A steady flow has velocity field

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

Conditions of Validity, Ch Gauss, Stokes and Green proved, and the first inversion of a differential operator

Problem

A steady flow has velocity field F(r)=q r/(r2+a2)3/2\mathbf F(\mathbf r)=q\,\mathbf r/(r^2+a^2)^{3/2}, where r\mathbf r is the position vector, r=∣r∣r=|\mathbf r|, q=2.0 m3 s−1q=2.0\,\mathrm{m^3\,s^{-1}} and a=0.50 ma=0.50\,\mathrm m. It is a point source smeared over a region of size aa. ((a)) Show that ∇×F=0\nabla\times\mathbf F=\mathbf 0 and ∇⋅F=3qa2/(r2+a2)5/2\nabla\cdot\mathbf F=3qa^2/(r^2+a^2)^{5/2}, and evaluate ∇⋅F\nabla\cdot\mathbf F at the origin. Explain why F\mathbf F satisfies every hypothesis of the Helmholtz key result. ((b)) Write down A\mathbf A from the integral formulas of the Helmholtz decomposition, and evaluate ϕ\phi at the origin from its integral formula, working in spherical coordinates about the origin. ((c)) Check that ψ=q/r2+a2\psi=q/\sqrt{r^2+a^2} satisfies −∇ψ=F-\nabla\psi=\mathbf F and that ψ(0)\psi(\mathbf 0) agrees with part (b).

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