The swirl field is defined everywhere except on the axis

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

Mathematical Methods for Physics, Ch The integral theorems: Gauss, Stokes and Green

Problem

The swirl field F=(−y,x,0)/(x2+y2)\mathbf{F} = (-y, x, 0)/(x^2+y^2) is defined everywhere except on the zz axis, and ∇×F=0\nabla\times\mathbf{F} = \mathbf{0} wherever it is defined. Treat all coordinates as pure numbers. Every loop below lies in the plane z=0z = 0 and is walked anticlockwise seen from +z+z. ((a)) Compute directly the circulation of F\mathbf{F} round the square with corners (±1,±1,0)(\pm1, \pm1, 0). ((b)) Without a further integral, give the circulation round the circle x2+y2=9x^2+y^2 = 9, and justify it. ((c)) Without a further integral, give the circulation round the square with corners (2,−1,0)(2,-1,0), (4,−1,0)(4,-1,0), (4,1,0)(4,1,0) and (2,1,0)(2,1,0), and justify it. ((d)) Someone argues that the answer to (a) must be zero, because ∇×F=0\nabla\times\mathbf{F} = \mathbf{0} at every point of the square and Stokes's theorem turns the circulation into the flux of ∇×F\nabla\times\mathbf{F}. Say exactly where the argument fails.

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