Curl of Vector Field is Solenoidal

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 curl of $\mathbf V$ is a solenoidal vector field.

Proof

By definition, a solenoidal vector field is one whose divergence is zero.

The result follows from Divergence of Curl is Zero.

$\blacksquare$

Sources

• 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {V}$: Further Applications of the Operator $\nabla$: $5$. The Operator $\operatorname {div} \curl$