The spherical unit vectors

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Kinematics and Newtonian dynamics of particles

Mathematical Methods for Physics, Ch Curvilinear and orthogonal coordinates

Problem

The spherical unit vectors, in Cartesian components, are r^=(sin⁡θcos⁡ϕ, sin⁡θsin⁡ϕ, cos⁡θ),θ^=(cos⁡θcos⁡ϕ, cos⁡θsin⁡ϕ, −sin⁡θ),ϕ^=(−sin⁡ϕ, cos⁡ϕ, 0).\begin{aligned}\hat{\mathbf r} &= (\sin\theta\cos\phi,\ \sin\theta\sin\phi,\ \cos\theta),\\ \hat{\boldsymbol\theta} &= (\cos\theta\cos\phi,\ \cos\theta\sin\phi,\ -\sin\theta),\\ \hat{\boldsymbol\phi} &= (-\sin\phi,\ \cos\phi,\ 0).\end{aligned} (a) Differentiate the components to find ∂θ^/∂θ\partial\hat{\boldsymbol\theta}/\partial\theta, ∂θ^/∂ϕ\partial\hat{\boldsymbol\theta}/\partial\phi and ∂ϕ^/∂ϕ\partial\hat{\boldsymbol\phi}/\partial\phi, and write each as a combination of r^\hat{\mathbf r}, θ^\hat{\boldsymbol\theta} and ϕ^\hat{\boldsymbol\phi}. (b) ∂r^/∂ϕ\partial\hat{\mathbf r}/\partial\phi has length sin⁡θ\sin\theta, yet ∂ϕ^/∂ϕ\partial\hat{\boldsymbol\phi}/\partial\phi has length 11 at every θ\theta. Explain the difference from the way each vector moves as ϕ\phi turns, and say which way ∂ϕ^/∂ϕ\partial\hat{\boldsymbol\phi}/\partial\phi points. (c) A point fixed on the Earth's surface, at r=6.37×106 mr = 6.37\times10^{6}\,\mathrm m and θ=π/3 rad\theta = \pi/3\,\mathrm{rad}, is carried round by the Earth's rotation, so that ϕ=Ωt\phi = \Omega t with Ω=7.29×10−5 rad s−1\Omega = 7.29\times10^{-5}\,\mathrm{rad\,s^{-1}} while rr and θ\theta stay fixed. Starting from r=rr^\mathbf r = r\hat{\mathbf r}, find its acceleration in terms of r^\hat{\mathbf r} and θ^\hat{\boldsymbol\theta}, give its magnitude, and say towards which point it is directed.

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