\chapter{Introduction}\label{sec:introduction}This chapter has not yet been written. In the meantime, referto~\cite{KirLog06,KirLog07} where the algorithms that \ffc{} is basedon are described in detail.%------------------------------------------------------------------------------%\section{The FEniCS project}%In preparation.% FIXME: Automation of CMM, other components of \fenics{}%------------------------------------------------------------------------------%\section{Multilinear forms}%In preparation.%Let $\{V_i\}_{i=1}^r$ be a given set of discrete function%spaces defined on a triangulation $\mathcal{T}$ of $\Omega \subset%\R^d$ and consider a general multilinear form $a$ defined on the%product space $V_1 \times V_2 \times \cdots \times V_r$:%\begin{equation}%  a : V_1 \times V_2 \times \cdots \times V_r \rightarrow \R.%\end{equation}%Typically, $r = 1$ (linear form) or $r = 2$ (bilinear form), but%\ffc{} can handle multilinear forms of arbitrary arity $r$.%Let now%$\{\varphi_i^1\}_{i=1}^{M_1},% \{\varphi_i^2\}_{i=1}^{M_2}, \ldots,% \{\varphi_i^r\}_{i=1}^{M_r}$%be bases of $V_1, V_2, \ldots, V_r$ and let $i = (i_1, i_2, \ldots,%i_r)$ be a multiindex. The multilinear form $a$ then%defines a rank $r$ tensor given by%\begin{equation}%  A_i = a(\varphi_{i_1}^1, \varphi_{i_2}^2, \ldots, \varphi_{i_r}^r).%\end{equation}%In the case of a bilinear form, the tensor $A$ is a matrix (the%stiffness matrix), and in the case of a linear form, the tensor $A$ is%a vector (the load vector).%------------------------------------------------------------------------------%\section{Tensor-representation of multilinear forms}%In preparation.%------------------------------------------------------------------------------%\section{Overview}%In preparation.