this is for holding javascript data
Brian Jackson edited Thus_begin_eqnarray_label_eqn__.tex
over 8 years ago
Commit id: 8dbd50bae799630db4000b1d6465feb840381690
deletions | additions
diff --git a/Thus_begin_eqnarray_label_eqn__.tex b/Thus_begin_eqnarray_label_eqn__.tex
index d892be5..2086a57 100644
--- a/Thus_begin_eqnarray_label_eqn__.tex
+++ b/Thus_begin_eqnarray_label_eqn__.tex
...
Thus,
\begin{eqnarray}
\label{eqn:convert_from_observed_to_actual_density}
& \dfrac{d}{db} \bigg( \rho(\Gamma_{\rm obs}, P_{\rm obs}) - \rho(\Gamma_0, P_{\rm min}) \bigg) & =
& \dfrac{d}{db}
& & \left( \int_{b^\prime = 0}^{b} f\ \rho({\rm act}) \dfrac{2b^\prime\ db^\prime}{b_{\rm max}^2}
\right)\\ \right) & \\
& \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} &,
\end{eqnarray}
where $\partial \Gamma_0/\partial P_{\rm obs}$ works out to be zero, and ${\rm obs \rightarrow act}$ indicates that ${\rm obs}$ quantities should be replaced with ${\rm act}$ quantities after the derivatives are taken.