From c0589e8ba715303ec550b7b7b5d3988bc35dceec Mon Sep 17 00:00:00 2001
From: Oliver Sander <sander@igpm.rwth-aachen.de>
Date: Thu, 2 Feb 2006 11:51:18 +0000
Subject: [PATCH] kleine Korrekturen
[[Imported from SVN: r9296]]
---
doc/neohooke_der.tex | 24 ++++++++++++++----------
1 file changed, 14 insertions(+), 10 deletions(-)
diff --git a/doc/neohooke_der.tex b/doc/neohooke_der.tex
index a107990..556ff80 100644
--- a/doc/neohooke_der.tex
+++ b/doc/neohooke_der.tex
@@ -43,22 +43,28 @@ Thus,
\begin{eqnarray*}
\parder{J}{u_i^0}
& = &
- \parder{\phi_i^0}{X^0} ( 1 + \sum_k u^1_k \parder{\phi^1_k}{X^1})
+ \parder{\phi_i^0}{X^0} \Big( 1 + \sum_k u^1_k \parder{\phi^1_k}{X^1} \Big)
-
- (\sum_k u^1_k \parder{\phi^1_k}{X^0}) \parder{\phi_i^0}{X^1} \\
+ \Big(\sum_k u^1_k \parder{\phi^1_k}{X^0} \Big) \parder{\phi_i^0}{X^1} \\
+ & = &
+ \parder{\phi_i^0}{X^0} \Big( 1 + \parder{u^1_k}{X^1} \Big)
+ -
+ \parder{u^1_k}{X^0} \parder{\phi_i^0}{X^1} \\
%
\parder{J}{u_i^1}
& = &
- \parder{\phi_i^1}{X^1} ( 1 + \sum_k u^0_k \parder{\phi^0_k}{X^0})
+ \parder{\phi_i^1}{X^1} \Big( 1 + \sum_k u^0_k \parder{\phi^0_k}{X^0} \Big)
+ -
+ \Big(\sum_k u^0_k \parder{\phi^0_k}{X^1} \Big) \parder{\phi_i^1}{X^0} \\
+ & = &
+ \parder{\phi_i^1}{X^1} \Big( 1 + \parder{u^0_k}{X^0} \Big)
-
- (\sum_k u^0_k \parder{\phi^0_k}{X^1}) \parder{\phi_i^1}{X^0}
+ \parder{u^0_k}{X^1} \parder{\phi_i^1}{X^0}
\end{eqnarray*}
The four second derivatives are
\begin{eqnarray*}
\secparder{J}{u_i^0}{u_j^0}
& = & 0 \\
- %\delta_{ij} \parder{\phi^0_i}{X^0} (1 + \sum_k u^1_k \parder{\phi^1_k}{X^1})
- %- \delta_{ij} \parder{\phi^0_i}{X^1} (\sum_k u^1_k \parder{\phi^1_k}{X^0})\\
%
\secparder{J}{u_i^0}{u_j^1}
& = &
@@ -67,13 +73,11 @@ The four second derivatives are
%
\secparder{J}{u_i^1}{u_j^0}
& = &
- \parder{\phi^1_j}{X^0} \parder{\phi^0_i}{X^1}
- - \parder{\phi_i^0}{X^0} \parder{\phi^1_j}{X^1} \\
+ \parder{\phi^0_j}{X^0} \parder{\phi^1_i}{X^1}
+ - \parder{\phi_i^1}{X^0} \parder{\phi^0_j}{X^1} \\
%
\secparder{J}{u_i^1}{u_j^1}
& = & 0
- %\delta_{ij} \parder{\phi^1_i}{X^1} (1 + \sum_k u^0_k \parder{\phi^0_k}{X^0})
- %- \delta_{ij} \parder{\phi^1_i}{X^0} (\sum_k u^0_k \parder{\phi^0_k}{X^1})
\end{eqnarray*}
\subsection{The Derivatives of $\tr E$}
--
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