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 .