A vector field is given by

PHYSICS · Subjective · 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

A vector field is given by F=(y,z,x)\mathbf{F} = (y, z, x), in which each component is numerically equal to the named coordinate expressed in metres (so every component of F\mathbf{F} carries the unit metre). A circular hoop of radius a=0.6 ma = 0.6\,\mathrm{m} lies in the xyxy-plane, centred on the origin, and is traversed anticlockwise as seen from the +z+z axis. (a) Find ∇×F\nabla\times\mathbf{F}. (b) Compute the circulation ∮CF⋅dr\oint_C\mathbf{F}\cdot\mathrm{d}\mathbf{r} round the hoop directly, and separately compute the flux of ∇×F\nabla\times\mathbf{F} through the flat disc bounded by the hoop, taking the disc's normal to be tied to the direction of travel round CC by the right-hand rule. Confirm the two numbers agree. (c) A dome shaped like the upper hemisphere of radius aa is built on the same hoop, its normal pointing away from the centre, which is tied to the hoop's direction of travel by the right-hand rule. Without evaluating a new surface integral over the dome, apply the divergence theorem to the closed surface made of the dome and the disc, together with the identity ∇⋅(∇×F)=0\nabla\cdot(\nabla\times\mathbf{F}) = 0, to find the flux of ∇×F\nabla\times\mathbf{F} through the dome.

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