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:
- $\nabla^2 F \equiv 0$
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
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {V}$: Further Applications of the Operator $\nabla$: $7$. The Classification of Vector Fields: $\text {(i)}$