The field , with a constant and the Cartesian coordinates of measured in metres from…

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

The field G(r)=k(x,y,−2z)\mathbf G(\mathbf r) = k(x, y, -2z), with k=1.0 s−1 m−1k = 1.0\,\mathrm{s^{-1}\,m^{-1}} a constant and x,y,zx,y,z the Cartesian coordinates of r\mathbf r measured in metres from the origin, satisfies ∇⋅G=k(1+1−2)=0\nabla\cdot\mathbf G = k(1+1-2) = 0 on all of space, which is star-shaped about the origin. ((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)=k(yz, −xz, 0)\mathbf A(\mathbf r) = k(yz,\,-xz,\,0). ((b)) Evaluate A\mathbf A at the point (1.0,2.0,3.0) m(1.0, 2.0, 3.0)\,\mathrm m.

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