In cylindrical coordinates a steady flow has coordinate (contravariant) velocity…

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 steady flow has coordinate (contravariant) velocity components Fr=2.0 m s−1F^r=2.0\ \mathrm{m\,s^{-1}}, Fθ=0.50 s−1F^\theta=0.50\ \mathrm{s^{-1}}, Fz=1.0 m s−1F^z=1.0\ \mathrm{m\,s^{-1}} at the point r=4.0 mr=4.0\ \mathrm{m}. ((a)) Write the physical (unit-vector) components Fr^F_{\hat r}, Fθ^F_{\hat\theta}, Fz^F_{\hat z} at this point, with their units. ((b)) State which of the six numbers above is directly what a velocity probe aligned along θ^\hat{\boldsymbol\theta} would read, and why.

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