The Curl in Cartesian Coordinates

On the other hand, we can also compute the curl in Cartesian coordinates. Again, we let $\vec{F} = F_1 \hat{i} + F_2 \hat{j} + F_3 \hat{k}$ and compute

$$\mbox{curl}\vec{F} = \nabla \times \vec{F} = \left\vert \beg... ...artial}{\partial z} \\ F_1 & F_2 & F_3 \end{array} \right\vert$$

$$\qquad = \hat{i} \left( \frac{\partial F_3}{\partial y} - \f... ...ial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right).$$

Not surprisingly, the curl is a vector quantity. It also will generally be a (vector valued) function.