Poisson's Differential Equation for Rotational and Solenoidal Field

Theorem
Let $R$ be a region of ordinary space.

Let $\mathbf V$ be a vector field over $R$.

Let $\mathbf V$ be both rotational and solenoidal.

Let $\mathbf A$ be a vector field such that $\mathbf V = \curl \mathbf A$.

Then $\mathbf V$ satisfies this version of Poisson's differential equation:


 * $\curl \mathbf V = -\nabla^2 \mathbf A \ne \bszero$

Proof
As $\mathbf V$ is rotational it is not conservative.

Hence from Vector Field is Expressible as Gradient of Scalar Field iff Conservative $\mathbf V$ cannot be the gradient of some scalar field.

However, by definition of rotational vector field:
 * $\curl \mathbf V \ne \bszero$

As $\mathbf V$ is solenoidal:
 * $\operatorname {div} \mathbf V = 0$

Hence from Divergence of Curl is Zero, for some vector field $\mathbf A$ over $R$:


 * $\operatorname {div} \mathbf V = \operatorname {div} \curl \mathbf A = 0$

and so:
 * $\mathbf V = \curl \mathbf A$

Then we have: