- #1
NexusN
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Homework Statement
[tex]\nabla \cdot(\vec E \times \vec H)[/tex]=[tex]\vec H\cdot(\nabla \times \vec E) - \vec E\cdot(\nabla \times \vec H)[/tex]
Homework Equations
The Attempt at a Solution
[tex]\begin{array}{l}
\nabla \cdot(\vec E \times \vec H)\\
= \left[ {\frac{1}{{{h_1}{h_2}{h_3}}}\left( {{{\hat a}_1}\frac{\partial }{{\partial {l_1}}}{h_2}{h_3} + {{\hat a}_2}\frac{\partial }{{\partial {l_2}}}{h_1}{h_3} + {{\hat a}_3}\frac{\partial }{{\partial {l_3}}}{h_1}{h_2}} \right)} \right]\cdot[({{\hat a}_1}{E_1} + {{\hat a}_2}{E_2} + {{\hat a}_3}{E_3}) \times ({{\hat a}_1}{H_1} + {{\hat a}_2}{H_2} + {{\hat a}_3}{H_3})]\\
= \left[ {\frac{1}{{{h_1}{h_2}{h_3}}}\left( {{{\hat a}_1}\frac{\partial }{{\partial {l_1}}}{h_2}{h_3} + {{\hat a}_2}\frac{\partial }{{\partial {l_2}}}{h_1}{h_3} + {{\hat a}_3}\frac{\partial }{{\partial {l_3}}}{h_1}{h_2}} \right)} \right]\cdot\left| {\begin{array}{*{20}{c}}
{{{\hat a}_1}}&{{{\hat a}_2}}&{{{\hat a}_3}}\\
{{E_1}}&{{E_2}}&{{E_3}}\\
{{H_1}}&{{H_2}}&{{H_3}}
\end{array}} \right|\\
= \left[ {\frac{1}{{{h_1}{h_2}{h_3}}}\left( {{{\hat a}_1}\frac{\partial }{{\partial {l_1}}}{h_2}{h_3} + {{\hat a}_2}\frac{\partial }{{\partial {l_2}}}{h_1}{h_3} + {{\hat a}_3}\frac{\partial }{{\partial {l_3}}}{h_1}{h_2}} \right)} \right]\cdot[{{\hat a}_1}({E_2}{H_3} - {H_2}{E_3}) + {{\hat a}_2}({E_3}{H_1} - {H_3}{E_1}) + {{\hat a}_3}({E_1}{H_2} - {H_1}{E_2})]\\
= \frac{1}{{{h_1}{h_2}{h_3}}}\left\{ {\frac{\partial }{{\partial {l_1}}}[{h_2}{h_3}({E_2}{H_3} - {H_2}{E_3})] + \frac{\partial }{{\partial {l_2}}}[{h_1}{h_3}({E_3}{H_1} - {H_3}{E_1})] + \frac{\partial }{{\partial {l_3}}}[{h_1}{h_2}({E_1}{H_2} - {H_1}{E_2})]} \right\}
\end{array}[/tex]
[tex]\begin{array}{l}
\vec H\cdot(\nabla \times \vec E) - \vec E\cdot(\nabla \times \vec H)\\
= ({{\hat a}_1}{H_1} + {{\hat a}_2}{H_2} + {{\hat a}_3}{H_3})\cdot\frac{1}{{{h_1}{h_2}{h_3}}}\left| {\begin{array}{*{20}{c}}
{{{\hat a}_1}{h_1}}&{{{\hat a}_2}{h_2}}&{{{\hat a}_3}{h_3}}\\
{\frac{\partial }{{\partial {l_1}}}}&{\frac{\partial }{{\partial {l_2}}}}&{\frac{\partial }{{\partial {l_3}}}}\\
{{h_1}{E_1}}&{{h_2}{E_2}}&{{h_3}{E_3}}
\end{array}} \right| - ({{\hat a}_1}{E_1} + {{\hat a}_2}{E_2} + {{\hat a}_3}{E_3})\cdot\frac{1}{{{h_1}{h_2}{h_3}}}\left| {\begin{array}{*{20}{c}}
{{{\hat a}_1}{h_1}}&{{{\hat a}_2}{h_2}}&{{{\hat a}_3}{h_3}}\\
{\frac{\partial }{{\partial {l_1}}}}&{\frac{\partial }{{\partial {l_2}}}}&{\frac{\partial }{{\partial {l_3}}}}\\
{{h_1}{H_1}}&{{h_2}{H_2}}&{{h_3}{H_3}}
\end{array}} \right|\\
= ({{\hat a}_1}{H_1} + {{\hat a}_2}{H_2} + {{\hat a}_3}{H_3})\cdot\frac{1}{{{h_1}{h_2}{h_3}}}[{{\hat a}_1}({h_1}\frac{\partial }{{\partial {l_2}}}{h_3}{E_3} - {h_1}\frac{\partial }{{\partial {l_3}}}{h_2}{E_2}) + {{\hat a}_2}({h_2}\frac{\partial }{{\partial {l_3}}}{h_1}{E_1} - {h_2}\frac{\partial }{{\partial {l_1}}}{h_3}{E_3}) + {{\hat a}_3}({h_3}\frac{\partial }{{\partial {l_1}}}{h_2}{E_2} - {h_3}\frac{\partial }{{\partial {l_2}}}{h_1}{E_1})]\\
- ({{\hat a}_1}{E_1} + {{\hat a}_2}{E_2} + {{\hat a}_3}{E_3})\cdot\frac{1}{{{h_1}{h_2}{h_3}}}[{{\hat a}_1}({h_1}\frac{\partial }{{\partial {l_2}}}{h_3}{H_3} - {h_1}\frac{\partial }{{\partial {l_3}}}{h_2}{H_2}) + {{\hat a}_2}({h_2}\frac{\partial }{{\partial {l_3}}}{h_1}{H_1} - {h_2}\frac{\partial }{{\partial {l_1}}}{h_3}{H_3}) + {{\hat a}_3}({h_3}\frac{\partial }{{\partial {l_1}}}{h_2}{H_2} - {h_3}\frac{\partial }{{\partial {l_2}}}{h_1}{H_1})]\\
= \frac{1}{{{h_1}{h_2}{h_3}}}({h_1}{H_1}\frac{\partial }{{\partial {l_2}}}{h_3}{E_3} - {h_1}{H_1}\frac{\partial }{{\partial {l_3}}}{h_2}{E_2} + {h_2}{H_2}\frac{\partial }{{\partial {l_3}}}{h_1}{E_1} - {h_2}{H_2}\frac{\partial }{{\partial {l_1}}}{h_3}{E_3} + {h_3}{H_3}\frac{\partial }{{\partial {l_1}}}{h_2}{E_2} - {h_3}{H_3}\frac{\partial }{{\partial {l_2}}}{h_1}{E_1})\\
- \frac{1}{{{h_1}{h_2}{h_3}}}({h_1}{E_1}\frac{\partial }{{\partial {l_2}}}{h_3}{H_3} - {h_1}{E_1}\frac{\partial }{{\partial {l_3}}}{h_2}{H_2} + {h_2}{E_2}\frac{\partial }{{\partial {l_3}}}{h_1}{H_1} - {h_2}{E_2}\frac{\partial }{{\partial {l_1}}}{h_3}{H_3} + {h_3}{E_3}\frac{\partial }{{\partial {l_1}}}{h_2}{H_2} - {h_3}{E_3}\frac{\partial }{{\partial {l_2}}}{h_1}{H_1})\\
= \frac{1}{{{h_1}{h_2}{h_3}}}[({h_1}{H_1}\frac{\partial }{{\partial {l_2}}}{h_3}{E_3} + {h_3}{E_3}\frac{\partial }{{\partial {l_2}}}{h_1}{H_1}) - ({h_1}{H_1}\frac{\partial }{{\partial {l_3}}}{h_2}{E_2} + {h_2}{E_2}\frac{\partial }{{\partial {l_3}}}{h_1}{H_1}) + ({h_2}{H_2}\frac{\partial }{{\partial {l_3}}}{h_1}{E_1} + {h_1}{E_1}\frac{\partial }{{\partial {l_3}}}{h_2}{H_2})\\
- ({h_2}{H_2}\frac{\partial }{{\partial {l_1}}}{h_3}{E_3} + {h_3}{E_3}\frac{\partial }{{\partial {l_1}}}{h_2}{H_2}) + ({h_3}{H_3}\frac{\partial }{{\partial {l_1}}}{h_2}{E_2} + {h_2}{E_2}\frac{\partial }{{\partial {l_1}}}{h_3}{H_3}) - ({h_3}{H_3}\frac{\partial }{{\partial {l_2}}}{h_1}{E_1} + {h_1}{E_1}\frac{\partial }{{\partial {l_2}}}{h_3}{H_3})]\\
= \frac{1}{{{h_1}{h_2}{h_3}}}(\frac{\partial }{{\partial {l_2}}}{h_1}{h_3}{E_3}{H_1} - \frac{\partial }{{\partial {l_3}}}{h_1}{h_2}{E_2}{H_1} + \frac{\partial }{{\partial {l_3}}}{h_1}{h_2}{E_1}{H_2} - \frac{\partial }{{\partial {l_1}}}{h_2}{h_3}{E_3}{H_2} + \frac{\partial }{{\partial {l_1}}}{h_2}{h_3}{E_2}{H_3} - \frac{\partial }{{\partial {l_2}}}{h_1}{h_3}{E_1}{H_3})\\
= \frac{1}{{{h_1}{h_2}{h_3}}}\{ \frac{\partial }{{\partial {l_1}}}[{h_2}{h_3}({E_2}{H_3} - {E_3}{H_2})] + \frac{\partial }{{\partial {l_2}}}[{h_1}{h_3}({E_3}{H_1} - {E_1}{H_3})] + \frac{\partial }{{\partial {l_3}}}[{h_1}{h_2}({E_1}{H_2} - {E_2}{H_1})]\} \\
= \nabla \cdot(\vec E \times \vec H)
\end{array}[/tex]
Thank you so much for your kind attention!