Curl of Gradient is Zero/Physical Interpretation

Physical Interpretation of Curl of Gradient is Zero

From Vector Field is Expressible as Gradient of Scalar Field iff Conservative, the vector field given rise to by $\grad F$ is conservative.

The characteristic of a conservative field is that the contour integral around every simple closed contour is zero.

Since the curl is defined as a particular closed contour contour integral, it follows that $\map \curl {\grad F}$ equals zero.

Sources

• 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {V}$: Further Applications of the Operator $\nabla$: $2$. The Operator $\curl \grad$