optics:res5.11

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optics:res5.11 [2020/05/23 09:41] – создано rootoptics:res5.11 [2020/05/23 09:43] (текущий) – root
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-$$F^{\mu\nu}=\frac{\partial A^{\nu}}{\partial x_{\mu}}-\frac{\partial A^{\mu}}{\partial x_{\nu}}\pause=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}\pause=-F^{\nu\mu}$$ +$$F^{\mu\nu}=\frac{\partial A^{\nu}}{\partial x_{\mu}}-\frac{\partial A^{\mu}}{\partial x_{\nu}}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}=-F^{\nu\mu}$$
-\pause+
 $$F^{\mu\nu}=\left(\begin{array}{cccc} $$F^{\mu\nu}=\left(\begin{array}{cccc}
 0 & -E_{x} & -E_{y} & -E_{z}\\ 0 & -E_{x} & -E_{y} & -E_{z}\\
Строка 12: Строка 11:
 E_{z} & -H_{y} & H_{x} & 0 E_{z} & -H_{y} & H_{x} & 0
 \end{array}\right)$$ \end{array}\right)$$
-\pause 
 $$F'^{\mu\nu}=L_{i}^{\mu}L_{j}^{\nu}F^{ij}$$ $$F'^{\mu\nu}=L_{i}^{\mu}L_{j}^{\nu}F^{ij}$$
  
Строка 21: Строка 19:
 0 & 0 & 0 & -1 0 & 0 & 0 & -1
 \end{array}\right)$$ \end{array}\right)$$
-\pause +$$A^{i}=\left(A_{0},A_{x},A_{y},A_{z}\right), \ \ A_{j}=A^{i}g_{ij}=\left(A_{0},-A_{x},-A_{y},-A_{z}\right)$$ 
-$$A^{i}=\left(A_{0},A_{x},A_{y},A_{z}\right), \ \ \pause A_{j}=A^{i}g_{ij}=\left(A_{0},-A_{x},-A_{y},-A_{z}\right)$$ +Инварианты --- скаляры, $$A^{i}A_{i}= A_{0}^{2}-A_{x}^{2}-A_{y}^{2}-A_{z}^{2}=inv$$ 
-\pause +$$A'^{i}A'_{i}=A'^{i}A'^{j}g_{ij}=A^{\mu}L_{\mu}^{i}A^{\nu}L_{\nu}^{j}g_{ij}=$$ 
-Инварианты \pause \textemdash{} скаляры,\pause $$A^{i}A_{i}=\pause A_{0}^{2}-A_{x}^{2}-A_{y}^{2}-A_{z}^{2}=inv$$ +$$A^{\mu}A^{\nu}L_{\mu}^{i}L_{\nu}^{j}g_{ij}=A^{\mu}A^{\nu}g_{\mu\nu}=A^{\mu}A_{\mu}$$
-\pause +
-$$A'^{i}A'_{i}=\pause A'^{i}A'^{j}g_{ij}\pause=A^{\mu}L_{\mu}^{i}A^{\nu}L_{\nu}^{j}g_{ij}=$$\pause +
-$$A^{\mu}A^{\nu}L_{\mu}^{i}L_{\nu}^{j}g_{ij}\pause=A^{\mu}A^{\nu}g_{\mu\nu}\pause=A^{\mu}A_{\mu}$$ +
-\pause+
 $$F^{\mu\nu}F_{\mu\nu}=F^{\mu\nu}F^{ij}g_{i\mu}g_{j\nu}$$ $$F^{\mu\nu}F_{\mu\nu}=F^{\mu\nu}F^{ij}g_{i\mu}g_{j\nu}$$
-$$F^{\mu\nu}=\frac{\partial A^{\nu}}{\partial x_{\mu}}-\frac{\partial A^{\mu}}{\partial x_{\nu}}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$$\pause+$$F^{\mu\nu}=\frac{\partial A^{\nu}}{\partial x_{\mu}}-\frac{\partial A^{\mu}}{\partial x_{\nu}}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$$
 $$\left(\begin{array}{cccc} $$\left(\begin{array}{cccc}
 0 & -E_{x} & -E_{y} & -E_{z}\\ 0 & -E_{x} & -E_{y} & -E_{z}\\
Строка 36: Строка 30:
 E_{y} & H_{z} & 0 & -H_{x}\\ E_{y} & H_{z} & 0 & -H_{x}\\
 E_{z} & -H_{y} & H_{x} & 0 E_{z} & -H_{y} & H_{x} & 0
-\end{array}\right)=F^{\mu\nu}\to $$\pause+\end{array}\right)=F^{\mu\nu}\to $$
 $$F_{\mu\nu}=\left(\begin{array}{cccc} $$F_{\mu\nu}=\left(\begin{array}{cccc}
 0 & E_{x} & E_{y} & E_{z}\\ 0 & E_{x} & E_{y} & E_{z}\\
Строка 43: Строка 37:
 -E_{z} & -H_{y} & H_{x} & 0 -E_{z} & -H_{y} & H_{x} & 0
 \end{array}\right)$$ \end{array}\right)$$
-\pause +$$F^{\mu\nu}F_{\mu\nu}=2\left(-E^{2}+H^{2}\right)=inv$$
-$$F^{\mu\nu}F_{\mu\nu}\pause=2\left(-E^{2}+H^{2}\right)=inv$$ +
-\pause+
 $$\left\{ \begin{array}{c} $$\left\{ \begin{array}{c}
 H_{x}=H_{x}'\\ H_{x}=H_{x}'\\
Строка 56: Строка 48:
 E_{z}=\gamma\left(E_{z}'-\beta H_{y}'\right) E_{z}=\gamma\left(E_{z}'-\beta H_{y}'\right)
 \end{array}\right.$$ \end{array}\right.$$
-\pause +$$H^{2}-E^{2}= H_{x}'^{2}+\gamma^{2}\left(H_{y}'-\beta E_{z}'\right)^{2}+\gamma^{2}\left(H_{z}'+\beta E_{y}'\right)^{2}-$$
-$$H^{2}-E^{2}=\pause H_{x}'^{2}+\gamma^{2}\left(H_{y}'-\beta E_{z}'\right)^{2}+\gamma^{2}\left(H_{z}'+\beta E_{y}'\right)^{2}\pause-$$+
 $$E_{x}'^{2}-\gamma^{2}\left(E_{y}'+\beta H_{z}'\right)^{2}-\gamma^{2}\left(E_{z}'-\beta H_{y}'\right)^{2}$$ $$E_{x}'^{2}-\gamma^{2}\left(E_{y}'+\beta H_{z}'\right)^{2}-\gamma^{2}\left(E_{z}'-\beta H_{y}'\right)^{2}$$