Curl of Vector Field is Solenoidal
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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 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 $\operatorname {div} \curl$