Cylindrical coordinates are , ,

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Vector calculus and field geometry

Mathematical Methods for Physics, Ch Curvilinear and orthogonal coordinates

Problem

Cylindrical coordinates are x=ρcos⁡ϕx = \rho\cos\phi, y=ρsin⁡ϕy = \rho\sin\phi, z=zz = z. Consider the point ρ=3 m\rho = 3\,\mathrm{m}, ϕ=π/6 rad\phi = \pi/6\,\mathrm{rad}, z=0 mz = 0\,\mathrm{m}. ((a)) Form the 3×33\times3 Jacobian matrix ∂(x,y,z)/∂(ρ,ϕ,z)\partial(x,y,z)/\partial(\rho,\phi,z) and evaluate its determinant JJ at this point. ((b)) Compute the product hρhϕhzh_\rho h_\phi h_z of the cylindrical scale factors at this point and confirm it equals ∣J∣|J|. ((c)) Write ρ^\hat{\boldsymbol\rho} and ϕ^\hat{\boldsymbol\phi} in Cartesian components at ϕ=π/6 rad\phi = \pi/6\,\mathrm{rad}, evaluate ρ^×ϕ^\hat{\boldsymbol\rho}\times\hat{\boldsymbol\phi} by the determinant rule, check its direction with the right-hand rule, and confirm the system is right-handed with ρ^×ϕ^=z^\hat{\boldsymbol\rho}\times\hat{\boldsymbol\phi} = \hat{\mathbf{z}}.

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