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Question Number 186077 by Mastermind last updated on 31/Jan/23

$$\mathrm{Prove}\mathrm{that}$$$$▽\bullet \left(\varnothing \stackrel{-}{\mathrm{A}}\right)=(▽\varnothing )\bullet \mathrm{A}+\varnothing (▽\bullet \stackrel{-}{\mathrm{A}}).$$$$$$$$$$$$\mathrm{Help}!$$

Answered by aleks041103 last updated on 30/Apr/23

$$Interestingchoiceofnotation...$$$$whatyouwantis$$$$div\left(a\overrightarrow{v}\right)=grad\left(a\right).\overrightarrow{v}+adiv\left(\overrightarrow{v}\right)$$$$$$$$Firstway:$$$$div\left(av\right)=\frac{\partial \left({av}_x\right)}{\partial x}+\frac{\partial \left({av}_y\right)}{\partial y}+\frac{\partial \left({av}_z\right)}{\partial z}=$$$$=v_x\frac{\partial a}{\partial x}+v_y\frac{\partial a}{\partial y}+v_z\frac{\partial a}{\partial z}+a(\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+\frac{\partial v_z}{\partial z})=$$$$=\left(\begin{array}{c}{v}_x\\ v_y\\ v_z\end{array}\right).\left(\begin{array}{c}\partial a/\partial x\\ \partial a/\partial y\\ \partial a/\partial z\end{array}\right)+adiv\left(v\right)=$$$$=v.grad\left(a\right)+adiv\left(v\right)=div\left(av\right)$$$$$$$$Secondway:usingeinsteinnotation$$$$div\left(av\right)={\partial }_i\left({av}^i\right)=v^i\left({\partial }_{i}a\right)+a\left({\partial }_i{v}^i\right)=$$$$=v.grad\left(a\right)+adiv\left(v\right)=div\left(av\right)$$$$$$$$Note:$$$$Idroppedthe\overrightarrow{}ontopofthevforeasy$$$$typesetting.$$

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