'Log-polar' cylindrical coordinates

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

Conditions of Validity, Ch Gradient, divergence and curl without coordinates, and with any coordinates

Problem

'Log-polar' cylindrical coordinates (τ,θ,z)(\tau,\theta,z), defined by r=eτr=e^\tau, have line element ds2=e2τ(dτ2+dθ2)+dz2.\mathrm{d}s^2=e^{2\tau}(\mathrm{d}\tau^2+\mathrm{d}\theta^2)+\mathrm{d}z^2. ((a)) State briefly why ∇⋅(∇×G)=0\nabla\cdot(\nabla\times\mathbf G)=0 for any twice-differentiable vector field G\mathbf G, in any coordinate system, without recomputing it here. ((b)) Write the general Laplacian ∇2f\nabla^2 f for (τ,θ,z)(\tau,\theta,z), using gijg^{ij} and g\sqrt g read off from the line element above. ((c)) Evaluate ∇2f\nabla^2 f for f=τf=\tau and check the result against the known Cartesian fact that ln⁡r\ln r is harmonic for r>0r>0.

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