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CURL of a Vector Field

If we form the vector (``cross'') product of $\Grad{}$ with a vector function $\Vec{A}(x,y,z)$ we get a vector result called the curl of $\Vec{A}$:

$$\hbox{\bf curl} \, \Vec{A} \; \equiv \; \Curl{A} \; \equiv . . . . . . \over \partial x} - {\partial{A}_x \over \partial y} \right)$$

This is a lot harder to visualize than the DIVERGENCE, but not impossible. Suppose you are in a boat in a huge river (or Pass) where the current flows mainly in the x direction but where the speed of the current (flux of water) varies with y. Then if we call the current $\Vec{J}$, we have a nonzero value for the derivative ${\partial J_x \over \partial y}$, which you will recognize as one of the terms in the formula for $\Curl{J}$. What does this imply? Well, if you are sitting in the boat, moving with the current, it means the current on your port side moves faster - i.e. forward relative to the boat - and the current on your starboard side moves slower - i.e. backward relative to the boat - and this implies a circulation of the water around the boat - i.e. a whirlpool!  So $\Curl{J}$ is a measure of the local ``swirliness'' of the current $\Vec{J}$, which means ``curl'' is not a bad name after all!