diffop expressions in various systems
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A short working reference document by Phil, dated 11/15/13, started because he wanted these formulas ready to paste into other documents. It lists grad, div and curl in Cartesian and cylindrical coordinates, plus the scalar and vector Laplacian in cylindrical form, and grad and div in spherical coordinates. The spherical section appears unfinished, and the extracted text has some garbled symbols.
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Diffop expressions in various systems PhL 11.15.13
Tired of not having this ready to paste. Starting 11/15/13 to build this doc.
Cartesian Coordinates (x,y,z0
F = Fx + Fx + Fx
grad ψ = ∂xψ + ∂yψ + ∂zψ
div F = ∂xFx + ∂yFy + ∂zFz
curl F = (∂yFz - ∂zFy) + (∂zFx - ∂xFz) + (∂xFy - ∂yFx) // each term cyclic x,y,z
Cylindrical Coordinates (r,θ,z)
F = Fr + Fθ + Fz
grad ψ = ∂rψ + r-1∂θψ + ∂zψ
div F = r-1∂r(rFr) + r-1∂θFθ + ∂zFz
curl F = [ r-1∂θFz - ∂zFθ] + [∂zFr - ∂rFz] + [ r-1∂r(rFθ) - r-1∂θFr ]
2ψ = r-1∂r(r∂rψ) + r-2∂θ2ψ + ∂z2ψ
2F = [2Fr -2 r-2∂θFθ - r-2Fr] + [2Fr +2 r-2∂θFr - r-2Fθ] + 2Fz
Spherical coordinates (r,θ,φ)
grad ψ = ∂rψ + (1/r)∂θψ + [rsinθ]-1 ∂φψ
div F = r-2∂r(r2Fr) + [rsinθ]-1∂θ[sinθFθ] + [rsinθ]-1∂φFφ