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Brian Jackson edited Thus_begin_equation_label_eqn__.tex
over 8 years ago
Commit id: d3c78b61ecbb5aa742c55db1f6337da7173e7fbb
deletions | additions
diff --git a/Thus_begin_equation_label_eqn__.tex b/Thus_begin_equation_label_eqn__.tex
index 8165790..0508f30 100644
--- a/Thus_begin_equation_label_eqn__.tex
+++ b/Thus_begin_equation_label_eqn__.tex
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\Rightarrow b^{-1}\ P_{\rm obs} \left( \dfrac{\partial \rho({\rm obs})}{\partial P_{\rm obs}} - \left( \dfrac{\Gamma_{\rm obs}}{2 P_{\rm obs}} \right) \dfrac{\partial \rho({\rm obs})}{\partial \Gamma_{\rm obs}} \right) = b_{\rm max}^{-2}\ f(b)\ \rho({\rm act}(b))\ b \\
\Rightarrow \rho(\Gamma_{\rm act}, P_{\rm act}) = \left[ \left( \dfrac{b_{\rm max}}{b} \right)^2\ f(b)^{-1}\ P_{\rm obs} \left( \dfrac{\partial \rho(\Gamma_{\rm obs}, P_{\rm obs})}{\partial P_{\rm obs}} - \left( \dfrac{\Gamma_{\rm obs}}{2 P_{\rm obs}} \right) \dfrac{\partial \rho(\Gamma_{\rm obs}, P_{\rm obs})}{\partial \Gamma_{\rm obs}} \right) \right]_{\rm obs \rightarrow
act}, act} \\
\Rightarrow k^\prime \Gamma_{\rm act}^{-11/3} \left( \dfrac{P_{\rm min}}{P_{\rm act} - P_{\rm min}} \right)^{3/2} \left[ \dfrac{\partial \rho(\Gamma_{\rm obs}, P_{\rm obs})}{\partial P_{\rm obs}} - \left( \dfrac{\Gamma_{\rm obs}}{2 P_{\rm obs}} \right) \dfrac{\partial \rho(\Gamma_{\rm obs}, P_{\rm obs})}{\partial \Gamma_{\rm obs}} \right) \right]_{\rm obs \rightarrow act},
\end{equation}
where
$k^\prime = 4k b_{\rm max}^2 A_{\rm max} \kappa^{-1}\ \upsilon^{-1}$, $\partial \Gamma_0/\partial P_{\rm obs} = 0$, and ${\rm obs \rightarrow act}$ indicates that ${\rm obs}$ quantities should be replaced with ${\rm act}$ quantities after the derivatives are taken.