= Radial vector potential of a solenoidal vector field
{title2=$A(x)=D(x)\times x$}
Let $B$ be a <continuously differentiable> <solenoidal vector field> on an open <star-shaped set> containing the origin. Then
$$
A(x)=\left(\int_0^1tB(tx)\,dt\right)\times x
$$
is a <vector potential> for $B$. For $D=\int_0^1tB(tx)\,dt$, differentiation gives $\nabla\cdot D=0$ and $\nabla\times(D\times x)=2D+(x\cdot\nabla)D=\int_0^1\partial_t(t^2B(tx))\,dt=B(x)$. Boundedness near the origin removes the lower endpoint. A field defined only on a punctured domain need not satisfy these hypotheses.
Back to article page
