deletions | additions       diff --git a/begin_flushleft_begin_align_S_2__1.tex b/begin_flushleft_begin_align_S_2__1.tex  index c0c964e..31854ca 100644  --- a/begin_flushleft_begin_align_S_2__1.tex  +++ b/begin_flushleft_begin_align_S_2__1.tex 
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S_2 &= \left< \left[ I(\vec{r},v) - I(\vec{r}+\delta \vec{r},v+\delta v) \right]^2 \right>\\  &= \left< \left[ I - I'\right]^2 \right>\\  &= \left<\left[ I_{signal}+I_{noise} - I'_{signal}-I'_{noise} \right]^2 \right>\\  &= \left< I_{signal}^2 + 2I_{signal}I_{noise} - 2I_{signal}I'_{signal} - 2I_{signal}I'_{noise} + I_{noise}^2 - 2I_{noise}I'_{signal} - 2I_{noise}I'_{noise} - I'_{signal}^2 + 2I'_{signal}I'_{noise} + I'_{noise} \right>\\    &= S_{2(signal)} + S_{2(noise)}  \end{align*}