PlanetPhysics/Vector Potential

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


 * The surface integrals of $$\vec{U}$$ over all contractible closed surfaces $$S$$ vanish: $$\oint_S\vec{U}\cdot d\vec{S} = 0$$
 * The divergence of $$\vec{U}$$ vanishes everywhere in the field: $$\nabla\!\cdot\!\vec{U} = 0$$
 * There exists the vector potential \, $$\vec{A} = \vec{A}(x,\,y,\,z)$$\, of $$\vec{U}$$: $$\nabla\!\times\!\vec{A} = \vec{U}$$

Under those conditions, the vector field $$\vec{U}$$ is called solenoidal.