Preface

Summary

Das Ziel der Wissenschaft ist einerseits, neue Tatsachen zu erobern, anderseits, bekannte unter hoheren Gesichtspunkten zusammenfassen.

S. Lie

One of the principal objects of theoretical research is to find the point of view from which the subject appears in its greatest simplicity.

J.W. Gibbs

Good mathematics = down-to-earth mathematics.

A. Andreotti

Everyone knows that the theory of PDE systems is enormously rich in results—but what about foundations? This monograph describes one approach, but there is no claim that it is the only one, or that it is the best possible; just that there is one, and moreover one that has been around for a long time without having been as recognized as it deserves.

The study is restricted to local solvability. If a PDE system S is defined in a domain D, and if it can be shown that S possesses local solutions at each point of D, the question of global solvability boils down to whether it is possible to glue together local solutions in order to form global ones—in analogy with the cohomology theory for coherent analytic sheaves. But first of all one has to know that there are local solutions.

A major idea is to regard PDE theory from the point of view of differential geometry—rather than basing it on analysis, say.

To illustrate this, let us first find a suitable angle from which ODE systems appear in a simple way.