From 508e7e3b93ca48e9ac3da695c7c8fff1cd26b7c6 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 23 Aug 2019 17:34:05 -0400
Subject: more inverse function verbosity

---
 main.tex | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index d0184d5..a458623 100644
--- a/main.tex
+++ b/main.tex
@@ -313,7 +313,8 @@ which implicitly gives $\eta[\epsilon]$ and $F_\e[\epsilon]=F[\eta[\epsilon],\ep
 cannot be solved explicitly, we can make use of the inverse function theorem.
 First, denote by $\eta^{-1}[\eta]$ the inverse functional of $\eta$ implied by
 \eqref{eq:implicit.eta}, which gives the function $\epsilon_\X$ corresponding
-to each solution of \eqref{eq:implicit.eta} it receives. Now, we use the inverse function
+to each solution of \eqref{eq:implicit.eta} it receives. This we can immediately identify from \eqref{eq:implicit.eta} as $\eta^{-1}[\eta](x)=-2/b(\delta F_\o[\eta]/\delta\eta(x))$.
+Now, we use the inverse function
 theorem to relate the functional reciprocal of the derivative of $\eta[\epsilon]$ with respect to $\epsilon_\X$ to the derivative of $\eta^{-1}[\eta]$ with respect to $\eta$, yielding
 \begin{equation}
   \begin{aligned}
-- 
cgit v1.2.3-70-g09d2

