\begin{align*}
\mb{u}'=\frac{\mr{d}\mb{r}'}{\mr{d}t'}&=(\frac{\mr{d}x'}{\dd t'},\frac{\mr{d}y'}{\dd t'},\frac{\mr{d}z'}{\dd t'})\\[10bp]
&=(\frac{-\dfrac{v}{\sqrt{1-\beta^2}}\dd t+\dfrac{1}{\sqrt{1-\beta^2}}\dd x}{\dfrac{1}{\sqrt{1-\beta^2}}\dd t-\dfrac{\dfrac{v}{c^2}}{\sqrt{1-\beta^2}}\dd x},\frac{\mr{d}y}{\dfrac{1}{\sqrt{1-\beta^2}}\dd t-\dfrac{\dfrac{v}{c^2}}{\sqrt{1-\beta^2}}\dd x},\frac{\mr{d}z}{\dfrac{1}{\sqrt{1-\beta^2}}\dd t-\dfrac{\dfrac{v}{c^2}}{\sqrt{1-\beta^2}}\dd x})\\[10bp]
&=(\dfrac{u_x-v}{1-\dfrac{vu_x}{c^2}},\dfrac{\sqrt{1-\beta^2}u_y}{1-\dfrac{vu_x}{c^2}},\dfrac{\sqrt{1-\beta^2}u_z}{1-\dfrac{vu_x}{c^2}})
\end{align*}
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这里面有些我自定义的命令