Question Number 219185 by fantastic last updated on 20/Apr/25
Answered by MrGaster last updated on 20/Apr/25
$$\bigtriangledown\centerdot\left(\overset{\rightarrow} {{F}}×\overset{\rightarrow} {{G}}\right)=\partial_{{i}} \left(\epsilon_{{ijk}} {F}_{{j}} {G}_{{k}} \right) \\ $$$$=\epsilon_{{ijk}} \left(\partial_{{i}} {F}_{{j}} \right){G}_{{k}} +\epsilon_{{ijk}} {F}_{{j}} \left(\partial_{{i}} {G}_{{k}} \right) \\ $$$$={G}_{{k}} \left(\epsilon_{{kij}} \partial_{{i}} {F}_{{j}} \right)−{F}_{{j}} \left(\epsilon_{{jik}} \partial_{{i}} {G}_{{k}} \right) \\ $$$$=\overset{\rightarrow} {{G}}\centerdot\left(\bigtriangledown×\overset{\rightarrow} {{F}}\right)−\overset{\rightarrow} {{F}}\centerdot\left(\bigtriangledown+\overset{\rightarrow} {{G}}\right) \\ $$$$\begin{array}{|c|}{\bigtriangledown\centerdot\left(\overset{\rightarrow} {{F}}×\overset{\rightarrow} {{G}}\right)=\overset{\rightarrow} {{G}}\centerdot\bigtriangledown×\overset{\rightarrow} {{F}}−\overset{\rightarrow} {{F}}\centerdot\bigtriangledown×\overset{\rightarrow} {{G}}\left({a}\right)\checkmark}\\\hline\end{array} \\ $$$$\bigtriangledown\centerdot\left({g}\overset{\rightarrow} {{F}}\right)=\partial_{{i}} \left({gF}_{{i}} \right) \\ $$$$={g}\bigtriangledown\centerdot\overset{\rightarrow} {{F}}+\bigtriangledown{g}\centerdot\overset{\rightarrow} {{F}} \\ $$$$\left(\mathrm{d}\right)\checkmark \\ $$$$\cancel{\bigtriangledown\centerdot\left(\overset{\rightarrow} {{F}}×\overset{\rightarrow} {{G}}\right)=\overset{\rightarrow} {{G}}\centerdot\bigtriangledown×\overset{\rightarrow} {{F}}−\overset{\rightarrow} {{F}}\centerdot\bigtriangledown×\overset{\rightarrow} {{G}}\left({a}\right)\checkmark} \\ $$$$\cancel{\bigtriangledown\centerdot\left({g}\overset{\rightarrow} {{F}}\right)=\partial_{{i}} \left({gF}_{{i}} \right)} \\ $$$$\therefore{d}\checkmark \\ $$
Answered by Nicholas666 last updated on 20/Apr/25
$${I}\:{think}\:{option}\:{a}\:{an}\:{d}\:{it}'{s}\:{true}\:{solution} \\ $$
Commented by Nicholas666 last updated on 20/Apr/25
Commented by Nicholas666 last updated on 20/Apr/25
$${I}\:{will}\:{send}\:{you}\:{link}\:{for}\:{the}\:{complete}\:{solution} \\ $$
Answered by fantastic last updated on 20/Apr/25
$${the}\:{real}\:{answers}\:{are}\:\left({a}\right)\:{and}\left(\:{d}\right) \\ $$
Answered by fantastic last updated on 20/Apr/25
Answered by fantastic last updated on 20/Apr/25
