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Display information for equation id:math.1439.249 on revision:1439

* Page found: Elektromagnetische Wellen (eq math.1439.249)

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TeX (original user input): 

\begin{align}
& \tilde{G}\left( \bar{r}-\bar{r}\acute{\ } \right)=\frac{1}{4\pi }\left( \frac{{{e}^{ik\left| \bar{r}-\bar{r}\acute{\ } \right|}}}{\left| \bar{r}-\bar{r}\acute{\ } \right|}-\frac{{{e}^{ik\left| \bar{r}-\bar{r}\acute{\ }\acute{\ } \right|}}}{\left| \bar{r}-\bar{r}\acute{\ }\acute{\ } \right|} \right) \\
& \Rightarrow {{\nabla }_{r}}\tilde{G}\left( \bar{r}-\bar{r}\acute{\ } \right)=\frac{1}{4\pi }\left( {{\nabla }_{r}}\frac{{{e}^{ik\left| \bar{r}-\bar{r}\acute{\ } \right|}}}{\left| \bar{r}-\bar{r}\acute{\ } \right|}-{{\nabla }_{r}}\frac{{{e}^{ik\left| \bar{r}-\bar{r}\acute{\ }\acute{\ } \right|}}}{\left| \bar{r}-\bar{r}\acute{\ }\acute{\ } \right|} \right):=\frac{1}{4\pi }\left( {{\nabla }_{r}}\frac{{{e}^{ikR}}}{R}-{{\nabla }_{r}}\frac{{{e}^{ikR\acute{\ }\acute{\ }}}}{R\acute{\ }\acute{\ }} \right) \\
\end{align}

TeX (checked): 

{\begin{aligned}&{\tilde {G}}\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)={\frac {1}{4\pi }}\left({\frac {{e}^{ik\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}-{\frac {{e}^{ik\left|{\bar {r}}-{\bar {r}}{\acute {\ }}{\acute {\ }}\right|}}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}{\acute {\ }}\right|}}\right)\\&\Rightarrow {{\nabla }_{r}}{\tilde {G}}\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)={\frac {1}{4\pi }}\left({{\nabla }_{r}}{\frac {{e}^{ik\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}-{{\nabla }_{r}}{\frac {{e}^{ik\left|{\bar {r}}-{\bar {r}}{\acute {\ }}{\acute {\ }}\right|}}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}{\acute {\ }}\right|}}\right):={\frac {1}{4\pi }}\left({{\nabla }_{r}}{\frac {{e}^{ikR}}{R}}-{{\nabla }_{r}}{\frac {{e}^{ikR{\acute {\ }}{\acute {\ }}}}{R{\acute {\ }}{\acute {\ }}}}\right)\\\end{aligned}}

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G~(r¯−r¯ ´)=14π(eik|r¯−r¯ ´||r¯−r¯ ´|−eik|r¯−r¯ ´ ´||r¯−r¯ ´ ´|)⇒∇rG~(r¯−r¯ ´)=14π(∇reik|r¯−r¯ ´||r¯−r¯ ´|−∇reik|r¯−r¯ ´ ´||r¯−r¯ ´ ´|):=14π(∇reikRR−∇reikR ´ ´R ´ ´)
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data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>k</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">−</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo data-mjx-texclass="CLOSE">|</mo></mrow></mrow></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">−</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo data-mjx-texclass="CLOSE">|</mo></mrow></mrow></mfrac></mrow><mo stretchy="false">−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>k</mi><mrow data-mjx-texclass="INNER"><mo 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data-mjx-texclass="ORD"><mi>r</mi></mrow></msub><mover><mi>G</mi><mo>~</mo></mover><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">−</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>4</mn><mi>π</mi></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mi mathvariant="normal">∇</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>k</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">−</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo data-mjx-texclass="CLOSE">|</mo></mrow></mrow></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">−</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo data-mjx-texclass="CLOSE">|</mo></mrow></mrow></mfrac></mrow><mo stretchy="false">−</mo><msub><mi mathvariant="normal">∇</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>k</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">−</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo data-mjx-texclass="CLOSE">|</mo></mrow></mrow></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">−</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo data-mjx-texclass="CLOSE">|</mo></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">:=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>4</mn><mi>π</mi></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo 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data-mjx-pseudoscript="true">´</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>

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Identifiers

  • G~
  • r¯
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  •  ´
  • π
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  • G~
  • r¯
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  • π
  • r
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  • r¯
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  • r¯
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  • π
  • r
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  • R
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  • R
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