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.