eCurl

Example Find the curl of ${\bf F}$ ．

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

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

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

Formally, we can write

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

In this problem, we have

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