The following technical report is now available at ftp.cwi.nl as pub/CWIreports/SEN/SEN-R0120.ps.Z (also via my home page: http://www.cwi.nl/~janr): J.J.M.M. Rutten Elements of stream calculus (an extensive exercise in coinduction) Technical Report SEN-R0120, CWI, Amsterdam, 2001. Abstract: Based on the presence of a final coalgebra structure on the set of streams (infinite sequences of real numbers), a coinductive calculus of streams is developed. The main ingredient is the notion of stream derivative, with which both coinductive proofs and definitions can be formulated. In close analogy to classical analysis, the latter are presented as behavioural differential equations. A number of applications of the calculus are presented, including difference equations, analytical differential equations, continued fractions, and some problems from discrete mathematics and combinatorics. The report will also appear in Volume 45 of Elsevier's ENTCS series, which contains the proceedings of MFPS 17.