PlanetPhysics/Vector Potential

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

• The surface integrals of $\overrightarrow{U}$ over all contractible closed surfaces $S$ vanish: ${\oint }_S\overrightarrow{U}\cdot d\overrightarrow{S}=0$
• The [[../DivergenceOfAVectorField/|divergence]] of $\overrightarrow{U}$ vanishes everywhere in the [[../VectorField/|field]]: $\mathrm{\nabla }\cdot \overrightarrow{U}=0$
• There exists the vector potential \, $\overrightarrow{A}=\overrightarrow{A}(x,y,z)$\, of $\overrightarrow{U}$: $\mathrm{\nabla }\times \overrightarrow{A}=\overrightarrow{U}$

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