diff --git a/doc/neohooke_der.tex b/doc/neohooke_der.tex
new file mode 100644
index 0000000000000000000000000000000000000000..a1079900aa9e30281fcdb181bb9c302770d77c80
--- /dev/null
+++ b/doc/neohooke_der.tex
@@ -0,0 +1,127 @@
+\documentclass{article}
+
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{graphicx}
+\usepackage{natbib}
+\usepackage{psfrag}
+
+\graphicspath{{gfx/}}
+
+\providecommand{\parder}[2]{\frac{\partial #1}{\partial #2}}
+\providecommand{\secparder}[3]{\frac{\partial {#1}^2}{\partial #2 \partial #3}}
+\newcommand{\tr}{\operatorname{tr}}
+
+\begin{document}
+
+\section{The Derivatives of the Energy Functional of a Neo-Hookean Material}
+
+\subsection{Preliminaries: The Derivative of the Determinant of the Deformation  Gradient}
+
+Let
+\begin{equation*}
+  J = \det F = \det \nabla (X + u) = \det (I + \nabla u) 
+  = \det (I + \frac{\partial u_i}{\partial X^j}).
+\end{equation*}
+The deformation $u$ is a finite element function $u = \sum_{i,j} u_i^j \phi_i^j$.
+
+\subsubsection{In Two Space Dimensions}
+
+\begin{eqnarray*}
+  \parder{J}{u_i^j}
+  & = & \parder{}{u_i^j}\det (I + \frac{\partial u_i}{\partial X^j}) \\
+  & = & \parder{}{u_i^j} \det 
+  \begin{bmatrix}
+    1 + \sum_k u^0_k \parder{\phi^0_k}{X^0} & \sum_k u^1_k \parder{\phi^1_k}{X^0}  \\
+    \sum_k u^0_k \parder{\phi^0_k}{X^1} & 1 + \sum_k u^1_k \parder{\phi^1_k}{X^1}
+  \end{bmatrix} \\
+  & = & \parder{}{u_i^j}
+  [ ( 1 + \sum_k u^0_k \parder{\phi^0_k}{X^0} ) (1 + \sum_k u^1_k \parder{\phi^1_k}{X^1})
+  - (\sum_k u^1_k \parder{\phi^1_k}{X^0}) (\sum_k u^0_k \parder{\phi^0_k}{X^1}) ]
+\end{eqnarray*}
+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})
+  -
+  (\sum_k u^1_k \parder{\phi^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})
+  -
+  (\sum_k u^0_k \parder{\phi^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}
+  & = &
+  \parder{\phi_i^0}{X^0} \parder{\phi^1_j}{X^1} 
+  - \parder{\phi^1_j}{X^0} \parder{\phi^0_i}{X^1}\\
+  % 
+ \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} \\
+  % 
+ \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 $\tr E$}
+
+\begin{eqnarray*}
+  \tr E = \frac 12 \tr (\nabla u + \nabla^T u + \nabla^T u \nabla u)
+  = \tr \nabla u + \frac 12 \tr  \nabla^T u \nabla u
+\end{eqnarray*}
+\subsection{First Derivatives of $W$}
+
+\begin{equation*}
+  W(u) = \frac{\lambda}{4} ( J^2 -1 ) - (\frac \lambda 2 + \mu) \ln J + \mu \tr E
+\end{equation*}
+
+Thus,
+\begin{eqnarray*}
+  \parder{W}{u_i^j}
+  & = &
+   \frac{\lambda}{4} \parder{}{u_i^j} J^2 
+   - (\frac \lambda 2 + \mu) \parder{}{u_i^j} \ln J 
+   + \mu \parder{}{u_i^j} \tr E \\
+  %
+   & = &
+   \frac{\lambda J}{2} \parder{J}{u_i^j}
+   - (\frac \lambda 2 + \mu) J^{-1} \parder{J}{u_i^j}
+   + \mu \tr \parder{}{u_i^j} E
+\end{eqnarray*}
+
+\subsection{Second Derivatives of $W$}
+
+\begin{eqnarray*}
+  \secparder{W}{u_i^j}{u_k^l}
+  & = &
+  \parder{}{u_k^l} \Big[ \frac{\lambda J}{2} \parder{J}{u_i^j}
+  - (\frac \lambda 2 + \mu) J^{-1} \parder{J}{u_i^j}
+  + \mu \tr \parder{}{u_i^j} E \Big] \\
+  % 
+  & = & 
+  \frac{\lambda}{2} \Bigg[ \parder{J}{u_i^j} \parder{J}{u_k^l} 
+  + J \secparder{J}{u_i^j}{u_k^l} \Bigg] \\
+  & &
+  - (\frac \lambda 2 + \mu) J^{-2} 
+  \Bigg[ \secparder{J}{u_i^j}{u_k^l} J - \parder{J}{u_i^j} \parder{J}{u_k^l} \Bigg]
+  + \mu \tr \secparder{}{u_i^j}{u_k^l} E
+\end{eqnarray*}
+\end{document}
+
+%%% Local Variables: 
+%%% mode: latex
+%%% TeX-master: t
+%%% End: