Definition:Curl Operator/Physical Interpretation
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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:
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