This chapter gathers a number of mathematical definitions and theorems that, to different extents, will be needed for the economic theory of the following seven chapters. It is not meant to be read systematically before the rest of the book. The chapter is divided into twelve sections. The ordering of the sections is one of convenience and not of intrinsic importance for later developments. This being a book on differentiable techniques, it stands to reason that the central sections are Sections B on linear algebra, C on differential calculus, D on optimization, H on differentiable manifolds, I on transversality theory, and J on degrees of functions and indices of zeros of vector fields.

The economic theory of this book is presented in Chapters 2 through 8 with a fairly strict adherence to the axiomatic method and without presupposing much. No similar claim is made, however, for this chapter. A quick reading of the headings of the different sections will convince the reader that their content cannot be a systematic, complete, or rigorous exposition. It serves to fix terminology and to facilitate reference, but if one wishes to go deeper into the purely mathematical aspects, this chapter is no substitute for the study of the pertinent mathematical sources cited at the end of every section. It should suffice to say that the chapter does not contain any proofs.