A sensor on a rotating drum sits at cylindrical polar coordinates , , .

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

Mathematical Methods for Physics, Ch Curvilinear and orthogonal coordinates

Problem

A sensor on a rotating drum sits at cylindrical polar coordinates ρ=2 m\rho = 2\,\mathrm{m}, ϕ=π/4 rad\phi = \pi/4\,\mathrm{rad}, z=1 mz = 1\,\mathrm{m}. ((a)) Write the three scale factors hρh_\rho, hϕh_\phi, hzh_z at this point. ((b)) Give the Cartesian components of ∂ϕ^/∂ϕ\partial\hat{\boldsymbol{\phi}}/\partial\phi at this point. ((c)) Using the scale factors, find the volume dV\mathrm{d}V of the small curvilinear cell swept out by changing the coordinates by dρ=0.050 m\mathrm{d}\rho = 0.050\,\mathrm{m}, dϕ=0.10 rad\mathrm{d}\phi = 0.10\,\mathrm{rad}, dz=0.20 m\mathrm{d}z = 0.20\,\mathrm{m} from this point.

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