In cylindrical coordinates a flow has physical components and

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

In cylindrical coordinates (r,θ,z)(r,\theta,z) a flow has physical components Fr^=Fz^=0F_{\hat r}=F_{\hat z}=0 and Fθ^=C/r+krF_{\hat\theta}=C/r+kr, with C=0.20 m2 s−1C=0.20\ \mathrm{m^2\,s^{-1}} and k=0.50 s−1k=0.50\ \mathrm{s^{-1}}, on the region r>0r>0. ((a)) Write the covariant components FrF_r, FθF_\theta, FzF_z, and compute ∇×F\nabla\times\mathbf F from the curl determinant over g\sqrt g. ((b)) Now set k=0k=0. Find the circulation of F\mathbf F once anticlockwise round the circle r=1.0 mr=1.0\ \mathrm{m}, z=0z=0. ((c)) With k=0k=0, say on which of the two regions (i) r>0r>0 and (ii) r>0r>0 with the half-plane θ=π\theta=\pi removed the condition for a curl-free field to have a single-valued potential holds, and give that potential where it exists.

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