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Display information for equation id:math.1444.65 on revision:1444

* Page found: Materie in elektrischen und magnetischen Feldern (eq math.1444.65)

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TeX (original user input): 

\begin{align}
& 3)\nabla \cdot \bar{E}=-\frac{\partial }{\partial t}\nabla \cdot \bar{A}\left( \bar{r},t \right)-\nabla \cdot \nabla \Phi  \\
& \nabla \cdot \bar{A}\left( \bar{r},t \right)=-\frac{1}{{{c}^{2}}}\frac{\partial }{\partial t}\Phi  \\
& \Rightarrow \nabla \cdot \bar{E}=-\frac{\partial }{\partial t}\nabla \cdot \bar{A}\left( \bar{r},t \right)-\nabla \cdot \nabla \Phi =\frac{1}{{{c}^{2}}}\frac{{{\partial }^{2}}}{\partial {{t}^{2}}}\Phi -\Delta \Phi =-\#\Phi  \\
\end{align}

TeX (checked): 

{\begin{aligned}&3)\nabla \cdot {\bar {E}}=-{\frac {\partial }{\partial t}}\nabla \cdot {\bar {A}}\left({\bar {r}},t\right)-\nabla \cdot \nabla \Phi \\&\nabla \cdot {\bar {A}}\left({\bar {r}},t\right)=-{\frac {1}{{c}^{2}}}{\frac {\partial }{\partial t}}\Phi \\&\Rightarrow \nabla \cdot {\bar {E}}=-{\frac {\partial }{\partial t}}\nabla \cdot {\bar {A}}\left({\bar {r}},t\right)-\nabla \cdot \nabla \Phi ={\frac {1}{{c}^{2}}}{\frac {{\partial }^{2}}{\partial {{t}^{2}}}}\Phi -\Delta \Phi =-\#\Phi \\\end{aligned}}

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MathML (32.681 KB / 3.891 KB) :

3 ) ∇ ⋅ E ¯ = - ∂ ∂ ⁡ t ∇ ⋅ A ¯ ( r ¯ , t ) - ∇ ⋅ ∇ Φ ∇ ⋅ A ¯ ⁢ ( r ¯ , t ) = - 1 c 2 ⁢ ∂ ∂ ⁡ t ⁢ Φ ⇒ ∇ ⋅ E ¯ = - ∂ ∂ ⁡ t ⁢ ∇ ⋅ A ¯ ⁢ ( r ¯ , t ) - ∇ ⋅ ∇ ⁡ Φ = 1 c 2 ⁢ ∂ 2 ∂ ⁡ t 2 ⁢ Φ - Δ ⁢ Φ = - # ⁢ Φ missing-subexpression fragments 3 ) ∇ ⋅ ¯ 𝐸 𝑡 ∇ ⋅ ¯ 𝐴 ( ¯ 𝑟 , t ) ∇ ⋅ ∇ Φ missing-subexpression ⋅ ∇ ¯ 𝐴 ¯ 𝑟 𝑡 1 superscript 𝑐 2 𝑡 Φ missing-subexpression ⇒ absent ⋅ ∇ ¯ 𝐸 ⋅ 𝑡 ∇ ¯ 𝐴 ¯ 𝑟 𝑡 ⋅ ∇ ∇ Φ 1 superscript 𝑐 2 superscript 2 superscript 𝑡 2 Φ Δ Φ # Φ {\displaystyle{\displaystyle\begin{aligned} &\displaystyle 3)\nabla\cdot\bar{E% }=-\frac{\partial}{\partial t}\nabla\cdot\bar{A}\left(\bar{r},t\right)-\nabla% \cdot\nabla\Phi\\ &\displaystyle\nabla\cdot\bar{A}\left(\bar{r},t\right)=-\frac{1}{{{c}^{2}}}% \frac{\partial}{\partial t}\Phi\\ &\displaystyle\Rightarrow\nabla\cdot\bar{E}=-\frac{\partial}{\partial t}\nabla% \cdot\bar{A}\left(\bar{r},t\right)-\nabla\cdot\nabla\Phi=\frac{1}{{{c}^{2}}}% \frac{{{\partial}^{2}}}{\partial{{t}^{2}}}\Phi-\Delta\Phi=-\#\Phi\\ \end{aligned}}}

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3)∇⋅E¯=−∂∂t∇⋅A¯(r¯,t)−∇⋅∇Φ∇⋅A¯(r¯,t)=−1c2∂∂tΦ⇒∇⋅E¯=−∂∂t∇⋅A¯(r¯,t)−∇⋅∇Φ=1c2∂2∂t2Φ−ΔΦ=−#Φ

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