Example Find the curl of ${\bf F}$．

$${\bf F} = z{\bf i} + x^2{\bf j} + 2y{\bf k}$$

Answer The curl of a vector field ${\bf F}$ is defined as follows:

$$\rm {curl} \boldsymbol{F} = \nabla \times \boldsymbol{F} = \left(... ...al F_{2}}{\partial x} - \frac{\partial F_{1}}{\partial y}\right)\boldsymbol{k}$$

Formally, we can write

$$\nabla \times \boldsymbol{F} = \left\vert\begin{array}{ccc} \bol... ...rac{\partial }{\partial z}\\ F_{1} & F_{2} & F_{3} \end{array} \right\vert$$

In this problem, we have

$${\rm curl} \boldsymbol{F} = \left\vert\begin{array}{ccc} \boldsy... ... \end{array} \right\vert = 2\boldsymbol{i} + \boldsymbol{j} + 2x\boldsymbol{k}$$