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  "title": "03 - Vector Algebra and Coordinate Systems",
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    "A": "\\mathbf{A}",
    "Amag": "\\left|\\mathbf{A}\\right|",
    "Ahat": "\\hat{\\mathbf{A}}",
    "Ahat_check": "\\left|\\hat{\\mathbf{A}}\\right|",
    "Avec": "\\mathbf{A}",
    "Bvec": "\\mathbf{B}",
    "AdotB": "\\mathbf{A}\\cdot\\mathbf{B}",
    "angleAB": "\\theta_{AB}",
    "B_parallel_scalar": "B_{\\parallel}",
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    "Evec": "\\mathbf{E}",
    "Hvec": "\\mathbf{H}",
    "Svec": "\\mathbf{S}",
    "rho0": "\\rho",
    "r0": "r",
    "drho": "\\mathrm{d}\\rho",
    "dphi": "\\mathrm{d}\\phi",
    "dz": "\\mathrm{d}z",
    "dtheta": "\\mathrm{d}\\theta",
    "dV_cyl": "\\mathrm{d}V_{\\mathrm{cyl}}",
    "dV_sph": "\\mathrm{d}V_{\\mathrm{sph}}"
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      "content": "Interactive M²S companion to EM Course Document 03.\nVector components, projections, dot/cross products and Cartesian/cylindrical/spherical geometry.\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": "1. Components, magnitude and unit direction",
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      "title": "Projection geometry",
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      "content": "3. Cross product and EM power direction",
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      "id": "ch03-table-021",
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      "title": "Poynting-vector direction",
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      "id": "ch03-head-022",
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      "content": "4. Cartesian vector field",
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      "id": "ch03-plot-023",
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      "title": "3D Cartesian vector samples: F=(x,y,z)",
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      "content": "5. Cylindrical coordinates: azimuthal field around a wire",
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      "content": "For cylindrical symmetry, the local azimuthal direction changes with position. The field below is proportional to φ-hat and illustrates why cylindrical unit vectors are not globally fixed.",
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      "content": "6. Spherical geometry and radial fields",
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      "target": "dV_sph",
      "expr": "r0^2*sin(pi/3)*drho*dtheta*dphi"
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      "id": "ch03-table-038",
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      "title": "Coordinate Jacobian examples",
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    {
      "id": "ch03-head-039",
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      "content": "Checkpoint / exploration",
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      "id": "ch03-note-040",
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      "w": 990,
      "content": "Change the coordinate-grid limits and compare the radial and azimuthal 3D/2D arrows. The field is physical; the coordinate description is a choice made to fit the geometry.",
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}