Someone argues: "Stokes's theorem converts a circulation into a flux through the…

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 The integral theorems: Gauss, Stokes and Green

Problem

Someone argues: "Stokes's theorem converts a circulation into a flux through the surface bounded by the curve, but this only works if that surface is the flat disc. For the field F=(z2/a,−x,y)\mathbf{F} = (z^2/a, -x, y) around the circle x2+y2=a2x^2+y^2=a^2, z=0z=0 (radius a=1.0 ma=1.0\,\mathrm{m}, traversed anticlockwise seen from +z+z), the flat disc gives one value; if I instead spanned the curve with the upper hemispherical cap of radius aa, I would get a different answer, because ∇×F\nabla\times\mathbf{F} is not purely along z^\hat{\mathbf{z}}." ((a)) Compute the circulation ∮CF⋅dr\oint_C \mathbf{F}\cdot\mathrm{d}\mathbf{r} directly, by parameterising the circle. ((b)) Compute the flux of ∇×F\nabla\times\mathbf{F} through the flat disc bounded by CC. ((c)) Compute the flux of ∇×F\nabla\times\mathbf{F} through the upper hemispherical cap of radius aa on the same rim, using the outward normal r^\hat{\mathbf{r}} consistent with the right-hand rule. ((d)) State whether the claim is correct, and explain in one sentence why.

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