kleine Korrekturen

[[Imported from SVN: r9296]]

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 $\tr E$}