "math",
      "x": 560,
      "y": 6991,
      "w": 475,
      "target": "xfd",
      "expr": "0.7"
    },
    {
      "id": "ch06-math-038",
      "type": "math",
      "x": 65,
      "y": 7156,
      "w": 990,
      "target": "fd_second",
      "expr": "((xfd+hfd)^2-2*xfd^2+(xfd-hfd)^2)/hfd^2"
    },
    {
      "id": "ch06-math-039",
      "type": "math",
      "x": 65,
      "y": 7321,
      "w": 990,
      "target": "fd_error",
      "expr": "fd_second-2"
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      "id": "ch06-table-040",
      "type": "table",
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      "y": 7486,
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      "h": 280,
      "title": "Laplacian / finite difference",
      "headersText": "Value",
      "rowLabels": "∇²(x²+0.5y²)\nFinite-difference d²(x²)/dx²\nFD error",
      "expr": "cols([lapV,fd_second,fd_error])",
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    },
    {
      "id": "ch06-head-041",
      "type": "text",
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      "w": 990,
      "content": "Checkpoint / exploration",
      "heading": true
    },
    {
      "id": "ch06-note-042",
      "type": "text",
      "x": 65,
      "y": 7961,
      "w": 990,
      "content": "M²S does not provide automatic grad/div/curl operators, so the component formulas are written explicitly. This is intentional: the worksheet makes the local derivatives visible rather than hiding them behind an operator.",
      "heading": false
    }
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}