  We consider the General Purpose Analog Computer (GPAC), introduced by Claude
Shannon in 1941 as a mathematical model of Differential Analysers, that is to
say as a model of continuous-time analog (mechanical, and later one electronic)
machines of that time. We extend the model properly to a model of computation
not restricted to univariate functions (i.e. functions $f: \mathbb{R} \to
\mathbb{R}$) but also to the multivariate case of (i.e. functions $f:
\mathbb{R}^n \to \mathbb{R}^m$), and establish some basic properties. In
particular, we prove that a very wide class of (continuous and discontinuous)
functions can be uniformly approximated over their full domain. Technically: we
generalize some known results about the GPAC to the multidimensional case: we
extend naturally the notion of \emph{generable} function, from the
unidimensional to the multidimensional case. We prove a few stability
properties of this class, mostly stability by arithmetic operations,
composition and ODE solving. We establish that generable functions are always
analytic. We prove that generable functions include some basic (useful)
generable functions, and that we can (uniformly) approximate a wide range of
functions this way. This extends some of the results from \cite{Sha41} to the
multidimensional case, and this also strengths the approximation result from
\cite{Sha41} over a compact domain to a uniform approximation result over
unbounded domains. We also discuss the issue of constants, and we prove that
involved constants can basically assumed to always be polynomial time
computable numbers.
