Divergence Operator is Invariant under Coordinate Transformation

Theorem

Let $R$ be a region of space in which there exists an vector field $\mathbf V$.

Let $\nabla \cdot \mathbf V$ denote the divergence of $\mathbf V$.

Then $\nabla \cdot \mathbf V$ is invariant under a change of coordinate system on $R$.

Proof

The result follows directly from the physical interpretation of the divergence operator.

$\blacksquare$

Sources

• 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {IV}$: The Operator $\nabla$ and its Uses: $3 a$. The Operation $\nabla \cdot \mathbf V$