# Thread: properites of curl operator proof

1. ## properites of curl operator proof

suppose F and G are vector functions of (x,y,z) and $\alpha , \beta$ are scalar functions of (x,y,z) then

prove

1) $\nabla \times (\alpha F) = \nabla \alpha \times F+ \alpha \nabla \times F$

2. ## Re: properites of curl operator proof

What does "G" have to do with this? Have you tried just writing $\vec{F}= f_1\vec{i}+ f_2\vec{j}+ f_3\vec{k}$ and just doing the calculations?
$\nabla\times (\alpha\vec{F}= \left|\begin{array}{ccc}\vec{i} & \vec{j} & \vec{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ \alpha\vec{f_1} & \alpha\vec{f_2} & \alpha\vec{f_3}\end{array}\right|$
$= \left(\frac{\partial\alpha f_3}{\partial y}- \frac{\partial \alpha f_2}{\partial y}\right)\vec{i}- \left(\frac{\partial \alpha f_3}{\partial x}- \frac{\partial\alpha f_1}{\partial z}\right)\vec{j}+ \left(\frac{\partial\alpha f_2}{\partial x}- \frac{\partial\alpha f_1}{\partial z}\right)\vec{k}$

Complete that, then do the same with the right side.