Gradient of Divergence is Conservative
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Theorem
Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions.
Let $\mathbf V$ be a vector field on $\R^3$:
Then the gradient of the divergence of $\mathbf V$ is a conservative vector field.
Proof
The divergence of $\mathbf V$ is by definition a scalar field.
Then from Vector Field is Expressible as Gradient of Scalar Field iff Conservative it follows that $\grad \operatorname {div} \mathbf V$ is a conservative vector field.
$\blacksquare$
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {V}$: Further Applications of the Operator $\nabla$: $4$. The Operator $\grad \operatorname {div}$