Condition for Vector Field to satisfy Laplace's Equation

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

Then:
 * $\mathbf V$ is both solenoidal and conservative


 * $\mathbf V$ is the gradient of a scalar field $F$ over $R$ which satisfies Laplace's equation:
 * $\nabla^2 F \equiv 0$
 * $\nabla^2 F \equiv 0$

Sufficient Condition
Let $\mathbf V$ be both solenoidal and conservative.

Then 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 solenoidal, we have:


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

that is:


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

Hence by Laplacian on Scalar Field is Divergence of Gradient:


 * $\nabla^2 F = 0$

where $\nabla^2$ is the Laplacian on $F$.

This holds throughout $R$, and so the equality is an equivalence:
 * $\nabla^2 F \equiv 0$

Necessary Condition
Let $\mathbf V$ be the gradient of a scalar field $F$ over $R$ which satisfies Laplace's equation:
 * $\nabla^2 F \equiv 0$

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:


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

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

and so by definition $\mathbf V$ is solenoidal.