Parabolic cylindrical coordinates have line element d s^2=(u^2+v^2)( d u^2+ d v^2)+ d z^2.

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

Parabolic cylindrical coordinates (u,v,z)(u,v,z) have line element ds2=(u2+v2)(du2+dv2)+dz2.\mathrm{d}s^2=(u^2+v^2)(\mathrm{d}u^2+\mathrm{d}v^2)+\mathrm{d}z^2. ((a)) State briefly why ∇×(∇f)=0\nabla\times(\nabla f)=\mathbf 0 for any twice-differentiable scalar function ff, in any coordinate system, without recomputing the identity here. ((b)) Write the general Laplacian ∇2f\nabla^2 f for this coordinate system, using gijg^{ij} and g\sqrt g read off from the line element above. ((c)) Evaluate ∇2f\nabla^2 f for f=u2−v2f=u^2-v^2 (with u2+v2≠0u^2+v^2\neq 0) and state what the result shows about ff as a candidate electrostatic potential in a charge-free region.

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