Definition:Curl Operator/Physical Interpretation

Definition

Let $\mathbf V$ be a vector field acting over a region of space $R$.

Let a small vector area $\mathbf a$ of any shape be placed at an arbitrary point $P$ in $R$.

Let the contour integral $L$ be computed around the boundary edge of $A$.

Then there will be an angle of direction of $\mathbf a$ to the direction of $\mathbf V$ for which $L$ is a maximum.

The curl of $\mathbf V$ at $P$ is defined as the vector:

whose magnitude is the amount of this maximum $L$ per unit area

whose direction is the direction of $\mathbf a$ at this maximum.

Also known as

The curl of a vector quantity is also known in some older works as its rotation, denoted $\operatorname {rot}$.

However, curl is now practically universal, being unambiguous and compact.

Also see

• Results about the curl operator can be found here.

Historical Note

During the course of development of vector analysis, various notations for the curl operator were introduced, as follows:

Symbol	Used by
$\nabla \times$ or $\curl$	Josiah Willard Gibbs and Edwin Bidwell Wilson
$\curl$	Oliver Heaviside Max Abraham
$\operatorname {rot}$	Vladimir Sergeyevitch Ignatowski Hendrik Antoon Lorentz Cesare Burali-Forti and Roberto Marcolongo

Sources

• 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {IV}$: The Operator $\nabla$ and its Uses: $4$. The Curl of a Vector Field