From 508e7e3b93ca48e9ac3da695c7c8fff1cd26b7c6 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias 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