Lot F be as before and C be the unit circle (once around, counter-clockwise). F

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Lot F be as before and C be the unit circle (once around, counter-clockwise). F . dr = 0, and so F must be conservative F . dr 0, and so F must be non-conservative F . dr = 0, yet F is non-conservative since other closed loops exist along which F . dr 0 None of the above [Hint: Use the fact that if k is an odd positive integer, then sinkt dt = 0 = cosk t dt.] Let F be a vector field defined on a region R, and consider the following three conditions: F . dr = 0 for all closed loops C lying in R. There exists a scalar field Phi defined on R for which F = Phi. curl F is identically zero. Which of the following four statements is not true: If (I) is true, then all three conditions must be true (I) and (II) are equivalent conditions, each of which implies (III) If (III) is true, then either (I) or (II) must be true Assuming R is simply connected, if (III) is true then all three conditions must be true

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C C

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