A uniform field with fills a star-shaped domain about the origin

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

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

Problem

A uniform field G=G0z^\mathbf{G} = G_0\hat{\mathbf{z}} with G0=2.0 TG_0 = 2.0\,\mathrm{T} fills a star-shaped domain about the origin. The star-shaped construction gives the vector potential A=∫01t G(tr)×r dt=12G×r=12G0(xy^−yx^).\mathbf{A} = \int_0^1 t\,\mathbf{G}(t\mathbf{r})\times\mathbf{r}\,\mathrm{d}t = \tfrac{1}{2}\mathbf{G}\times\mathbf{r} = \tfrac{1}{2}G_0(x\hat{\mathbf{y}} - y\hat{\mathbf{x}}). Someone then checks A′=−G0y x^\mathbf{A}' = -G_0y\,\hat{\mathbf{x}} and argues: "Since the star-shaped formula is the construction proved to give the vector potential, and A′≠A\mathbf{A}'\ne\mathbf{A}, A′\mathbf{A}' cannot be a valid vector potential for G\mathbf{G}." ((a)) Check directly whether ∇×A′=G\nabla\times\mathbf{A}' = \mathbf{G}. ((b)) Find a scalar function χ(x,y)\chi(x,y) such that A′=A+∇χ\mathbf{A}' = \mathbf{A} + \nabla\chi, and use it to say what is wrong with the claim.

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