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An irrotational field \( \mathbf{F} \) may always be written locally as

  • A.\( \nabla \times \mathbf{A} \) for some vector \( \mathbf{A} \)
  • B.a constant vector field
  • C.the sum of a gradient and a curl in equal parts
  • D.\( \nabla\phi \) for some scalar \( \phi \)

Answer: D

Irrotational means \( \nabla \times \mathbf{F} = \mathbf{0} \), and since the curl of any gradient vanishes identically, a scalar potential is the natural representation. A vector potential belongs to the solenoidal case, where the divergence rather than the curl is zero

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