Cylindrical coordinates are related to Cartesian coordinates by , ,

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Tensors and differential geometry

Invariance, Ch Differential forms and the exterior derivative

Problem

Cylindrical coordinates (ρ,φ,z)(\rho,\varphi,z) are related to Cartesian coordinates by x=ρcos⁡φx=\rho\cos\varphi, y=ρsin⁡φy=\rho\sin\varphi, z=zz=z, with ρ>0\rho \gt 0. The line element is ds2=dρ2+ρ2dφ2+dz2ds^2=d\rho^2+\rho^2d\varphi^2+dz^2 (held from the curvilinear-coordinates course), so the orthonormal coframe is e1=dρe^1=d\rho, e2=ρ dφe^2=\rho\,d\varphi, e3=dze^3=dz. A vector field is written in this frame as F=Fρ(ρ,z) eρ+Fz(ρ,z) ez\mathbf F=F_\rho(\rho,z)\,\mathbf e_\rho+F_z(\rho,z)\,\mathbf e_z (no φ\varphi-dependence, and Fφ=0F_\varphi=0), with dual one-form α=Fρe1+Fφe2+Fze3\alpha=F_\rho e^1+F_\varphi e^2+F_z e^3. ((a)) Starting from dxdx and dydy in terms of dρd\rho and dφd\varphi, derive dx∧dy=ρ dρ∧dφdx\wedge dy=\rho\,d\rho\wedge d\varphi, and hence write the volume form dx∧dy∧dzdx\wedge dy\wedge dz in cylindrical coordinates. ((b)) Using the Hodge star of R3\mathbb R^3, with ⋆e1=e2∧e3{\star}e^1=e^2\wedge e^3, ⋆e2=e3∧e1{\star}e^2=e^3\wedge e^1, ⋆e3=e1∧e2{\star}e^3=e^1\wedge e^2 and ⋆⋆=1{\star}{\star}=1, derive the formula for ∇⋅F=⋆d⋆α\nabla\cdot\mathbf F={\star}d{\star}\alpha in terms of ∂ρ,∂φ,∂z\partial_\rho,\partial_\varphi,\partial_z acting on ρFρ\rho F_\rho, FφF_\varphi and ρFz\rho F_z. ((c)) Apply the formula to Fρ=ρ2F_\rho=\rho^2, Fφ=0F_\varphi=0, Fz=z2F_z=z^2 (coordinates and components taken as pure numbers for this exercise) and evaluate ∇⋅F\nabla\cdot\mathbf F at ρ=2\rho=2, z=3z=3.

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