Let U = U(x, y, z) be a vector field in ℝ³ with continuous partial derivatives. Then the following three conditions are equivalent:

• The surface integrals of U over all contractible closed surfaces S vanish:
  
• The divergence of U vanishes everywhere in the field:
  
• There exists the vector potential A = A(x, y, z) of U:
  

Under those conditions, the vector field U is called solenoidal.

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