**DIPS 5/98**[tex source, PostScript, PDF file, dvi]-
**Title:**INTRODUCTION TO SECONDARY CALCULUS**Author:**Alexandre VINOGRADOVFirst we exhibit some basic notions and constructions of modern geometry of partial differential equations that lead to the concept of diffiety, an analogue of affine algebraic varieties for partial differential equations. Then it is shown how the differential calculus on diffieties which respects the underlying infinite order contact structure is self-organized into Secondary Calculus in such a way that higher symmetries of PDE's become secondary vector fields and the first term of the ${\cal C}$-spectral sequence becomes the algebra of secondary differential forms. Then the general secondarization problem is formulated. Its solution for modules and multi-vector-valued differential forms is proposed and the relevant homological algebra is discussed. Eventually, relations with gauge theories are briefly outlined at the end.

**To appear**in Proceedings of Conference on "Secondary Calculus and Cohomological Physics", Contemporary Mathematics, vol. 219, 1998.

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