Condition for Vector Field to satisfy Poisson's Equation

Theorem
Let $\mathbf V$ be a vector field over a region of space $R$.

Then:
 * $\mathbf V$ is conservative but not solenoidal


 * $\mathbf V$ is the gradient of a scalar field $F$ over $R$ which satisfies Poisson's equation over $R$:
 * $\nabla^2 F = \phi$
 * where $\phi$ is a function which is not identically zero.
 * where $\phi$ is a function which is not identically zero.

Sufficient Condition
Let $\mathbf V$ be conservative but not solenoidal.

From Vector Field is Expressible as Gradient of Scalar Field iff Conservative:
 * $\mathbf V = \grad F$

for some scalar field $F$ over $R$.

Because $\mathbf V$ is not solenoidal, we have:


 * $\exists \mathbf v \in R: \operatorname {div} \mathbf v \ne 0$

that is:


 * $\operatorname {div} \grad F \ne 0$

for at least some $\mathbf v \in R$.

Hence by Laplacian on Scalar Field is Divergence of Gradient:


 * $\nabla^2 F = \phi$

where:
 * $\phi$ is not identically zero
 * $\nabla^2$ is the Laplacian on $F$.

Hence $F$ satisfies Poisson's equation.

Necessary Condition
Let $\mathbf V$ be the gradient of a scalar field $F$ over $R$ which satisfies Poisson's equation:
 * $\nabla^2 F = \phi$

where $\phi$ is not identically zero.

Thus $F$ is such that:
 * $\mathbf V = \grad F$

and from Vector Field is Expressible as Gradient of Scalar Field iff Conservative it follows that $\mathbf V$ is conservative.

Then by Laplacian on Scalar Field is Divergence of Gradient:


 * $\exists \mathbf v \in R: \operatorname {div} \grad F \ne 0$

That is:
 * $\exists \mathbf v \in R: \operatorname {div} \mathbf V \ne 0$

and so by definition $\mathbf V$ is specifically not solenoidal.