Condition for Vector Field to satisfy Laplace's Equation

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Theorem

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

Then:

$\mathbf V$ is both solenoidal and conservative

if and only if:

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


Proof

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:

$\nabla^2 F \equiv 0$

$\Box$


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.

$\blacksquare$


Sources