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Math Help - properites of curl operator proof

  1. #1
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    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
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  2. #2
    MHF Contributor

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    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.
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