Gradient of Divergence is Conservative

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 $\grad \operatorname {div}$
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.