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 $\vec{U}$ $$\nabla\!\times\!\vec{A} = \vec{U}$$

Bibliography

1

K. V¨AISÄLÄ: Vektorianalyysi. Werner Söderström Osakeyhtiö, Helsinki (1961).