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+\pdfoutput=0
+\documentclass[a4paper,11pt]{article}
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+\usepackage{ae}
+\usepackage{aecompl}
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+\usepackage{color}
+\usepackage{pstricks}
+\usepackage{pst-node}
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+
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+\renewcommand{\labelenumiii}{\roman{enumiii})}
+
+\begin{document}
+
+% header
+\begin{center}
+ {\LARGE {\bf Materials Physics II}\\}
+ \vspace{8pt}
+ Prof. B. Stritzker\\
+ SS 2008\\
+ \vspace{8pt}
+ {\Large\bf Tutorial 4}
+\end{center}
+
+\vspace{8pt}
+
+\section{Legendre transformation and Maxwell relations}
+
+\begin{enumerate}
+ \item Consider the total differential
+       \[
+       df= \sum_{i=1}^{n} u_i dx_i
+       \]
+       with the state function $f=f(x_1,\ldots,x_n)$ and its partial derivatives
+       $u_i=\frac{\partial f}{\partial x_i}$.
+       Rewrite the total differential of the function $g$ defined as
+       \[
+       g=f-\sum_{i=r+1}^{n} u_i x_i
+       \]
+       in such a way that $g$ is immediately identified to be a function of
+       the variables $x_1,\ldots,x_r$ and $u_{r+1},\ldots,u_n$,
+       where $u_i$ is called the conjugate variable of $x_i$.
+       The transformation is called Legendre transformation.
+ \item By taking the derivatives of transformed thermodynamic potentials
+       with respect to the variables they depend on,
+       relations between intensive and extensive variables can be gained.
+
+       Start with the internal energy $E=E(S,V)$.
+       Write down the total differential using the equalities
+       $T=\left.\frac{\partial E}{\partial S}\right|_V$ and
+       $-p=\left.\frac{\partial E}{\partial V}\right|_S$.
+       Find more relations by doing the transformation to the potentials
+       \begin{itemize}
+        \item $H=E+pV$ (Enthalpy)
+        \item $F=E-TS$ (Helmholtz free energy)
+        \item $G=H-TS=E+pV-TS$ (Gibbs free energy)
+       \end{itemize}
+       and taking the appropriate derivatives.
+ \item For a thermodynamic potential $\Phi(X,Y)$ the following identity
+       expressing the permutability of derivatives holds:
+       \[
+       \frac{\partial^2 \Phi}{\partial X \partial Y} =
+       \frac{\partial^2 \Phi}{\partial Y \partial X}
+       \]
+       Derive the Maxwell relations by taking the mixed derivatives of the
+       potentials in (b) with respect to the variables they depend on.
+       Exchange the sequence of derivation and use the identities gained in (b).
+\end{enumerate}
+
+\section{Thermal expansion of solids}
+
+It is well known that solids change their length $L$ and volume $V$ respectively
+if there is a change in temperature $T$ or in pressure $p$ of the system.
+
+\begin{enumerate}
+ \item The coefficient of thermal expansion of a solid is given by
+       $\alpha_L=\frac{1}{L}\left.\frac{\partial L}{\partial T}\right|_p$.
+       Show that the coefficient of thermal expansion of the volume
+       $\alpha_V=\frac{1}{V}\left.\frac{\partial V}{\partial T}\right|_p$
+       equals $3\alpha_L$ for isotropic materials.
+ \item 
+ \item
+\end{enumerate}
+
+\end{document}