Discussion
True or false: a field with zero curl throughout an annular region must be conservative there.
Answer: C
On a simply connected region a vanishing curl does give a potential, but an annulus has a hole, so a loop encircling it bounds no surface lying wholly inside. The field \( (-y, x)/(x^{2}+y^{2}) \) is the standard case: its curl is zero yet its circulation round the hole is \( 2\pi \)
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