A ground station computes the spherical unit vectors at the point ,

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

Mathematical Methods for Physics, Ch Curvilinear and orthogonal coordinates

Problem

A ground station computes the spherical unit vectors at the point r=2 mr = 2\,\mathrm{m}, θ=π/3 rad\theta = \pi/3\,\mathrm{rad}, ϕ=π/2 rad\phi = \pi/2\,\mathrm{rad} from the standard formulas and reports the following Cartesian components: r^=(0,32,12)\hat{\mathbf{r}} = (0, \tfrac{\sqrt3}{2}, \tfrac12), θ^=(0,12,−32)\hat{\boldsymbol{\theta}} = (0, \tfrac12, -\tfrac{\sqrt3}{2}), ϕ^=(−1,0,0)\hat{\boldsymbol{\phi}} = (-1, 0, 0). ((a)) Check that the three reported vectors are mutually orthogonal and each of unit length, by computing the three pairwise dot products and the length of each vector. ((b)) Using the scale factors, find the area dA\mathrm{d}A of the small patch on the sphere r=2 mr = 2\,\mathrm{m} swept out by changing θ\theta and ϕ\phi by dθ=0.020 rad\mathrm{d}\theta = 0.020\,\mathrm{rad} and dϕ=0.030 rad\mathrm{d}\phi = 0.030\,\mathrm{rad} from this point.

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