A field is uniform (the same vector everywhere) on all of space

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 field G=(2.0,−3.0,1.0) m s−1\mathbf G = (2.0, -3.0, 1.0)\,\mathrm{m\,s^{-1}} is uniform (the same vector everywhere) on all of space, which is star-shaped about the origin, and satisfies ∇⋅G=0\nabla\cdot\mathbf G = 0 since its components are constant. ((a)) Using A(r)=∫01t G(tr)×r dt\mathbf A(\mathbf r) = \int_0^1 t\,\mathbf G(t\mathbf r)\times\mathbf r\,\mathrm dt, show that A(r)=12G×r\mathbf A(\mathbf r) = \tfrac12\mathbf G\times\mathbf r. ((b)) Evaluate A\mathbf A at r=(1.0,0,0) m\mathbf r = (1.0, 0, 0)\,\mathrm m.

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