vector potential

Let  $\overrightarrow{U}=\overrightarrow{U}(x,y,z)$  be a vector field in ${ℝ}^3$ with continuous partial derivatives.  Then the following three conditions are equivalent (http://planetmath.org/Equivalent3):

• The surface integrals of $\overrightarrow{U}$ over all contractible closed surfaces (http://planetmath.org/Sphere) $S$ vanish:

  $${\oint }_S\overrightarrow{U}\cdot 𝑑\overrightarrow{S}=0$$

• The divergence of $\overrightarrow{U}$ vanishes everywhere in the field (http://planetmath.org/VectorField):

  $$\nabla \cdot \overrightarrow{U}=0$$

• There exists the vector potential  $\overrightarrow{A}=\overrightarrow{A}(x,y,z)$  of $\overrightarrow{U}$:

  $$\nabla \times \overrightarrow{A}=\overrightarrow{U}$$